UNIVERSITY OF SWAZILAND
SUPPLEMENTARY EXAMINATION PAPER 2014
TITLE OF PAPER
COURSE CODE
TIME ALLOWED
REQUIRMENTS
INSTRUCTIONS
NON-PARAMETRIC ANALYSIS
ST409
2 (TWO) HOURS
STATISTICAL TABLES AND CALCULATOR
ANSWER ANY FOUR (4) QUESTIONS ALL QUESTIONS CARRY EQUAL MARKS
THIS PAPER IS NOT TO BE OPENED UNTIL PERMISSION HAS BEEN GRANTED BY THE INVIGILATOR
Page 1
For all questions clearly state the nuU amp alternaJe hypotheses the test statistics the decision rule the levelofsigniticance the decision the conclusions
QUESTION ONE [25 marks]
Eight volunteers for an experiment are divided randomly into two groups to see if a telescopic sight improves the ability to hit a target under twilight conditions Group A is given rifles with telescopic sights while group B has the same kind of rifle but with open sights After a learning period they are given a shooting test at twilight These are their scores (100 is perfect)
Group A 96 93 88 85 GroupB 89 83 80 77
Use the Mann-Whitney test to conclude about the experiment Use 5 level ofsignificance
QUESTION TWO [25 marks]
An Olympic diver is rated on ten practice dives with the following measurements 1753 76 89 9091939699 and 99 Test the hypothesis that the distribution function ofher scores is given by F(x) where
F(x) = o ifxltO F(x) 100 if0 SxS 10 F(x) 1 iflO ltx
QUESTION THREE [ 15 + 10 marks ]
a It is desired to design a given automobile to allow enough headroom to accommodate comfortably all but the tallest 5 of the people who drive Former studies indicate that the 95th percentile was 17856 cm In order to see if the former studies are still valid a random sample of size 100 is selected It is found that the 12 tallest persons in the sample have the following heights
18480 17780 18110 17907 17983 19304 17805 18415 18059 17932 18263 18491
Is it reasonable to use 17856 as the 95th percentile Use Quantile test with a = 005
b Twenty customers in a grocery store were asked to taste each of two types of cheese and declare their preference Sis customers preferred one kind 11 preferred the other kind and 3 had no preference Does this indicate a significant difference in preference Use Sign test with a = 010
Page 2
QUESTION FOUR [25 marks]
Four job training programs were tried on 20 new employees where 5 employees were randomly assigned to each training program The 20 employees were then placed under the same supervisor and at the end of a certain period the supervisor ranked the employees according to job ability with the lowest ranks being assigned to those employees with the lowest job ability
Program Ranks 1 467210 2 18 123 11 3 20 19 16 145 4 181517139
Do these data indicate a difference in the effectiveness ofthe various training programs
QUESTION FIVE [25 marks]
A new worker is assigned to a machine that manufactures bolts Each day a sample of bolts is examined and the percent defective is recorded Do the following data indicate a significant improvement over time for the worker
Day Percent Day Percent Day Percent 1 61 6 61 10 46 2 75 7 53 11 30 3 77 8 45 12 40 4 59 9 49 13 37 5 52
Use either Spearmans p test or Kendalls Ttest
e lit
TABLE AI Nonnal Distribution
z1 -37110 z- - -32105 oas -19600 rea -16449 4 - 37110 ZcIm1 32905 ZltIm -= 19600 rea 16+49 0000 0-001 0002 0001 0004 0005 0006 0007 0008 O-OOt
000 -3o1Ol -28782 -27478 -26521 -25758 -25121 -24573 -2ltI0Il9 -1l656
001 -13263 -ll9Of -llS71 -lll62 -21973 -21701 -21 -21201 -2OK9 -20749
0-02 -lOS37 -20335 -2041 -19954 -19774 -19600 -19431 -19268 -19110 -18957
0-01 -188D8 -18663 -18522 -18384 -18250 -18119 -17991 -17866 -177 -17624
0-04 -17507 -17392 -1n79 -17169 -17060 -16954 -16849 -16747 -16646 -165ltt6
0-05 -16449 -16352 -16258 -16164 -160n -15982 -153 -15805 -15718 -15632
006 -15t8 -15464 -15382 -15301 -15220 -15141 -15063 -149B5 -1 909 -IlttIll
007 -14758 -1-4684 -14611 -14538 -1441gt1 -14395 -14325 -1ltt2S5 -14187 -14118
001 -1laquo151 -13984 -13917 -13852 -13787 -13722 -13658 -13595 -13532 -13469
Oot -1)408 -13346 -13285 -13225 -13165 -13106 -13047 -12988 -12930 -12873
DI0 -12816 -12759 -Il702 -12646 -12591 -12536 -12ltt11 -12426 -123n -12319
DII -12265 -12212 -12160 -12107 -12055 -IlOO4 -11952 -11901 -11850 -11800
012 -11750 -11700 -116S0 -11601 -11552 -11503 -1455 -117 -11359 -11311
011 -11264 -11217 -11170 -11123 -11077 -11031 -10985 -10939 -1D893 -10amp48
014 -IoB03 -10758 -G7H -10669 -1D625 -10581 -Ios37 -10494 -11)0450 -1DlttD7
US -10364 -Iol22 -llI279 -10237 -10194 -GI52 -10110 -1006 -10027 -09 016 -099lt45 -09904 -03 -U822 -09782 -0974 -09701 -096amp1 -021 -09581
017 -09542 -09502 -03 -09424 -09385 -09346 -09307 -09269 -09230 -09192
011 -09154 -09116 -01078 -09040 -01002 -08965 -08927 -08890 -08853 -D8816
01t -08779 -08742 -08705 -089 -08pound33 -085 -08560 -DB524 -08488 -08452
OlO -08416 -0838 -08345 -08310 -08274 -08239 -08204 -08169 -08134 -08099
021 -08064 -08030 -07995 -07961 -D7926 -07892 -07858 -01824 -07790 -07756
Oll -07722 -07688 -07655 -07621 -07588 -07554 -07521 -07488 -07454 -07421
023 -07388 -07356 -07323 -on1O -07257 -0n25 -07192 -D716O -07128 -07095
024 -07063 -07031 -D6999 -D6967 -06935 -06103 -06871 -068lttD -06808 -D6776
TABLE AI (Continued)
0000 0001 0001 0001 0004 0005 0006 0007 0001 OOOt
025 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -06526 -06495 -06464 026 -06433 -D6ltt03 -O63n -06341 -06311 -06280 -06250 -06219 -06189 -06158 827 -06128 -06098 -06068 -06038 -06008 -05978 -O59ltt8 -05918 -05888 -05858 828 -05828 -05799 -05769 -05740 -05710 -05681 -05651 -05622 -05592 -05S63 O2t -05534 -05505 -05476 -DSi46 -05417 -05388 -0535 -05330 -05302 -05273 030 -052 -05215 -05187 -05158 -05129 -05101 -05072 -0SD44 -05015 -01987 031 -04959 -04930 -004902 -0lttI74 -04845 -004817 -04m -04761 -04733 -0A705 032 -077 -004649 -04621 -04593 -04565 -0045311 -04510 -04482 -04454 -04427 033 -04399 -04372 -043 -01316 -04289 -01261 -04234 -04207 -04179 -04152 034 -04IlS -04097 -01070 -04043 -D40 16 -039119 -031 -03934 -03907 -038BO lUi -031153 -03826 -0379~ -03m -03745 -037[9 -03poundn -03665 -li3638 -03611 I)l~ -03585 -03558 -oml -03505 -01478 -03451 -03425 -03398 -03372 -033-4S QlT -03319 -032~2 -03266 -0n~9 -03113 -03 [86 -03160 -03131 -03107 -1130ff rlt -03055 -O3Q2~ -03002 -027euroshy -02950 -0292pound -O2pound9G -02B71 -O2E-ltE -02lt1 ~~ -O2n3 -02767 -017lt[ -Ol7I~ -O~~6poundSmiddot -026euro -O2euroi -0261 [ -025pound -0l5~7
Q4~ -02523 -C2508 -024euro2 -02pound5pound -023(1 -0240lt -0237amp -02m -02027 -023QI ct bull -02275 -02250 -0222lt -021 -0217 -02[laquo7 -02121 -020 -02070 -02045 1147 -02019 -0193 -OIB -01gt42 -(U7 -OJgt -01866 -01840 -01815 -i17S 1141 -01714 -01738 -01713 -01687 -01662 -01637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -Ollll -01206 -01181 -01156 -01130 -01105 -01080 -01055 -01030 046 -01004 -00979 -00954 -00929 -Oo9tH -00878 -001153 -0DII28 -0DII03 -00778 047 -00753 -007211 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0tH76 -0tH51 -0tH26 -00401 -00376 -00351 -00326 -00301 -00276 049 -OOlSl -00226 -00201 -00175 -00150 -OOllS -00100 -00075 -00050 -OOOlS 050 00000 00025 00050 00075 00100 00125 00150 00175 00201 00226 051 OOlSl 00276 00301 00326 00351 00376 00401 OtH26 0tH51 0Oltt76 051 00502 00527 00552 00577 00602 00627 00652 00677 00702 00728 053 00753 00778 00803 00828 001153 00878 0090-4 00929 00954 00979 054 01004 01030 01055 01080 01105 01130 01156 01181 01206 01231
lit CI
Table AI (Continued) 0000 0001 0002 0003 0004 0005 0007 looa DDD
055 01257 OllIl 01307 01332 01358 01383 01408 01434 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738 057 017( 01789 01815 01amp40 01866 01891 01917 019-42 01968 01993 o5a 02019 020-45 02070 D1096 Dllll Dl147 01173 01198 0lll4 Ol25O 05 1U275 01301 01327 0l353 01378 DllaquoH 01 30 Dl456 DM81 01508 061 02533 02559 D2585 02611 Dl637 0l663 02689 02715 Dl741 01767 061 02793 02819 02845 02871 02898 Dl914 Dl950 Dl976 03002 03019 062 O3OSS 03081 03107 031304 03160 03186 03213 03239 03266 03292 063 03319 0335 03372 03398 03-425 03451 03(78 03505 03531 03558 06 03585 03611 03638 03665 03692 03719 03745 03772 03799 03826 065 03853 03880 03907 039304 03961 03989 0-4016 0laquoHl 0-4070 0-4097 066 ollS 0151 0179 0207 0134 0261 04219 0-4316 043 04372 067 04399 0 27 0 5-4 0+482 0510 04538 04565 0lt693 0lt1611 046-49 068 0-4677 04705 04733 0761 0 789 0-4817 0-4amp45 04874 04901 04930 069 04959 04987 DSOI5 050+1 05072 05101 05129 05158 05187 05215 070 051 05l73 05302 05330 O5lS9 05388 05417 05-4-46 D5476 O5SOS 071 055304 05563 05592 056ll 05651 05681 05710 057-40 05769 O5m Dn 05828 05858 05888 05918 05948 05918 06Q08 06038 06068 D6098 071 06121 06158 06189 D6219 0625Q 06l1D osil 06341 06372 064Q3 074 033 064 0 5 D6526 D6557 06588 066lQ 06651 0668l 06713 075 06745 06776 06808 06amp40 06871 06903 06935 0697 069 07031 176 07063 07095 07121 07160 07192 07225 07257 07290 07313 07356 077 07388 07411 0745-4 07-488 07Sl1 07554 07588 07621 07655 07688 071 077ll 07756 07790 07n4 07858 07892 07926 07961 07995 D8030 079 OllOM 08099 08134 Ul69 08204 08239 08174 08310 08345 D8381 OID 0110416 0amp451 08488 D85l 08560 08596 08633 08669 08705 D87-42 081 urn 08816 08853 08890 08927 08965 09002 090-40 09078 09116 082 0915-4 09192 09l30 09269 09307 093lt16 09385 0942lt1 09463 09502
Table AI (Continued) 0000 IUD I 0002 DDOJ 0004 0005 DDH 0007 DoDa 0009
Dl3 09542 09581 09611 09661 09701 09741 0978l 09822 09863 09904 084 099-45 09986 10017 10069 10110 10152 1094 1D237 0179 10322 085 1036-4 1()407 1D450 1D494 10537 10581 10615 10669 1071lt4 10758 D86 10803 1eIM8 10893 10939 10985 Ll031 11077 11123 11170 11217 087 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 118amp I750 11800 11850 1I~01 UITS2 f20Q4 12055 1207 L2160 12212 lIa~ 22euroS 123 f ~ 12372 12426 124BI (Elf 1251[ 12(46 12702 12759 bN 121H6 12en 12S30 t29se 1~CK7 I r06 316S IJru f2es flJ~C
M[ [gt4lS 14pound0 [3502 IlS~5 lJpoundSL [T21 12787 13E1 lS [7 IAon ~ II [ 11[17 Llt25E 1l2 r (E 1lt46 1lt5~t 14f II LCfilt 1475( r4poundZ3 [9(j rA9e~ IS(I(~ 151lt[ S22( L5301 1~3n L54H 155ltamp 15632 1573 15005 1513B [~ 16Q 161 G-t 162Sfl 161pound2 I~ 16546 6G46 167(7 ICIH~ [654 17060 17[6~ 17279- [73)2
175ar 17624 177 17S66 179gt1 LSI 19 18250 1pound384 18522 18663 18808 IB~5r 19J 10 19268 19431 19600 um4 19954 20141 20335 20537 20749 10969 11201 114+1 11701 11973 22262 11571 11904 13263 13656 24089 14573 l5111 l5758 26521 17478 18782 30902
Souka GeneIllted by Il L lman Used wiIh permission
The entries in his table are quand Z of the standard normal random variable Z selected so P(Z z) = p and P(Z gt z) = I - p Note that the value of P to two decimal places determines which row to USC the third decimal pia of detennines which col to use to find z
III o
APPENDIX 511 510 APPENDIX
TABLE Al ChI5qud Dlstributlon- TABLE Al Binomial Distributionmiddot
= 0750 000 0950 0975 1990 05 09 I I = 005 010 015 aI11 O~l~ 030 035 0-40 0-45
11=1 2 3 -4 5 6 7 8
10 II 12 13 1-4 15 16 17 18 I 10 11 11 13 1-4 15 16 17 28 l 30 -40 50 60 70 80 0
100
1323 2m 108 5385 6626 7HI 037
1012 113 1255 1370 185 1598 1711 1825 1937 20 2160 nn 2383
193 260-4 271 282 293 303 3153 3262 3371 30 561 5633 6698 7158 8813 865
1091 0675
2706 605 6251 7779 9136
106-4 1201 1l36 168 1599 1728 1855 1981 2106 1231 US 177 2599 1720 181 1961 3UI 3201 3310 338 3556 367 3791 390 fO16 5181 6317 n bull fO 8553 9658
1076 1185
1281
3HI 5991 7815
bull fBI 1107 115 107 1551 1691 1831 1968 1103 1236 1368 2500 1630 1759 1187 301 3 1 3267 3392 3517 362 3765 3889 011 113-4 256 4377 5576 6750 7908 9053
1019 1131 1243
16-45
502 7378 348
1114 1283 1445 1601 1753 1902 2048 2192 233-4 2474 2611 1749 2885 3019 3153 3285 3-417 3548 3678 3808 3937 fO65 412 4319 +6 1572 -4698 5U4 7142 8330 9502
1066 1181 1196
1960
6635 9110
11l4 1318 1509 1681 1848 2009 1167 2311 2473 2612 27 211 3U8 3200 3311 3-481 3619 3757 383 fO19 416-4 17-8 +131 156-4 -4696 4828
550 6369 7615 8838
1004 1123 1241 1358
2326
7179 10J00 1211 1116 1675 1855 203 219 235 2519 2676 1830 1982 3132 3210 3417 3572 3716 3858 fOOO 41fO 1210 441 155 3 1 44 SO99 513-4 5367 6677 791 US
10-42 1163 1113 1fO2
2576
1083 1312 1627 1847 1051 2246 2432 1613 1788 15 3116 32tI 3453 3611 3770 3915 fO79 4231 4312 532 80 27 19n 5118 5261 5405 55 568 5830 5970 n fO 666 61
1123 1118 1372 111
30t0
for gt 100 eIIllflPlmadon III shy Q)(z + lit-I) or ell I1IQA acane w -
k (I shy -l +z ~) whu 11 ch from ell IGlldudiud normal dlmibudon shown In ch bottOm
2
3
-4
5
6
7
o
o I 2
o I 1 3
o
1 3 -4
o I 2 3 -4 5
o
1 3 -4 5 6
o 1 2 3 -4 5 6
7
09500 10000 09025 09975 10000
08574 09928
099 10000
0815 09860 09995 10000 10000
07738 0977ltf 09988 10000 10000 10000
07351 09672 09978 099 10000 10000 10000
06983 0556 0992 09998 10000 10000 10000 10000
09000 10000 08100 0900 10000
07290 09720 09990 10000
06561 09477 09963 09999 10000
05905 09185 Q991-4 09995 10000 10000
05314 08857 098-42 09987 09999 10000 10000
0783 08503 09713 099n 09998 10000 10000 10000
08500 10000
07TIS 09775 10000
061041 09392 09966 10000
05220 08905 09880 09995 10000
0 37 08352 09734 09978 09999 10000
03771 07765 09527 09HI 09996 10000 10000
03206 07166 09162 09879 09988 09999 10000 10000
OBOOO 10000
06400 09600
100011
0512u
080 09920
10000
OA09 0B191 0971(J 099N 10000
OJ7
0737)
09411 09933
09997 10000
(llbLI
06554 09011
09830
099fH
09999
10000
02097
057a 0B520
09667
09953
099
10000 10000
01500 luoOO
iL5625
09375
10000
OAII
08138 0-844
10000
lIj I 6-1 O)(j
OJ4~gt
091 10000
Il2J7J 06]28 08965 01844 ti9990
10000
01780 05339 08306
014 09954
u9999 10UOO
UI335
0-149
07pound4 0911middot)
09971
0991l7
09999
10000
07000 10000
004900 09100 10000
03430 078-40 Q9730 10000
02401 06517 09163 09919 10000
01681 05281 08369 09692 09976 10000
01176 OA202 07 3 09295 09891 09993 10000
0082-4 0329-4 06-471
0870 09712 09962 09998 10000
06500 10000
OATIS 08775 10000
027-46 07182 09571 10000
01785 05630 08735 09850 10000
01160 OA18-4 076-48 09460 099-47 10000
0075 03191 06-471 08826 09777 09982 10000
00-490 02338 05323 0S002 09+44 09910 09994 10000
06000 10000
03600 011lt100 10000
02160 O6-4SO 09360 10000
01196 OA751 08208 097 10000
00778
03370 06826 09130 09898 10000 00-467 02l11 05 3 08208 09590 09959 10000
002SO 01586 OAI99 07102 09037 09812 O99H 10000
05500 10000
03025 07975 10000
0166-4 057-48 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00177 01636 0 15 07 7 09308 09917 10000
00152
0101 0316-4 06083 0H71 096-43 09963 10000
ofch abl SouacI AbrldSiOd from Tab VoL I of Pearson and HaM) (I 76) wilh ImlulOR from ell llatnetrllro Tr bullThe uMI1 In chis abl ar quanllle IIIr of a chllquarOd random varlabl W wilh It Uv- of frMdom Clad SO (W $ wJ ~ p and P(W gt wpl - I - p
III w
w w - bull
0 U P bullIII
P III III
fbull
f III
p III
bull
I n
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lCI W mIII Z
gtC
bull w
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O UI WaJ
o o W
0 II o U o
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gtlt o ~ o
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w
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pbullo
p
=
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b po III o
po III
c
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bull
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520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
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00000 00000
00000
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00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
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00001
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00000 00000
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00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
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00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
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00003
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00000 00000
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00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
Page 1
For all questions clearly state the nuU amp alternaJe hypotheses the test statistics the decision rule the levelofsigniticance the decision the conclusions
QUESTION ONE [25 marks]
Eight volunteers for an experiment are divided randomly into two groups to see if a telescopic sight improves the ability to hit a target under twilight conditions Group A is given rifles with telescopic sights while group B has the same kind of rifle but with open sights After a learning period they are given a shooting test at twilight These are their scores (100 is perfect)
Group A 96 93 88 85 GroupB 89 83 80 77
Use the Mann-Whitney test to conclude about the experiment Use 5 level ofsignificance
QUESTION TWO [25 marks]
An Olympic diver is rated on ten practice dives with the following measurements 1753 76 89 9091939699 and 99 Test the hypothesis that the distribution function ofher scores is given by F(x) where
F(x) = o ifxltO F(x) 100 if0 SxS 10 F(x) 1 iflO ltx
QUESTION THREE [ 15 + 10 marks ]
a It is desired to design a given automobile to allow enough headroom to accommodate comfortably all but the tallest 5 of the people who drive Former studies indicate that the 95th percentile was 17856 cm In order to see if the former studies are still valid a random sample of size 100 is selected It is found that the 12 tallest persons in the sample have the following heights
18480 17780 18110 17907 17983 19304 17805 18415 18059 17932 18263 18491
Is it reasonable to use 17856 as the 95th percentile Use Quantile test with a = 005
b Twenty customers in a grocery store were asked to taste each of two types of cheese and declare their preference Sis customers preferred one kind 11 preferred the other kind and 3 had no preference Does this indicate a significant difference in preference Use Sign test with a = 010
Page 2
QUESTION FOUR [25 marks]
Four job training programs were tried on 20 new employees where 5 employees were randomly assigned to each training program The 20 employees were then placed under the same supervisor and at the end of a certain period the supervisor ranked the employees according to job ability with the lowest ranks being assigned to those employees with the lowest job ability
Program Ranks 1 467210 2 18 123 11 3 20 19 16 145 4 181517139
Do these data indicate a difference in the effectiveness ofthe various training programs
QUESTION FIVE [25 marks]
A new worker is assigned to a machine that manufactures bolts Each day a sample of bolts is examined and the percent defective is recorded Do the following data indicate a significant improvement over time for the worker
Day Percent Day Percent Day Percent 1 61 6 61 10 46 2 75 7 53 11 30 3 77 8 45 12 40 4 59 9 49 13 37 5 52
Use either Spearmans p test or Kendalls Ttest
e lit
TABLE AI Nonnal Distribution
z1 -37110 z- - -32105 oas -19600 rea -16449 4 - 37110 ZcIm1 32905 ZltIm -= 19600 rea 16+49 0000 0-001 0002 0001 0004 0005 0006 0007 0008 O-OOt
000 -3o1Ol -28782 -27478 -26521 -25758 -25121 -24573 -2ltI0Il9 -1l656
001 -13263 -ll9Of -llS71 -lll62 -21973 -21701 -21 -21201 -2OK9 -20749
0-02 -lOS37 -20335 -2041 -19954 -19774 -19600 -19431 -19268 -19110 -18957
0-01 -188D8 -18663 -18522 -18384 -18250 -18119 -17991 -17866 -177 -17624
0-04 -17507 -17392 -1n79 -17169 -17060 -16954 -16849 -16747 -16646 -165ltt6
0-05 -16449 -16352 -16258 -16164 -160n -15982 -153 -15805 -15718 -15632
006 -15t8 -15464 -15382 -15301 -15220 -15141 -15063 -149B5 -1 909 -IlttIll
007 -14758 -1-4684 -14611 -14538 -1441gt1 -14395 -14325 -1ltt2S5 -14187 -14118
001 -1laquo151 -13984 -13917 -13852 -13787 -13722 -13658 -13595 -13532 -13469
Oot -1)408 -13346 -13285 -13225 -13165 -13106 -13047 -12988 -12930 -12873
DI0 -12816 -12759 -Il702 -12646 -12591 -12536 -12ltt11 -12426 -123n -12319
DII -12265 -12212 -12160 -12107 -12055 -IlOO4 -11952 -11901 -11850 -11800
012 -11750 -11700 -116S0 -11601 -11552 -11503 -1455 -117 -11359 -11311
011 -11264 -11217 -11170 -11123 -11077 -11031 -10985 -10939 -1D893 -10amp48
014 -IoB03 -10758 -G7H -10669 -1D625 -10581 -Ios37 -10494 -11)0450 -1DlttD7
US -10364 -Iol22 -llI279 -10237 -10194 -GI52 -10110 -1006 -10027 -09 016 -099lt45 -09904 -03 -U822 -09782 -0974 -09701 -096amp1 -021 -09581
017 -09542 -09502 -03 -09424 -09385 -09346 -09307 -09269 -09230 -09192
011 -09154 -09116 -01078 -09040 -01002 -08965 -08927 -08890 -08853 -D8816
01t -08779 -08742 -08705 -089 -08pound33 -085 -08560 -DB524 -08488 -08452
OlO -08416 -0838 -08345 -08310 -08274 -08239 -08204 -08169 -08134 -08099
021 -08064 -08030 -07995 -07961 -D7926 -07892 -07858 -01824 -07790 -07756
Oll -07722 -07688 -07655 -07621 -07588 -07554 -07521 -07488 -07454 -07421
023 -07388 -07356 -07323 -on1O -07257 -0n25 -07192 -D716O -07128 -07095
024 -07063 -07031 -D6999 -D6967 -06935 -06103 -06871 -068lttD -06808 -D6776
TABLE AI (Continued)
0000 0001 0001 0001 0004 0005 0006 0007 0001 OOOt
025 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -06526 -06495 -06464 026 -06433 -D6ltt03 -O63n -06341 -06311 -06280 -06250 -06219 -06189 -06158 827 -06128 -06098 -06068 -06038 -06008 -05978 -O59ltt8 -05918 -05888 -05858 828 -05828 -05799 -05769 -05740 -05710 -05681 -05651 -05622 -05592 -05S63 O2t -05534 -05505 -05476 -DSi46 -05417 -05388 -0535 -05330 -05302 -05273 030 -052 -05215 -05187 -05158 -05129 -05101 -05072 -0SD44 -05015 -01987 031 -04959 -04930 -004902 -0lttI74 -04845 -004817 -04m -04761 -04733 -0A705 032 -077 -004649 -04621 -04593 -04565 -0045311 -04510 -04482 -04454 -04427 033 -04399 -04372 -043 -01316 -04289 -01261 -04234 -04207 -04179 -04152 034 -04IlS -04097 -01070 -04043 -D40 16 -039119 -031 -03934 -03907 -038BO lUi -031153 -03826 -0379~ -03m -03745 -037[9 -03poundn -03665 -li3638 -03611 I)l~ -03585 -03558 -oml -03505 -01478 -03451 -03425 -03398 -03372 -033-4S QlT -03319 -032~2 -03266 -0n~9 -03113 -03 [86 -03160 -03131 -03107 -1130ff rlt -03055 -O3Q2~ -03002 -027euroshy -02950 -0292pound -O2pound9G -02B71 -O2E-ltE -02lt1 ~~ -O2n3 -02767 -017lt[ -Ol7I~ -O~~6poundSmiddot -026euro -O2euroi -0261 [ -025pound -0l5~7
Q4~ -02523 -C2508 -024euro2 -02pound5pound -023(1 -0240lt -0237amp -02m -02027 -023QI ct bull -02275 -02250 -0222lt -021 -0217 -02[laquo7 -02121 -020 -02070 -02045 1147 -02019 -0193 -OIB -01gt42 -(U7 -OJgt -01866 -01840 -01815 -i17S 1141 -01714 -01738 -01713 -01687 -01662 -01637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -Ollll -01206 -01181 -01156 -01130 -01105 -01080 -01055 -01030 046 -01004 -00979 -00954 -00929 -Oo9tH -00878 -001153 -0DII28 -0DII03 -00778 047 -00753 -007211 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0tH76 -0tH51 -0tH26 -00401 -00376 -00351 -00326 -00301 -00276 049 -OOlSl -00226 -00201 -00175 -00150 -OOllS -00100 -00075 -00050 -OOOlS 050 00000 00025 00050 00075 00100 00125 00150 00175 00201 00226 051 OOlSl 00276 00301 00326 00351 00376 00401 OtH26 0tH51 0Oltt76 051 00502 00527 00552 00577 00602 00627 00652 00677 00702 00728 053 00753 00778 00803 00828 001153 00878 0090-4 00929 00954 00979 054 01004 01030 01055 01080 01105 01130 01156 01181 01206 01231
lit CI
Table AI (Continued) 0000 0001 0002 0003 0004 0005 0007 looa DDD
055 01257 OllIl 01307 01332 01358 01383 01408 01434 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738 057 017( 01789 01815 01amp40 01866 01891 01917 019-42 01968 01993 o5a 02019 020-45 02070 D1096 Dllll Dl147 01173 01198 0lll4 Ol25O 05 1U275 01301 01327 0l353 01378 DllaquoH 01 30 Dl456 DM81 01508 061 02533 02559 D2585 02611 Dl637 0l663 02689 02715 Dl741 01767 061 02793 02819 02845 02871 02898 Dl914 Dl950 Dl976 03002 03019 062 O3OSS 03081 03107 031304 03160 03186 03213 03239 03266 03292 063 03319 0335 03372 03398 03-425 03451 03(78 03505 03531 03558 06 03585 03611 03638 03665 03692 03719 03745 03772 03799 03826 065 03853 03880 03907 039304 03961 03989 0-4016 0laquoHl 0-4070 0-4097 066 ollS 0151 0179 0207 0134 0261 04219 0-4316 043 04372 067 04399 0 27 0 5-4 0+482 0510 04538 04565 0lt693 0lt1611 046-49 068 0-4677 04705 04733 0761 0 789 0-4817 0-4amp45 04874 04901 04930 069 04959 04987 DSOI5 050+1 05072 05101 05129 05158 05187 05215 070 051 05l73 05302 05330 O5lS9 05388 05417 05-4-46 D5476 O5SOS 071 055304 05563 05592 056ll 05651 05681 05710 057-40 05769 O5m Dn 05828 05858 05888 05918 05948 05918 06Q08 06038 06068 D6098 071 06121 06158 06189 D6219 0625Q 06l1D osil 06341 06372 064Q3 074 033 064 0 5 D6526 D6557 06588 066lQ 06651 0668l 06713 075 06745 06776 06808 06amp40 06871 06903 06935 0697 069 07031 176 07063 07095 07121 07160 07192 07225 07257 07290 07313 07356 077 07388 07411 0745-4 07-488 07Sl1 07554 07588 07621 07655 07688 071 077ll 07756 07790 07n4 07858 07892 07926 07961 07995 D8030 079 OllOM 08099 08134 Ul69 08204 08239 08174 08310 08345 D8381 OID 0110416 0amp451 08488 D85l 08560 08596 08633 08669 08705 D87-42 081 urn 08816 08853 08890 08927 08965 09002 090-40 09078 09116 082 0915-4 09192 09l30 09269 09307 093lt16 09385 0942lt1 09463 09502
Table AI (Continued) 0000 IUD I 0002 DDOJ 0004 0005 DDH 0007 DoDa 0009
Dl3 09542 09581 09611 09661 09701 09741 0978l 09822 09863 09904 084 099-45 09986 10017 10069 10110 10152 1094 1D237 0179 10322 085 1036-4 1()407 1D450 1D494 10537 10581 10615 10669 1071lt4 10758 D86 10803 1eIM8 10893 10939 10985 Ll031 11077 11123 11170 11217 087 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 118amp I750 11800 11850 1I~01 UITS2 f20Q4 12055 1207 L2160 12212 lIa~ 22euroS 123 f ~ 12372 12426 124BI (Elf 1251[ 12(46 12702 12759 bN 121H6 12en 12S30 t29se 1~CK7 I r06 316S IJru f2es flJ~C
M[ [gt4lS 14pound0 [3502 IlS~5 lJpoundSL [T21 12787 13E1 lS [7 IAon ~ II [ 11[17 Llt25E 1l2 r (E 1lt46 1lt5~t 14f II LCfilt 1475( r4poundZ3 [9(j rA9e~ IS(I(~ 151lt[ S22( L5301 1~3n L54H 155ltamp 15632 1573 15005 1513B [~ 16Q 161 G-t 162Sfl 161pound2 I~ 16546 6G46 167(7 ICIH~ [654 17060 17[6~ 17279- [73)2
175ar 17624 177 17S66 179gt1 LSI 19 18250 1pound384 18522 18663 18808 IB~5r 19J 10 19268 19431 19600 um4 19954 20141 20335 20537 20749 10969 11201 114+1 11701 11973 22262 11571 11904 13263 13656 24089 14573 l5111 l5758 26521 17478 18782 30902
Souka GeneIllted by Il L lman Used wiIh permission
The entries in his table are quand Z of the standard normal random variable Z selected so P(Z z) = p and P(Z gt z) = I - p Note that the value of P to two decimal places determines which row to USC the third decimal pia of detennines which col to use to find z
III o
APPENDIX 511 510 APPENDIX
TABLE Al ChI5qud Dlstributlon- TABLE Al Binomial Distributionmiddot
= 0750 000 0950 0975 1990 05 09 I I = 005 010 015 aI11 O~l~ 030 035 0-40 0-45
11=1 2 3 -4 5 6 7 8
10 II 12 13 1-4 15 16 17 18 I 10 11 11 13 1-4 15 16 17 28 l 30 -40 50 60 70 80 0
100
1323 2m 108 5385 6626 7HI 037
1012 113 1255 1370 185 1598 1711 1825 1937 20 2160 nn 2383
193 260-4 271 282 293 303 3153 3262 3371 30 561 5633 6698 7158 8813 865
1091 0675
2706 605 6251 7779 9136
106-4 1201 1l36 168 1599 1728 1855 1981 2106 1231 US 177 2599 1720 181 1961 3UI 3201 3310 338 3556 367 3791 390 fO16 5181 6317 n bull fO 8553 9658
1076 1185
1281
3HI 5991 7815
bull fBI 1107 115 107 1551 1691 1831 1968 1103 1236 1368 2500 1630 1759 1187 301 3 1 3267 3392 3517 362 3765 3889 011 113-4 256 4377 5576 6750 7908 9053
1019 1131 1243
16-45
502 7378 348
1114 1283 1445 1601 1753 1902 2048 2192 233-4 2474 2611 1749 2885 3019 3153 3285 3-417 3548 3678 3808 3937 fO65 412 4319 +6 1572 -4698 5U4 7142 8330 9502
1066 1181 1196
1960
6635 9110
11l4 1318 1509 1681 1848 2009 1167 2311 2473 2612 27 211 3U8 3200 3311 3-481 3619 3757 383 fO19 416-4 17-8 +131 156-4 -4696 4828
550 6369 7615 8838
1004 1123 1241 1358
2326
7179 10J00 1211 1116 1675 1855 203 219 235 2519 2676 1830 1982 3132 3210 3417 3572 3716 3858 fOOO 41fO 1210 441 155 3 1 44 SO99 513-4 5367 6677 791 US
10-42 1163 1113 1fO2
2576
1083 1312 1627 1847 1051 2246 2432 1613 1788 15 3116 32tI 3453 3611 3770 3915 fO79 4231 4312 532 80 27 19n 5118 5261 5405 55 568 5830 5970 n fO 666 61
1123 1118 1372 111
30t0
for gt 100 eIIllflPlmadon III shy Q)(z + lit-I) or ell I1IQA acane w -
k (I shy -l +z ~) whu 11 ch from ell IGlldudiud normal dlmibudon shown In ch bottOm
2
3
-4
5
6
7
o
o I 2
o I 1 3
o
1 3 -4
o I 2 3 -4 5
o
1 3 -4 5 6
o 1 2 3 -4 5 6
7
09500 10000 09025 09975 10000
08574 09928
099 10000
0815 09860 09995 10000 10000
07738 0977ltf 09988 10000 10000 10000
07351 09672 09978 099 10000 10000 10000
06983 0556 0992 09998 10000 10000 10000 10000
09000 10000 08100 0900 10000
07290 09720 09990 10000
06561 09477 09963 09999 10000
05905 09185 Q991-4 09995 10000 10000
05314 08857 098-42 09987 09999 10000 10000
0783 08503 09713 099n 09998 10000 10000 10000
08500 10000
07TIS 09775 10000
061041 09392 09966 10000
05220 08905 09880 09995 10000
0 37 08352 09734 09978 09999 10000
03771 07765 09527 09HI 09996 10000 10000
03206 07166 09162 09879 09988 09999 10000 10000
OBOOO 10000
06400 09600
100011
0512u
080 09920
10000
OA09 0B191 0971(J 099N 10000
OJ7
0737)
09411 09933
09997 10000
(llbLI
06554 09011
09830
099fH
09999
10000
02097
057a 0B520
09667
09953
099
10000 10000
01500 luoOO
iL5625
09375
10000
OAII
08138 0-844
10000
lIj I 6-1 O)(j
OJ4~gt
091 10000
Il2J7J 06]28 08965 01844 ti9990
10000
01780 05339 08306
014 09954
u9999 10UOO
UI335
0-149
07pound4 0911middot)
09971
0991l7
09999
10000
07000 10000
004900 09100 10000
03430 078-40 Q9730 10000
02401 06517 09163 09919 10000
01681 05281 08369 09692 09976 10000
01176 OA202 07 3 09295 09891 09993 10000
0082-4 0329-4 06-471
0870 09712 09962 09998 10000
06500 10000
OATIS 08775 10000
027-46 07182 09571 10000
01785 05630 08735 09850 10000
01160 OA18-4 076-48 09460 099-47 10000
0075 03191 06-471 08826 09777 09982 10000
00-490 02338 05323 0S002 09+44 09910 09994 10000
06000 10000
03600 011lt100 10000
02160 O6-4SO 09360 10000
01196 OA751 08208 097 10000
00778
03370 06826 09130 09898 10000 00-467 02l11 05 3 08208 09590 09959 10000
002SO 01586 OAI99 07102 09037 09812 O99H 10000
05500 10000
03025 07975 10000
0166-4 057-48 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00177 01636 0 15 07 7 09308 09917 10000
00152
0101 0316-4 06083 0H71 096-43 09963 10000
ofch abl SouacI AbrldSiOd from Tab VoL I of Pearson and HaM) (I 76) wilh ImlulOR from ell llatnetrllro Tr bullThe uMI1 In chis abl ar quanllle IIIr of a chllquarOd random varlabl W wilh It Uv- of frMdom Clad SO (W $ wJ ~ p and P(W gt wpl - I - p
III w
w w - bull
0 U P bullIII
P III III
fbull
f III
p III
bull
I n
-= w o
lCI W mIII Z
gtC
bull w
C
O UI WaJ
o o W
0 II o U o
o U o 0shyo
o 0shy
o
o
o o Do o
o Do o o
o
w
O UI W~_OW
0 II
o
o m Z
C
gtlt o ~ o
o ~
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
Page 2
QUESTION FOUR [25 marks]
Four job training programs were tried on 20 new employees where 5 employees were randomly assigned to each training program The 20 employees were then placed under the same supervisor and at the end of a certain period the supervisor ranked the employees according to job ability with the lowest ranks being assigned to those employees with the lowest job ability
Program Ranks 1 467210 2 18 123 11 3 20 19 16 145 4 181517139
Do these data indicate a difference in the effectiveness ofthe various training programs
QUESTION FIVE [25 marks]
A new worker is assigned to a machine that manufactures bolts Each day a sample of bolts is examined and the percent defective is recorded Do the following data indicate a significant improvement over time for the worker
Day Percent Day Percent Day Percent 1 61 6 61 10 46 2 75 7 53 11 30 3 77 8 45 12 40 4 59 9 49 13 37 5 52
Use either Spearmans p test or Kendalls Ttest
e lit
TABLE AI Nonnal Distribution
z1 -37110 z- - -32105 oas -19600 rea -16449 4 - 37110 ZcIm1 32905 ZltIm -= 19600 rea 16+49 0000 0-001 0002 0001 0004 0005 0006 0007 0008 O-OOt
000 -3o1Ol -28782 -27478 -26521 -25758 -25121 -24573 -2ltI0Il9 -1l656
001 -13263 -ll9Of -llS71 -lll62 -21973 -21701 -21 -21201 -2OK9 -20749
0-02 -lOS37 -20335 -2041 -19954 -19774 -19600 -19431 -19268 -19110 -18957
0-01 -188D8 -18663 -18522 -18384 -18250 -18119 -17991 -17866 -177 -17624
0-04 -17507 -17392 -1n79 -17169 -17060 -16954 -16849 -16747 -16646 -165ltt6
0-05 -16449 -16352 -16258 -16164 -160n -15982 -153 -15805 -15718 -15632
006 -15t8 -15464 -15382 -15301 -15220 -15141 -15063 -149B5 -1 909 -IlttIll
007 -14758 -1-4684 -14611 -14538 -1441gt1 -14395 -14325 -1ltt2S5 -14187 -14118
001 -1laquo151 -13984 -13917 -13852 -13787 -13722 -13658 -13595 -13532 -13469
Oot -1)408 -13346 -13285 -13225 -13165 -13106 -13047 -12988 -12930 -12873
DI0 -12816 -12759 -Il702 -12646 -12591 -12536 -12ltt11 -12426 -123n -12319
DII -12265 -12212 -12160 -12107 -12055 -IlOO4 -11952 -11901 -11850 -11800
012 -11750 -11700 -116S0 -11601 -11552 -11503 -1455 -117 -11359 -11311
011 -11264 -11217 -11170 -11123 -11077 -11031 -10985 -10939 -1D893 -10amp48
014 -IoB03 -10758 -G7H -10669 -1D625 -10581 -Ios37 -10494 -11)0450 -1DlttD7
US -10364 -Iol22 -llI279 -10237 -10194 -GI52 -10110 -1006 -10027 -09 016 -099lt45 -09904 -03 -U822 -09782 -0974 -09701 -096amp1 -021 -09581
017 -09542 -09502 -03 -09424 -09385 -09346 -09307 -09269 -09230 -09192
011 -09154 -09116 -01078 -09040 -01002 -08965 -08927 -08890 -08853 -D8816
01t -08779 -08742 -08705 -089 -08pound33 -085 -08560 -DB524 -08488 -08452
OlO -08416 -0838 -08345 -08310 -08274 -08239 -08204 -08169 -08134 -08099
021 -08064 -08030 -07995 -07961 -D7926 -07892 -07858 -01824 -07790 -07756
Oll -07722 -07688 -07655 -07621 -07588 -07554 -07521 -07488 -07454 -07421
023 -07388 -07356 -07323 -on1O -07257 -0n25 -07192 -D716O -07128 -07095
024 -07063 -07031 -D6999 -D6967 -06935 -06103 -06871 -068lttD -06808 -D6776
TABLE AI (Continued)
0000 0001 0001 0001 0004 0005 0006 0007 0001 OOOt
025 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -06526 -06495 -06464 026 -06433 -D6ltt03 -O63n -06341 -06311 -06280 -06250 -06219 -06189 -06158 827 -06128 -06098 -06068 -06038 -06008 -05978 -O59ltt8 -05918 -05888 -05858 828 -05828 -05799 -05769 -05740 -05710 -05681 -05651 -05622 -05592 -05S63 O2t -05534 -05505 -05476 -DSi46 -05417 -05388 -0535 -05330 -05302 -05273 030 -052 -05215 -05187 -05158 -05129 -05101 -05072 -0SD44 -05015 -01987 031 -04959 -04930 -004902 -0lttI74 -04845 -004817 -04m -04761 -04733 -0A705 032 -077 -004649 -04621 -04593 -04565 -0045311 -04510 -04482 -04454 -04427 033 -04399 -04372 -043 -01316 -04289 -01261 -04234 -04207 -04179 -04152 034 -04IlS -04097 -01070 -04043 -D40 16 -039119 -031 -03934 -03907 -038BO lUi -031153 -03826 -0379~ -03m -03745 -037[9 -03poundn -03665 -li3638 -03611 I)l~ -03585 -03558 -oml -03505 -01478 -03451 -03425 -03398 -03372 -033-4S QlT -03319 -032~2 -03266 -0n~9 -03113 -03 [86 -03160 -03131 -03107 -1130ff rlt -03055 -O3Q2~ -03002 -027euroshy -02950 -0292pound -O2pound9G -02B71 -O2E-ltE -02lt1 ~~ -O2n3 -02767 -017lt[ -Ol7I~ -O~~6poundSmiddot -026euro -O2euroi -0261 [ -025pound -0l5~7
Q4~ -02523 -C2508 -024euro2 -02pound5pound -023(1 -0240lt -0237amp -02m -02027 -023QI ct bull -02275 -02250 -0222lt -021 -0217 -02[laquo7 -02121 -020 -02070 -02045 1147 -02019 -0193 -OIB -01gt42 -(U7 -OJgt -01866 -01840 -01815 -i17S 1141 -01714 -01738 -01713 -01687 -01662 -01637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -Ollll -01206 -01181 -01156 -01130 -01105 -01080 -01055 -01030 046 -01004 -00979 -00954 -00929 -Oo9tH -00878 -001153 -0DII28 -0DII03 -00778 047 -00753 -007211 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0tH76 -0tH51 -0tH26 -00401 -00376 -00351 -00326 -00301 -00276 049 -OOlSl -00226 -00201 -00175 -00150 -OOllS -00100 -00075 -00050 -OOOlS 050 00000 00025 00050 00075 00100 00125 00150 00175 00201 00226 051 OOlSl 00276 00301 00326 00351 00376 00401 OtH26 0tH51 0Oltt76 051 00502 00527 00552 00577 00602 00627 00652 00677 00702 00728 053 00753 00778 00803 00828 001153 00878 0090-4 00929 00954 00979 054 01004 01030 01055 01080 01105 01130 01156 01181 01206 01231
lit CI
Table AI (Continued) 0000 0001 0002 0003 0004 0005 0007 looa DDD
055 01257 OllIl 01307 01332 01358 01383 01408 01434 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738 057 017( 01789 01815 01amp40 01866 01891 01917 019-42 01968 01993 o5a 02019 020-45 02070 D1096 Dllll Dl147 01173 01198 0lll4 Ol25O 05 1U275 01301 01327 0l353 01378 DllaquoH 01 30 Dl456 DM81 01508 061 02533 02559 D2585 02611 Dl637 0l663 02689 02715 Dl741 01767 061 02793 02819 02845 02871 02898 Dl914 Dl950 Dl976 03002 03019 062 O3OSS 03081 03107 031304 03160 03186 03213 03239 03266 03292 063 03319 0335 03372 03398 03-425 03451 03(78 03505 03531 03558 06 03585 03611 03638 03665 03692 03719 03745 03772 03799 03826 065 03853 03880 03907 039304 03961 03989 0-4016 0laquoHl 0-4070 0-4097 066 ollS 0151 0179 0207 0134 0261 04219 0-4316 043 04372 067 04399 0 27 0 5-4 0+482 0510 04538 04565 0lt693 0lt1611 046-49 068 0-4677 04705 04733 0761 0 789 0-4817 0-4amp45 04874 04901 04930 069 04959 04987 DSOI5 050+1 05072 05101 05129 05158 05187 05215 070 051 05l73 05302 05330 O5lS9 05388 05417 05-4-46 D5476 O5SOS 071 055304 05563 05592 056ll 05651 05681 05710 057-40 05769 O5m Dn 05828 05858 05888 05918 05948 05918 06Q08 06038 06068 D6098 071 06121 06158 06189 D6219 0625Q 06l1D osil 06341 06372 064Q3 074 033 064 0 5 D6526 D6557 06588 066lQ 06651 0668l 06713 075 06745 06776 06808 06amp40 06871 06903 06935 0697 069 07031 176 07063 07095 07121 07160 07192 07225 07257 07290 07313 07356 077 07388 07411 0745-4 07-488 07Sl1 07554 07588 07621 07655 07688 071 077ll 07756 07790 07n4 07858 07892 07926 07961 07995 D8030 079 OllOM 08099 08134 Ul69 08204 08239 08174 08310 08345 D8381 OID 0110416 0amp451 08488 D85l 08560 08596 08633 08669 08705 D87-42 081 urn 08816 08853 08890 08927 08965 09002 090-40 09078 09116 082 0915-4 09192 09l30 09269 09307 093lt16 09385 0942lt1 09463 09502
Table AI (Continued) 0000 IUD I 0002 DDOJ 0004 0005 DDH 0007 DoDa 0009
Dl3 09542 09581 09611 09661 09701 09741 0978l 09822 09863 09904 084 099-45 09986 10017 10069 10110 10152 1094 1D237 0179 10322 085 1036-4 1()407 1D450 1D494 10537 10581 10615 10669 1071lt4 10758 D86 10803 1eIM8 10893 10939 10985 Ll031 11077 11123 11170 11217 087 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 118amp I750 11800 11850 1I~01 UITS2 f20Q4 12055 1207 L2160 12212 lIa~ 22euroS 123 f ~ 12372 12426 124BI (Elf 1251[ 12(46 12702 12759 bN 121H6 12en 12S30 t29se 1~CK7 I r06 316S IJru f2es flJ~C
M[ [gt4lS 14pound0 [3502 IlS~5 lJpoundSL [T21 12787 13E1 lS [7 IAon ~ II [ 11[17 Llt25E 1l2 r (E 1lt46 1lt5~t 14f II LCfilt 1475( r4poundZ3 [9(j rA9e~ IS(I(~ 151lt[ S22( L5301 1~3n L54H 155ltamp 15632 1573 15005 1513B [~ 16Q 161 G-t 162Sfl 161pound2 I~ 16546 6G46 167(7 ICIH~ [654 17060 17[6~ 17279- [73)2
175ar 17624 177 17S66 179gt1 LSI 19 18250 1pound384 18522 18663 18808 IB~5r 19J 10 19268 19431 19600 um4 19954 20141 20335 20537 20749 10969 11201 114+1 11701 11973 22262 11571 11904 13263 13656 24089 14573 l5111 l5758 26521 17478 18782 30902
Souka GeneIllted by Il L lman Used wiIh permission
The entries in his table are quand Z of the standard normal random variable Z selected so P(Z z) = p and P(Z gt z) = I - p Note that the value of P to two decimal places determines which row to USC the third decimal pia of detennines which col to use to find z
III o
APPENDIX 511 510 APPENDIX
TABLE Al ChI5qud Dlstributlon- TABLE Al Binomial Distributionmiddot
= 0750 000 0950 0975 1990 05 09 I I = 005 010 015 aI11 O~l~ 030 035 0-40 0-45
11=1 2 3 -4 5 6 7 8
10 II 12 13 1-4 15 16 17 18 I 10 11 11 13 1-4 15 16 17 28 l 30 -40 50 60 70 80 0
100
1323 2m 108 5385 6626 7HI 037
1012 113 1255 1370 185 1598 1711 1825 1937 20 2160 nn 2383
193 260-4 271 282 293 303 3153 3262 3371 30 561 5633 6698 7158 8813 865
1091 0675
2706 605 6251 7779 9136
106-4 1201 1l36 168 1599 1728 1855 1981 2106 1231 US 177 2599 1720 181 1961 3UI 3201 3310 338 3556 367 3791 390 fO16 5181 6317 n bull fO 8553 9658
1076 1185
1281
3HI 5991 7815
bull fBI 1107 115 107 1551 1691 1831 1968 1103 1236 1368 2500 1630 1759 1187 301 3 1 3267 3392 3517 362 3765 3889 011 113-4 256 4377 5576 6750 7908 9053
1019 1131 1243
16-45
502 7378 348
1114 1283 1445 1601 1753 1902 2048 2192 233-4 2474 2611 1749 2885 3019 3153 3285 3-417 3548 3678 3808 3937 fO65 412 4319 +6 1572 -4698 5U4 7142 8330 9502
1066 1181 1196
1960
6635 9110
11l4 1318 1509 1681 1848 2009 1167 2311 2473 2612 27 211 3U8 3200 3311 3-481 3619 3757 383 fO19 416-4 17-8 +131 156-4 -4696 4828
550 6369 7615 8838
1004 1123 1241 1358
2326
7179 10J00 1211 1116 1675 1855 203 219 235 2519 2676 1830 1982 3132 3210 3417 3572 3716 3858 fOOO 41fO 1210 441 155 3 1 44 SO99 513-4 5367 6677 791 US
10-42 1163 1113 1fO2
2576
1083 1312 1627 1847 1051 2246 2432 1613 1788 15 3116 32tI 3453 3611 3770 3915 fO79 4231 4312 532 80 27 19n 5118 5261 5405 55 568 5830 5970 n fO 666 61
1123 1118 1372 111
30t0
for gt 100 eIIllflPlmadon III shy Q)(z + lit-I) or ell I1IQA acane w -
k (I shy -l +z ~) whu 11 ch from ell IGlldudiud normal dlmibudon shown In ch bottOm
2
3
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5
6
7
o
o I 2
o I 1 3
o
1 3 -4
o I 2 3 -4 5
o
1 3 -4 5 6
o 1 2 3 -4 5 6
7
09500 10000 09025 09975 10000
08574 09928
099 10000
0815 09860 09995 10000 10000
07738 0977ltf 09988 10000 10000 10000
07351 09672 09978 099 10000 10000 10000
06983 0556 0992 09998 10000 10000 10000 10000
09000 10000 08100 0900 10000
07290 09720 09990 10000
06561 09477 09963 09999 10000
05905 09185 Q991-4 09995 10000 10000
05314 08857 098-42 09987 09999 10000 10000
0783 08503 09713 099n 09998 10000 10000 10000
08500 10000
07TIS 09775 10000
061041 09392 09966 10000
05220 08905 09880 09995 10000
0 37 08352 09734 09978 09999 10000
03771 07765 09527 09HI 09996 10000 10000
03206 07166 09162 09879 09988 09999 10000 10000
OBOOO 10000
06400 09600
100011
0512u
080 09920
10000
OA09 0B191 0971(J 099N 10000
OJ7
0737)
09411 09933
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06554 09011
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09999
10000
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09953
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10000 10000
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10000
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10000
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10000
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07pound4 0911middot)
09971
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10000
07000 10000
004900 09100 10000
03430 078-40 Q9730 10000
02401 06517 09163 09919 10000
01681 05281 08369 09692 09976 10000
01176 OA202 07 3 09295 09891 09993 10000
0082-4 0329-4 06-471
0870 09712 09962 09998 10000
06500 10000
OATIS 08775 10000
027-46 07182 09571 10000
01785 05630 08735 09850 10000
01160 OA18-4 076-48 09460 099-47 10000
0075 03191 06-471 08826 09777 09982 10000
00-490 02338 05323 0S002 09+44 09910 09994 10000
06000 10000
03600 011lt100 10000
02160 O6-4SO 09360 10000
01196 OA751 08208 097 10000
00778
03370 06826 09130 09898 10000 00-467 02l11 05 3 08208 09590 09959 10000
002SO 01586 OAI99 07102 09037 09812 O99H 10000
05500 10000
03025 07975 10000
0166-4 057-48 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00177 01636 0 15 07 7 09308 09917 10000
00152
0101 0316-4 06083 0H71 096-43 09963 10000
ofch abl SouacI AbrldSiOd from Tab VoL I of Pearson and HaM) (I 76) wilh ImlulOR from ell llatnetrllro Tr bullThe uMI1 In chis abl ar quanllle IIIr of a chllquarOd random varlabl W wilh It Uv- of frMdom Clad SO (W $ wJ ~ p and P(W gt wpl - I - p
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w w - bull
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f III
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b po III o
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c
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bull
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520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
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00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
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00000 00000 00000
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00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
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099)
10000
u~-JI
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0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
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00001
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00000
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00000 00000
00000
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00000 20 o
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07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
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00003
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00000 00000
00000
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00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
e lit
TABLE AI Nonnal Distribution
z1 -37110 z- - -32105 oas -19600 rea -16449 4 - 37110 ZcIm1 32905 ZltIm -= 19600 rea 16+49 0000 0-001 0002 0001 0004 0005 0006 0007 0008 O-OOt
000 -3o1Ol -28782 -27478 -26521 -25758 -25121 -24573 -2ltI0Il9 -1l656
001 -13263 -ll9Of -llS71 -lll62 -21973 -21701 -21 -21201 -2OK9 -20749
0-02 -lOS37 -20335 -2041 -19954 -19774 -19600 -19431 -19268 -19110 -18957
0-01 -188D8 -18663 -18522 -18384 -18250 -18119 -17991 -17866 -177 -17624
0-04 -17507 -17392 -1n79 -17169 -17060 -16954 -16849 -16747 -16646 -165ltt6
0-05 -16449 -16352 -16258 -16164 -160n -15982 -153 -15805 -15718 -15632
006 -15t8 -15464 -15382 -15301 -15220 -15141 -15063 -149B5 -1 909 -IlttIll
007 -14758 -1-4684 -14611 -14538 -1441gt1 -14395 -14325 -1ltt2S5 -14187 -14118
001 -1laquo151 -13984 -13917 -13852 -13787 -13722 -13658 -13595 -13532 -13469
Oot -1)408 -13346 -13285 -13225 -13165 -13106 -13047 -12988 -12930 -12873
DI0 -12816 -12759 -Il702 -12646 -12591 -12536 -12ltt11 -12426 -123n -12319
DII -12265 -12212 -12160 -12107 -12055 -IlOO4 -11952 -11901 -11850 -11800
012 -11750 -11700 -116S0 -11601 -11552 -11503 -1455 -117 -11359 -11311
011 -11264 -11217 -11170 -11123 -11077 -11031 -10985 -10939 -1D893 -10amp48
014 -IoB03 -10758 -G7H -10669 -1D625 -10581 -Ios37 -10494 -11)0450 -1DlttD7
US -10364 -Iol22 -llI279 -10237 -10194 -GI52 -10110 -1006 -10027 -09 016 -099lt45 -09904 -03 -U822 -09782 -0974 -09701 -096amp1 -021 -09581
017 -09542 -09502 -03 -09424 -09385 -09346 -09307 -09269 -09230 -09192
011 -09154 -09116 -01078 -09040 -01002 -08965 -08927 -08890 -08853 -D8816
01t -08779 -08742 -08705 -089 -08pound33 -085 -08560 -DB524 -08488 -08452
OlO -08416 -0838 -08345 -08310 -08274 -08239 -08204 -08169 -08134 -08099
021 -08064 -08030 -07995 -07961 -D7926 -07892 -07858 -01824 -07790 -07756
Oll -07722 -07688 -07655 -07621 -07588 -07554 -07521 -07488 -07454 -07421
023 -07388 -07356 -07323 -on1O -07257 -0n25 -07192 -D716O -07128 -07095
024 -07063 -07031 -D6999 -D6967 -06935 -06103 -06871 -068lttD -06808 -D6776
TABLE AI (Continued)
0000 0001 0001 0001 0004 0005 0006 0007 0001 OOOt
025 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -06526 -06495 -06464 026 -06433 -D6ltt03 -O63n -06341 -06311 -06280 -06250 -06219 -06189 -06158 827 -06128 -06098 -06068 -06038 -06008 -05978 -O59ltt8 -05918 -05888 -05858 828 -05828 -05799 -05769 -05740 -05710 -05681 -05651 -05622 -05592 -05S63 O2t -05534 -05505 -05476 -DSi46 -05417 -05388 -0535 -05330 -05302 -05273 030 -052 -05215 -05187 -05158 -05129 -05101 -05072 -0SD44 -05015 -01987 031 -04959 -04930 -004902 -0lttI74 -04845 -004817 -04m -04761 -04733 -0A705 032 -077 -004649 -04621 -04593 -04565 -0045311 -04510 -04482 -04454 -04427 033 -04399 -04372 -043 -01316 -04289 -01261 -04234 -04207 -04179 -04152 034 -04IlS -04097 -01070 -04043 -D40 16 -039119 -031 -03934 -03907 -038BO lUi -031153 -03826 -0379~ -03m -03745 -037[9 -03poundn -03665 -li3638 -03611 I)l~ -03585 -03558 -oml -03505 -01478 -03451 -03425 -03398 -03372 -033-4S QlT -03319 -032~2 -03266 -0n~9 -03113 -03 [86 -03160 -03131 -03107 -1130ff rlt -03055 -O3Q2~ -03002 -027euroshy -02950 -0292pound -O2pound9G -02B71 -O2E-ltE -02lt1 ~~ -O2n3 -02767 -017lt[ -Ol7I~ -O~~6poundSmiddot -026euro -O2euroi -0261 [ -025pound -0l5~7
Q4~ -02523 -C2508 -024euro2 -02pound5pound -023(1 -0240lt -0237amp -02m -02027 -023QI ct bull -02275 -02250 -0222lt -021 -0217 -02[laquo7 -02121 -020 -02070 -02045 1147 -02019 -0193 -OIB -01gt42 -(U7 -OJgt -01866 -01840 -01815 -i17S 1141 -01714 -01738 -01713 -01687 -01662 -01637 -01611 -01586 -01560 -01535 044 -01510 -01-484 -01459 -01434 -01408 -01383 -01358 -01332 -01307 -01282 045 -0llS7 -Ollll -01206 -01181 -01156 -01130 -01105 -01080 -01055 -01030 046 -01004 -00979 -00954 -00929 -Oo9tH -00878 -001153 -0DII28 -0DII03 -00778 047 -00753 -007211 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -0tH76 -0tH51 -0tH26 -00401 -00376 -00351 -00326 -00301 -00276 049 -OOlSl -00226 -00201 -00175 -00150 -OOllS -00100 -00075 -00050 -OOOlS 050 00000 00025 00050 00075 00100 00125 00150 00175 00201 00226 051 OOlSl 00276 00301 00326 00351 00376 00401 OtH26 0tH51 0Oltt76 051 00502 00527 00552 00577 00602 00627 00652 00677 00702 00728 053 00753 00778 00803 00828 001153 00878 0090-4 00929 00954 00979 054 01004 01030 01055 01080 01105 01130 01156 01181 01206 01231
lit CI
Table AI (Continued) 0000 0001 0002 0003 0004 0005 0007 looa DDD
055 01257 OllIl 01307 01332 01358 01383 01408 01434 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738 057 017( 01789 01815 01amp40 01866 01891 01917 019-42 01968 01993 o5a 02019 020-45 02070 D1096 Dllll Dl147 01173 01198 0lll4 Ol25O 05 1U275 01301 01327 0l353 01378 DllaquoH 01 30 Dl456 DM81 01508 061 02533 02559 D2585 02611 Dl637 0l663 02689 02715 Dl741 01767 061 02793 02819 02845 02871 02898 Dl914 Dl950 Dl976 03002 03019 062 O3OSS 03081 03107 031304 03160 03186 03213 03239 03266 03292 063 03319 0335 03372 03398 03-425 03451 03(78 03505 03531 03558 06 03585 03611 03638 03665 03692 03719 03745 03772 03799 03826 065 03853 03880 03907 039304 03961 03989 0-4016 0laquoHl 0-4070 0-4097 066 ollS 0151 0179 0207 0134 0261 04219 0-4316 043 04372 067 04399 0 27 0 5-4 0+482 0510 04538 04565 0lt693 0lt1611 046-49 068 0-4677 04705 04733 0761 0 789 0-4817 0-4amp45 04874 04901 04930 069 04959 04987 DSOI5 050+1 05072 05101 05129 05158 05187 05215 070 051 05l73 05302 05330 O5lS9 05388 05417 05-4-46 D5476 O5SOS 071 055304 05563 05592 056ll 05651 05681 05710 057-40 05769 O5m Dn 05828 05858 05888 05918 05948 05918 06Q08 06038 06068 D6098 071 06121 06158 06189 D6219 0625Q 06l1D osil 06341 06372 064Q3 074 033 064 0 5 D6526 D6557 06588 066lQ 06651 0668l 06713 075 06745 06776 06808 06amp40 06871 06903 06935 0697 069 07031 176 07063 07095 07121 07160 07192 07225 07257 07290 07313 07356 077 07388 07411 0745-4 07-488 07Sl1 07554 07588 07621 07655 07688 071 077ll 07756 07790 07n4 07858 07892 07926 07961 07995 D8030 079 OllOM 08099 08134 Ul69 08204 08239 08174 08310 08345 D8381 OID 0110416 0amp451 08488 D85l 08560 08596 08633 08669 08705 D87-42 081 urn 08816 08853 08890 08927 08965 09002 090-40 09078 09116 082 0915-4 09192 09l30 09269 09307 093lt16 09385 0942lt1 09463 09502
Table AI (Continued) 0000 IUD I 0002 DDOJ 0004 0005 DDH 0007 DoDa 0009
Dl3 09542 09581 09611 09661 09701 09741 0978l 09822 09863 09904 084 099-45 09986 10017 10069 10110 10152 1094 1D237 0179 10322 085 1036-4 1()407 1D450 1D494 10537 10581 10615 10669 1071lt4 10758 D86 10803 1eIM8 10893 10939 10985 Ll031 11077 11123 11170 11217 087 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 118amp I750 11800 11850 1I~01 UITS2 f20Q4 12055 1207 L2160 12212 lIa~ 22euroS 123 f ~ 12372 12426 124BI (Elf 1251[ 12(46 12702 12759 bN 121H6 12en 12S30 t29se 1~CK7 I r06 316S IJru f2es flJ~C
M[ [gt4lS 14pound0 [3502 IlS~5 lJpoundSL [T21 12787 13E1 lS [7 IAon ~ II [ 11[17 Llt25E 1l2 r (E 1lt46 1lt5~t 14f II LCfilt 1475( r4poundZ3 [9(j rA9e~ IS(I(~ 151lt[ S22( L5301 1~3n L54H 155ltamp 15632 1573 15005 1513B [~ 16Q 161 G-t 162Sfl 161pound2 I~ 16546 6G46 167(7 ICIH~ [654 17060 17[6~ 17279- [73)2
175ar 17624 177 17S66 179gt1 LSI 19 18250 1pound384 18522 18663 18808 IB~5r 19J 10 19268 19431 19600 um4 19954 20141 20335 20537 20749 10969 11201 114+1 11701 11973 22262 11571 11904 13263 13656 24089 14573 l5111 l5758 26521 17478 18782 30902
Souka GeneIllted by Il L lman Used wiIh permission
The entries in his table are quand Z of the standard normal random variable Z selected so P(Z z) = p and P(Z gt z) = I - p Note that the value of P to two decimal places determines which row to USC the third decimal pia of detennines which col to use to find z
III o
APPENDIX 511 510 APPENDIX
TABLE Al ChI5qud Dlstributlon- TABLE Al Binomial Distributionmiddot
= 0750 000 0950 0975 1990 05 09 I I = 005 010 015 aI11 O~l~ 030 035 0-40 0-45
11=1 2 3 -4 5 6 7 8
10 II 12 13 1-4 15 16 17 18 I 10 11 11 13 1-4 15 16 17 28 l 30 -40 50 60 70 80 0
100
1323 2m 108 5385 6626 7HI 037
1012 113 1255 1370 185 1598 1711 1825 1937 20 2160 nn 2383
193 260-4 271 282 293 303 3153 3262 3371 30 561 5633 6698 7158 8813 865
1091 0675
2706 605 6251 7779 9136
106-4 1201 1l36 168 1599 1728 1855 1981 2106 1231 US 177 2599 1720 181 1961 3UI 3201 3310 338 3556 367 3791 390 fO16 5181 6317 n bull fO 8553 9658
1076 1185
1281
3HI 5991 7815
bull fBI 1107 115 107 1551 1691 1831 1968 1103 1236 1368 2500 1630 1759 1187 301 3 1 3267 3392 3517 362 3765 3889 011 113-4 256 4377 5576 6750 7908 9053
1019 1131 1243
16-45
502 7378 348
1114 1283 1445 1601 1753 1902 2048 2192 233-4 2474 2611 1749 2885 3019 3153 3285 3-417 3548 3678 3808 3937 fO65 412 4319 +6 1572 -4698 5U4 7142 8330 9502
1066 1181 1196
1960
6635 9110
11l4 1318 1509 1681 1848 2009 1167 2311 2473 2612 27 211 3U8 3200 3311 3-481 3619 3757 383 fO19 416-4 17-8 +131 156-4 -4696 4828
550 6369 7615 8838
1004 1123 1241 1358
2326
7179 10J00 1211 1116 1675 1855 203 219 235 2519 2676 1830 1982 3132 3210 3417 3572 3716 3858 fOOO 41fO 1210 441 155 3 1 44 SO99 513-4 5367 6677 791 US
10-42 1163 1113 1fO2
2576
1083 1312 1627 1847 1051 2246 2432 1613 1788 15 3116 32tI 3453 3611 3770 3915 fO79 4231 4312 532 80 27 19n 5118 5261 5405 55 568 5830 5970 n fO 666 61
1123 1118 1372 111
30t0
for gt 100 eIIllflPlmadon III shy Q)(z + lit-I) or ell I1IQA acane w -
k (I shy -l +z ~) whu 11 ch from ell IGlldudiud normal dlmibudon shown In ch bottOm
2
3
-4
5
6
7
o
o I 2
o I 1 3
o
1 3 -4
o I 2 3 -4 5
o
1 3 -4 5 6
o 1 2 3 -4 5 6
7
09500 10000 09025 09975 10000
08574 09928
099 10000
0815 09860 09995 10000 10000
07738 0977ltf 09988 10000 10000 10000
07351 09672 09978 099 10000 10000 10000
06983 0556 0992 09998 10000 10000 10000 10000
09000 10000 08100 0900 10000
07290 09720 09990 10000
06561 09477 09963 09999 10000
05905 09185 Q991-4 09995 10000 10000
05314 08857 098-42 09987 09999 10000 10000
0783 08503 09713 099n 09998 10000 10000 10000
08500 10000
07TIS 09775 10000
061041 09392 09966 10000
05220 08905 09880 09995 10000
0 37 08352 09734 09978 09999 10000
03771 07765 09527 09HI 09996 10000 10000
03206 07166 09162 09879 09988 09999 10000 10000
OBOOO 10000
06400 09600
100011
0512u
080 09920
10000
OA09 0B191 0971(J 099N 10000
OJ7
0737)
09411 09933
09997 10000
(llbLI
06554 09011
09830
099fH
09999
10000
02097
057a 0B520
09667
09953
099
10000 10000
01500 luoOO
iL5625
09375
10000
OAII
08138 0-844
10000
lIj I 6-1 O)(j
OJ4~gt
091 10000
Il2J7J 06]28 08965 01844 ti9990
10000
01780 05339 08306
014 09954
u9999 10UOO
UI335
0-149
07pound4 0911middot)
09971
0991l7
09999
10000
07000 10000
004900 09100 10000
03430 078-40 Q9730 10000
02401 06517 09163 09919 10000
01681 05281 08369 09692 09976 10000
01176 OA202 07 3 09295 09891 09993 10000
0082-4 0329-4 06-471
0870 09712 09962 09998 10000
06500 10000
OATIS 08775 10000
027-46 07182 09571 10000
01785 05630 08735 09850 10000
01160 OA18-4 076-48 09460 099-47 10000
0075 03191 06-471 08826 09777 09982 10000
00-490 02338 05323 0S002 09+44 09910 09994 10000
06000 10000
03600 011lt100 10000
02160 O6-4SO 09360 10000
01196 OA751 08208 097 10000
00778
03370 06826 09130 09898 10000 00-467 02l11 05 3 08208 09590 09959 10000
002SO 01586 OAI99 07102 09037 09812 O99H 10000
05500 10000
03025 07975 10000
0166-4 057-48 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00177 01636 0 15 07 7 09308 09917 10000
00152
0101 0316-4 06083 0H71 096-43 09963 10000
ofch abl SouacI AbrldSiOd from Tab VoL I of Pearson and HaM) (I 76) wilh ImlulOR from ell llatnetrllro Tr bullThe uMI1 In chis abl ar quanllle IIIr of a chllquarOd random varlabl W wilh It Uv- of frMdom Clad SO (W $ wJ ~ p and P(W gt wpl - I - p
III w
w w - bull
0 U P bullIII
P III III
fbull
f III
p III
bull
I n
-= w o
lCI W mIII Z
gtC
bull w
C
O UI WaJ
o o W
0 II o U o
o U o 0shyo
o 0shy
o
o
o o Do o
o Do o o
o
w
O UI W~_OW
0 II
o
o m Z
C
gtlt o ~ o
o ~
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
Table AI (Continued) 0000 0001 0002 0003 0004 0005 0007 looa DDD
055 01257 OllIl 01307 01332 01358 01383 01408 01434 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738 057 017( 01789 01815 01amp40 01866 01891 01917 019-42 01968 01993 o5a 02019 020-45 02070 D1096 Dllll Dl147 01173 01198 0lll4 Ol25O 05 1U275 01301 01327 0l353 01378 DllaquoH 01 30 Dl456 DM81 01508 061 02533 02559 D2585 02611 Dl637 0l663 02689 02715 Dl741 01767 061 02793 02819 02845 02871 02898 Dl914 Dl950 Dl976 03002 03019 062 O3OSS 03081 03107 031304 03160 03186 03213 03239 03266 03292 063 03319 0335 03372 03398 03-425 03451 03(78 03505 03531 03558 06 03585 03611 03638 03665 03692 03719 03745 03772 03799 03826 065 03853 03880 03907 039304 03961 03989 0-4016 0laquoHl 0-4070 0-4097 066 ollS 0151 0179 0207 0134 0261 04219 0-4316 043 04372 067 04399 0 27 0 5-4 0+482 0510 04538 04565 0lt693 0lt1611 046-49 068 0-4677 04705 04733 0761 0 789 0-4817 0-4amp45 04874 04901 04930 069 04959 04987 DSOI5 050+1 05072 05101 05129 05158 05187 05215 070 051 05l73 05302 05330 O5lS9 05388 05417 05-4-46 D5476 O5SOS 071 055304 05563 05592 056ll 05651 05681 05710 057-40 05769 O5m Dn 05828 05858 05888 05918 05948 05918 06Q08 06038 06068 D6098 071 06121 06158 06189 D6219 0625Q 06l1D osil 06341 06372 064Q3 074 033 064 0 5 D6526 D6557 06588 066lQ 06651 0668l 06713 075 06745 06776 06808 06amp40 06871 06903 06935 0697 069 07031 176 07063 07095 07121 07160 07192 07225 07257 07290 07313 07356 077 07388 07411 0745-4 07-488 07Sl1 07554 07588 07621 07655 07688 071 077ll 07756 07790 07n4 07858 07892 07926 07961 07995 D8030 079 OllOM 08099 08134 Ul69 08204 08239 08174 08310 08345 D8381 OID 0110416 0amp451 08488 D85l 08560 08596 08633 08669 08705 D87-42 081 urn 08816 08853 08890 08927 08965 09002 090-40 09078 09116 082 0915-4 09192 09l30 09269 09307 093lt16 09385 0942lt1 09463 09502
Table AI (Continued) 0000 IUD I 0002 DDOJ 0004 0005 DDH 0007 DoDa 0009
Dl3 09542 09581 09611 09661 09701 09741 0978l 09822 09863 09904 084 099-45 09986 10017 10069 10110 10152 1094 1D237 0179 10322 085 1036-4 1()407 1D450 1D494 10537 10581 10615 10669 1071lt4 10758 D86 10803 1eIM8 10893 10939 10985 Ll031 11077 11123 11170 11217 087 11264 11311 11359 11407 11455 11503 11552 11601 11650 11700 118amp I750 11800 11850 1I~01 UITS2 f20Q4 12055 1207 L2160 12212 lIa~ 22euroS 123 f ~ 12372 12426 124BI (Elf 1251[ 12(46 12702 12759 bN 121H6 12en 12S30 t29se 1~CK7 I r06 316S IJru f2es flJ~C
M[ [gt4lS 14pound0 [3502 IlS~5 lJpoundSL [T21 12787 13E1 lS [7 IAon ~ II [ 11[17 Llt25E 1l2 r (E 1lt46 1lt5~t 14f II LCfilt 1475( r4poundZ3 [9(j rA9e~ IS(I(~ 151lt[ S22( L5301 1~3n L54H 155ltamp 15632 1573 15005 1513B [~ 16Q 161 G-t 162Sfl 161pound2 I~ 16546 6G46 167(7 ICIH~ [654 17060 17[6~ 17279- [73)2
175ar 17624 177 17S66 179gt1 LSI 19 18250 1pound384 18522 18663 18808 IB~5r 19J 10 19268 19431 19600 um4 19954 20141 20335 20537 20749 10969 11201 114+1 11701 11973 22262 11571 11904 13263 13656 24089 14573 l5111 l5758 26521 17478 18782 30902
Souka GeneIllted by Il L lman Used wiIh permission
The entries in his table are quand Z of the standard normal random variable Z selected so P(Z z) = p and P(Z gt z) = I - p Note that the value of P to two decimal places determines which row to USC the third decimal pia of detennines which col to use to find z
III o
APPENDIX 511 510 APPENDIX
TABLE Al ChI5qud Dlstributlon- TABLE Al Binomial Distributionmiddot
= 0750 000 0950 0975 1990 05 09 I I = 005 010 015 aI11 O~l~ 030 035 0-40 0-45
11=1 2 3 -4 5 6 7 8
10 II 12 13 1-4 15 16 17 18 I 10 11 11 13 1-4 15 16 17 28 l 30 -40 50 60 70 80 0
100
1323 2m 108 5385 6626 7HI 037
1012 113 1255 1370 185 1598 1711 1825 1937 20 2160 nn 2383
193 260-4 271 282 293 303 3153 3262 3371 30 561 5633 6698 7158 8813 865
1091 0675
2706 605 6251 7779 9136
106-4 1201 1l36 168 1599 1728 1855 1981 2106 1231 US 177 2599 1720 181 1961 3UI 3201 3310 338 3556 367 3791 390 fO16 5181 6317 n bull fO 8553 9658
1076 1185
1281
3HI 5991 7815
bull fBI 1107 115 107 1551 1691 1831 1968 1103 1236 1368 2500 1630 1759 1187 301 3 1 3267 3392 3517 362 3765 3889 011 113-4 256 4377 5576 6750 7908 9053
1019 1131 1243
16-45
502 7378 348
1114 1283 1445 1601 1753 1902 2048 2192 233-4 2474 2611 1749 2885 3019 3153 3285 3-417 3548 3678 3808 3937 fO65 412 4319 +6 1572 -4698 5U4 7142 8330 9502
1066 1181 1196
1960
6635 9110
11l4 1318 1509 1681 1848 2009 1167 2311 2473 2612 27 211 3U8 3200 3311 3-481 3619 3757 383 fO19 416-4 17-8 +131 156-4 -4696 4828
550 6369 7615 8838
1004 1123 1241 1358
2326
7179 10J00 1211 1116 1675 1855 203 219 235 2519 2676 1830 1982 3132 3210 3417 3572 3716 3858 fOOO 41fO 1210 441 155 3 1 44 SO99 513-4 5367 6677 791 US
10-42 1163 1113 1fO2
2576
1083 1312 1627 1847 1051 2246 2432 1613 1788 15 3116 32tI 3453 3611 3770 3915 fO79 4231 4312 532 80 27 19n 5118 5261 5405 55 568 5830 5970 n fO 666 61
1123 1118 1372 111
30t0
for gt 100 eIIllflPlmadon III shy Q)(z + lit-I) or ell I1IQA acane w -
k (I shy -l +z ~) whu 11 ch from ell IGlldudiud normal dlmibudon shown In ch bottOm
2
3
-4
5
6
7
o
o I 2
o I 1 3
o
1 3 -4
o I 2 3 -4 5
o
1 3 -4 5 6
o 1 2 3 -4 5 6
7
09500 10000 09025 09975 10000
08574 09928
099 10000
0815 09860 09995 10000 10000
07738 0977ltf 09988 10000 10000 10000
07351 09672 09978 099 10000 10000 10000
06983 0556 0992 09998 10000 10000 10000 10000
09000 10000 08100 0900 10000
07290 09720 09990 10000
06561 09477 09963 09999 10000
05905 09185 Q991-4 09995 10000 10000
05314 08857 098-42 09987 09999 10000 10000
0783 08503 09713 099n 09998 10000 10000 10000
08500 10000
07TIS 09775 10000
061041 09392 09966 10000
05220 08905 09880 09995 10000
0 37 08352 09734 09978 09999 10000
03771 07765 09527 09HI 09996 10000 10000
03206 07166 09162 09879 09988 09999 10000 10000
OBOOO 10000
06400 09600
100011
0512u
080 09920
10000
OA09 0B191 0971(J 099N 10000
OJ7
0737)
09411 09933
09997 10000
(llbLI
06554 09011
09830
099fH
09999
10000
02097
057a 0B520
09667
09953
099
10000 10000
01500 luoOO
iL5625
09375
10000
OAII
08138 0-844
10000
lIj I 6-1 O)(j
OJ4~gt
091 10000
Il2J7J 06]28 08965 01844 ti9990
10000
01780 05339 08306
014 09954
u9999 10UOO
UI335
0-149
07pound4 0911middot)
09971
0991l7
09999
10000
07000 10000
004900 09100 10000
03430 078-40 Q9730 10000
02401 06517 09163 09919 10000
01681 05281 08369 09692 09976 10000
01176 OA202 07 3 09295 09891 09993 10000
0082-4 0329-4 06-471
0870 09712 09962 09998 10000
06500 10000
OATIS 08775 10000
027-46 07182 09571 10000
01785 05630 08735 09850 10000
01160 OA18-4 076-48 09460 099-47 10000
0075 03191 06-471 08826 09777 09982 10000
00-490 02338 05323 0S002 09+44 09910 09994 10000
06000 10000
03600 011lt100 10000
02160 O6-4SO 09360 10000
01196 OA751 08208 097 10000
00778
03370 06826 09130 09898 10000 00-467 02l11 05 3 08208 09590 09959 10000
002SO 01586 OAI99 07102 09037 09812 O99H 10000
05500 10000
03025 07975 10000
0166-4 057-48 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00177 01636 0 15 07 7 09308 09917 10000
00152
0101 0316-4 06083 0H71 096-43 09963 10000
ofch abl SouacI AbrldSiOd from Tab VoL I of Pearson and HaM) (I 76) wilh ImlulOR from ell llatnetrllro Tr bullThe uMI1 In chis abl ar quanllle IIIr of a chllquarOd random varlabl W wilh It Uv- of frMdom Clad SO (W $ wJ ~ p and P(W gt wpl - I - p
III w
w w - bull
0 U P bullIII
P III III
fbull
f III
p III
bull
I n
-= w o
lCI W mIII Z
gtC
bull w
C
O UI WaJ
o o W
0 II o U o
o U o 0shyo
o 0shy
o
o
o o Do o
o Do o o
o
w
O UI W~_OW
0 II
o
o m Z
C
gtlt o ~ o
o ~
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
APPENDIX 511 510 APPENDIX
TABLE Al ChI5qud Dlstributlon- TABLE Al Binomial Distributionmiddot
= 0750 000 0950 0975 1990 05 09 I I = 005 010 015 aI11 O~l~ 030 035 0-40 0-45
11=1 2 3 -4 5 6 7 8
10 II 12 13 1-4 15 16 17 18 I 10 11 11 13 1-4 15 16 17 28 l 30 -40 50 60 70 80 0
100
1323 2m 108 5385 6626 7HI 037
1012 113 1255 1370 185 1598 1711 1825 1937 20 2160 nn 2383
193 260-4 271 282 293 303 3153 3262 3371 30 561 5633 6698 7158 8813 865
1091 0675
2706 605 6251 7779 9136
106-4 1201 1l36 168 1599 1728 1855 1981 2106 1231 US 177 2599 1720 181 1961 3UI 3201 3310 338 3556 367 3791 390 fO16 5181 6317 n bull fO 8553 9658
1076 1185
1281
3HI 5991 7815
bull fBI 1107 115 107 1551 1691 1831 1968 1103 1236 1368 2500 1630 1759 1187 301 3 1 3267 3392 3517 362 3765 3889 011 113-4 256 4377 5576 6750 7908 9053
1019 1131 1243
16-45
502 7378 348
1114 1283 1445 1601 1753 1902 2048 2192 233-4 2474 2611 1749 2885 3019 3153 3285 3-417 3548 3678 3808 3937 fO65 412 4319 +6 1572 -4698 5U4 7142 8330 9502
1066 1181 1196
1960
6635 9110
11l4 1318 1509 1681 1848 2009 1167 2311 2473 2612 27 211 3U8 3200 3311 3-481 3619 3757 383 fO19 416-4 17-8 +131 156-4 -4696 4828
550 6369 7615 8838
1004 1123 1241 1358
2326
7179 10J00 1211 1116 1675 1855 203 219 235 2519 2676 1830 1982 3132 3210 3417 3572 3716 3858 fOOO 41fO 1210 441 155 3 1 44 SO99 513-4 5367 6677 791 US
10-42 1163 1113 1fO2
2576
1083 1312 1627 1847 1051 2246 2432 1613 1788 15 3116 32tI 3453 3611 3770 3915 fO79 4231 4312 532 80 27 19n 5118 5261 5405 55 568 5830 5970 n fO 666 61
1123 1118 1372 111
30t0
for gt 100 eIIllflPlmadon III shy Q)(z + lit-I) or ell I1IQA acane w -
k (I shy -l +z ~) whu 11 ch from ell IGlldudiud normal dlmibudon shown In ch bottOm
2
3
-4
5
6
7
o
o I 2
o I 1 3
o
1 3 -4
o I 2 3 -4 5
o
1 3 -4 5 6
o 1 2 3 -4 5 6
7
09500 10000 09025 09975 10000
08574 09928
099 10000
0815 09860 09995 10000 10000
07738 0977ltf 09988 10000 10000 10000
07351 09672 09978 099 10000 10000 10000
06983 0556 0992 09998 10000 10000 10000 10000
09000 10000 08100 0900 10000
07290 09720 09990 10000
06561 09477 09963 09999 10000
05905 09185 Q991-4 09995 10000 10000
05314 08857 098-42 09987 09999 10000 10000
0783 08503 09713 099n 09998 10000 10000 10000
08500 10000
07TIS 09775 10000
061041 09392 09966 10000
05220 08905 09880 09995 10000
0 37 08352 09734 09978 09999 10000
03771 07765 09527 09HI 09996 10000 10000
03206 07166 09162 09879 09988 09999 10000 10000
OBOOO 10000
06400 09600
100011
0512u
080 09920
10000
OA09 0B191 0971(J 099N 10000
OJ7
0737)
09411 09933
09997 10000
(llbLI
06554 09011
09830
099fH
09999
10000
02097
057a 0B520
09667
09953
099
10000 10000
01500 luoOO
iL5625
09375
10000
OAII
08138 0-844
10000
lIj I 6-1 O)(j
OJ4~gt
091 10000
Il2J7J 06]28 08965 01844 ti9990
10000
01780 05339 08306
014 09954
u9999 10UOO
UI335
0-149
07pound4 0911middot)
09971
0991l7
09999
10000
07000 10000
004900 09100 10000
03430 078-40 Q9730 10000
02401 06517 09163 09919 10000
01681 05281 08369 09692 09976 10000
01176 OA202 07 3 09295 09891 09993 10000
0082-4 0329-4 06-471
0870 09712 09962 09998 10000
06500 10000
OATIS 08775 10000
027-46 07182 09571 10000
01785 05630 08735 09850 10000
01160 OA18-4 076-48 09460 099-47 10000
0075 03191 06-471 08826 09777 09982 10000
00-490 02338 05323 0S002 09+44 09910 09994 10000
06000 10000
03600 011lt100 10000
02160 O6-4SO 09360 10000
01196 OA751 08208 097 10000
00778
03370 06826 09130 09898 10000 00-467 02l11 05 3 08208 09590 09959 10000
002SO 01586 OAI99 07102 09037 09812 O99H 10000
05500 10000
03025 07975 10000
0166-4 057-48 09089 10000
00915 03910 07585 09590 10000
00503 02562 05931 08688 09815 10000
00177 01636 0 15 07 7 09308 09917 10000
00152
0101 0316-4 06083 0H71 096-43 09963 10000
ofch abl SouacI AbrldSiOd from Tab VoL I of Pearson and HaM) (I 76) wilh ImlulOR from ell llatnetrllro Tr bullThe uMI1 In chis abl ar quanllle IIIr of a chllquarOd random varlabl W wilh It Uv- of frMdom Clad SO (W $ wJ ~ p and P(W gt wpl - I - p
III w
w w - bull
0 U P bullIII
P III III
fbull
f III
p III
bull
I n
-= w o
lCI W mIII Z
gtC
bull w
C
O UI WaJ
o o W
0 II o U o
o U o 0shyo
o 0shy
o
o
o o Do o
o Do o o
o
w
O UI W~_OW
0 II
o
o m Z
C
gtlt o ~ o
o ~
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
III w
w w - bull
0 U P bullIII
P III III
fbull
f III
p III
bull
I n
-= w o
lCI W mIII Z
gtC
bull w
C
O UI WaJ
o o W
0 II o U o
o U o 0shyo
o 0shy
o
o
o o Do o
o Do o o
o
w
O UI W~_OW
0 II
o
o m Z
C
gtlt o ~ o
o ~
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
O UI WaJ
o o W
0 II o U o
o U o 0shyo
o 0shy
o
o
o o Do o
o Do o o
o
w
O UI W~_OW
0 II
o
o m Z
C
gtlt o ~ o
o ~
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
w
II o
== p
e
pbullo
p
=
iii a
O w _oC
p o
o W o
p w
o
III
I ~
n
1
m Z
~
us
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
~iii~i~i~iiiiiiii~i~~~~iisectii~iiii
Isectii~~~~iiiiiii~iii~i~~iiiiiiii
ccocpcooo-oooooococcoooco
~111sect11sect11~~~m~~sectIsectsectI~111
iiiiiii
b po III o
po III
c
=p
bull
I
=i c
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
520 AENDIX APPENDIX 521
TABLE A3 (Continued)
TABLE Al (Continued) n y p 005 010 015 CUll 111amp 0]0 035 040 045
n y p =050 OSS 060 065 070 075 080 085 090 895 19 o I
03774
07s-t7
01351
0201
000156
01985
OUI~H
00829
00041
00310
0001 I
001001
00003
00031
00001
00008
00000
00002 17 o 00000
00001
00000 00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 2 3
09]35
09868 070s-t 08850
04413
06841
ODgt 00455
0111)
uL63 I
00462
01332
00170 00591
00055 00230
00015 00077
1 l
4
00012 0006-4
0025
0000]
00019
00086
00001 00005
00025
00000 00001
00006
00000 00000
00001
00000 00000
00000
00000
00000
00000
00000 4)0000 00000
00000 00000
00000
00000
00000 00000
4 5 6
09980
09998
10000
096018
0991 09983
08556
09163
09837
OlID OBJf~1
09TH
0151
O6LnJ
ufll~1
02821
04739
06655
01500
02968
OA812
00696
01629
03081
00280
00777
01727 5 6
00717
01662
00301
00826
00106
003amp
00030
00120
00007
00032
00001
00006
00000
00001
00000
00000 00000 00000
00000
00000 7 8
10000
10000
09997
10000
09959
09992
090 09933
U9JJS
a971~
08180
09161
06656
081-15
0878
06675
03169 0940
7 8
9
0]15 05000
06855
01834 0ll74
05257
00919
01989
03595
00383
00994
02128
00127
00103
010016
00031
00121
00402
00005 00026
00109
00000 00003
00017
00000 00000
00001
00000
00000 00000
9 10 II
10000middot 10000
10000
10000 10000
10000
09999
10000
10000
O99ii-
099)
10000
u~-JI
u)17 (J~0l$
0967-1
09895
09972
09125
09653
09886
08139
09115
096-18
06710
08159
09129 10
II 0811amp
09283
07098
08529
05522
07l61
03812
05803
0l118 040]2
01071
Ql347
00377
01057
00083 00]19
00008 000017
00000 00001
12
13
10000 10000
10000
10000
10000
10000
10000 10000
u99iJ
10000 09991
09999
09969 09993
09881 09969
09658
09891 12 09755 094001 08740 07652 06113 0261 02 18 00981 00221 00012 14 10000 10000 10000 10000 LUOOO 10000 09999 09994 09972 11 099]6 09816 09536 O89n 07981 06470 04511 02 008l6 0008amp 15 10000 10000 10000 LOUilh LOOOO 10000 10000 09999 09995 14 09988 09959 09877 09673 09226 08363 069001 04802 02382 00503 16 10000 10000 10000 10000 Louno 10000 10000 10000 09999 15 09999 09994 09979 09933 09807 099 08818 01475 05182 02078 17 10000 10000 10000 1000l IUllOO 10000 10000 10000 10000 16 10000 10000 098 09993 09977 09925 09775 Q9369 083]2 05819 18 10000 10000 10000 100Di UUOO 10000 10000 10000 10000 17 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 19 10000 10000 10000 LOOllO 10000 10000 10000 10000 10000
18 o
I
00000
00001
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000
00000
00000 20 o
I 03SS5
07358
01216
0]917
00381
01756
0011
006Y
OUon 00243
00008
00076
00002
00021
00000
00005
00000
00001 2
3
4
5
00007
0003amp
00154
000181
00001
00010
000019 001amp1
00000 00002
0001l 00058
00000 00000 0000] 00014
00000
00000
00000
00003
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000
00000 00000
00000
00000
00000 00000 00000
2 3 4 5
0925 01
0997 09997
06769
08670
09568
09887
0A049
06177 08299
09327
0201
poundIAII-
OQ2~~
080~2
IJ0913 01251 (JHl
06172
00355
01071
02375
OAI64
00121
00444
01182
OHs-t
00036
00160
00510
01256
00009
00049
00189
00553 6 01189 00517 00203 0G062 00011 00002 00000 00000 00000 00000 6 10000 09916 09781 09135 07tl5fJ 06080 001166 02500 01299 7 0203 01280 00576 00212 00061 00012 00002 00000 00000 00000 7 10000 09996 099-11 U9611 OIl981 07723 06010 0 59 02520 8 007l 02527 01347 00597 00210 00054 00009 00001 00000 00000 8 10000 09999 09997 09900 091 08867 0762-1 05956 0-11-13 9 05927 0222 02632 01391 00596 00193 00001] 00005 00000 00000 9 10000 10000 09998 0)9ii u9il61 09520 08782 07553 0591
10 07597 06085 001366 02717 01407 00569 00163 00027 00002 00000 10 10000 10000 10000 0911- 091 09829 0968 08n5 07507
12
08811
09519
0772 0amp923
06257
07912
04509
06450
027amp3
01656
01390
02825
00513
01329
00118
000119
00012
00064 00000
00002 II 12
10000 10000
10000 10000
10000
10000
0)999
10000
U9911
09190
099-19
09987
098001
099-10
09-135
09790
08692
020 11 09 6 09589 09058 0amp114 06673 01813 028]6 01206 00282 00015 13 10000 10000 10000 10000 10000 09997 09985 09935 09786 14 09962 09amp80 O96n 09217 08351 Q63 04990 02798 00982 00109 14 10000 10000 10000 1000il 10000 10000 09997 09 09936 15 09993 09975 09918 0976-4 09iOO 08647 0n87 0520] 02662 005amp1 15 10000 10000 10000 10000 10000 10000 10000 09997 09985 16 09999 09997 09987 09951 09858 09605 09009 07759 05497 02265 16 10000 10000 10000 10000 10000 10000 10000 10000 09997 17 10000 10000 099 09996 099 09 09820 064 0 99 06028 17 10000 10000 10000 10000 LiJOOO 10000 10000 10000 10000 18 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 18 10000 10000 10000 IUOOO IOOllO 10000 10000 10000 10000
19 10000 10000 10000 10000 10000 10000 10000 10000 10000
20 10000 10000 10000 10000 10000 10000 10000 10000 10000
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
101---------- 101 Do w shy Do W - 101
~~~~~~~~ee~~eg~~pp~osectsect8~~~ ~==-~ ~J~-88oo8 -www-~ ~wa8
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
TABLE A7 Quantiles of the Mann-Whitney Test Statistic 12 13 I~ IS 16 17 18 19 20
9 10 II =2 5 n 3 3 33 3 3 3
3 3 3 3 3 3 3 3 3
3 3 30001 3 3
3 3 3 3 3 3 3 3 3 3 4 4 0005 3 3 3 3 4 4 4 4 4 4 5 5
3 3 3 3 3 3 3 001 3 3 3 5 5 5 5 6 6 6 6
3 3 3 4 4 4 5 5 olIlS 3 3 3
5 5 6 6 7 7 7 7 8 8 8 4 4 -4 5 5
0115 3 3 3 7 7 8 8 8 9 10 10 II II 4 5 5 5 6 64
6 6 6 6 6 7 7 7 7 6 6 6 6 6 6 6 6 010 3
110111 6 6 8 8 8 9 9 9 10 10 6 6 6 6 7 7 7
00115 6 6 6 8 9 9 9 10 10 II II II 11 6 6 7 7 a 8
6 6 6 14 15 6 6 7 B 8 9 9 IS
001 10 10 II II 12 12 13 13 14 oOlS 6 14 14 15 16 16 17 ODS 6 7 7 8 9 9 10 II II 12 12 13
20 21 2215 16 17 17 18 199 10 II 12 12 II 14
0111 7 8 8 14 14 1412 12 12 13 1310 10 10 10 10 II II II
11001 10 10 10 14 15 16 16 17 17 18 19 10 10 10 II II 12 12 13 13 14
11oODS 10 14 15 16 16 17 18 18 19 10 20
II 12 12 13 141101 10 10 10
17 18 19 20 21 II 22 23 24 25 12 13 14 15 15 16
111115 10 10 II 27 28 2920 21 22 23 lS 26 12 13 14 15 16 17 18
005 10 II 31 32 3324 26 27 111 2915 16 17 18 20 21 II 23
010 II 12 14 22 23 2319 19 20 21 2115 15 15 1( 17 17 18 18
0001 15 15 15 25 26 17 111 2920 21 II 23 23 14 0005 15 15 15 16 17 17 Ie
23 24 25 26 27 18 29 30 31 32 15 16 17 18 19 20 21 II
001 15 33 31 35 3627 111 19 30 315 18 19 21 II 23 24 lS0015 15 16 17 38 39 41
lS 27 111 29 31 32 31 35 36 17 18 20 21 II 24005 16 43 41 46
29 31 33 31 36 38 39 41 21 23 24 26 1817 18 20 33 34
21 21 21 21 21 23 24 lS 40 29 30 31 320 36 26 27 111
0001 21 H 35 37 38 3929 31 II 3336 27 11121 II 23 24 lS0005 21 38 ~o 41 41 4133 31 35 3724 lS 26 28 19 30 31
001 21 21 23 36 38 39 41 43 44 46 47 49 24 lS 27 18 30 32 33 35
0015 21 23 50 52 5441 43 45 47 48 27 29 30 32 31 36 38 39
ODS II 24 25 45 47 49 51 53 56 58 6039 41 4329 31 33 35 37
35 36 37 38 39 olD ~2 43 41 45010 23 lS 27
111 111 18 29 30 31 32 310001 18 41 42 44 4t- 47 ~8 50 51 53
19 30 32 33 35 36 38 390005 111 18 43 45 46 18 50 51 53 55 57
30 II 33 35 36 38 40 411101 111 29 59 61 6349 51 53 55 5741 43 45 4734 35 37 390015 18 30 II 66 6853 55 57 59 52 6446 18 5035 37 40 42 44005 29 31 33 65 67 70 n 75
50 52 55 57 60 62 30 33 35 37 40 41 45 47
10 52 54 55 57 5845 46 18 49 51 IIMH 36 36 52 54 55 57 59 61 63 65 6736 37 38 39 41 42 43
bullbull105 36 36 67 69 7138 29 oil 43 44 46 4B 50
oil 43 44 46 ole 50 52 54 56 5 61 63 65 01 36 37 39
52 54 56 59 61 63 66 68 71 73 75 78 43 45 47 SO0115 37 39 41 70 73 76 78 81 8457 60 63 loS 6845 47 50 52 55115 38 40 41 79 82 85 88 91(1 64 67 70 73 76
44 47 50 53 59011 39 42 ~
TABLE A7 (Continued)
II 5 6 7 9 10 II 12 11 14 15 16 17 18 If 10
0001 45 45 ~5 47 18 49 51 53 51 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 ~9 51 53 55 57 59 52 61 66 68 70 73 75 77 79 B2 ocil 15 47 49 51 53 55 57 60 52 64 67 69 72 7 77 79 92 84 86 0015 46 48 50 53 56 58 61 63 6( 69 72 74 77 BO 83 B5 Ba 91 94 005 010 0001
47 48 5S
SO 51 55
52 55 56
55 58 57
58 61
59
61 64
61
64 68 52
67 71 64
70 74 6(
73 77
68
76 81
70
79 84
73
82 87 75
85
9 77
88 94
79
91 98
BI
94 101 83
97 104 B5
100 lOB
B8 0005 55 56 SB 60 62 loS 67 69 72 74 77 80 82 85 B7 90 93 95 98
10 001 0015
55 56
57 59
59 61
52 64
64 (7
67 70
69 73
72 76
75 79
7B 82
80 85
83 89
86 91
89 95
92 98
9~ 101
97 104
100 lOB
103 III
005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III II~ liB 0 0001
59
66
62
66
66 67
69
69
73
71
77
73
80
7S
84
77
88 79
92
82
95
84
99 87
103 89
107
91 110 114
96
liB 99
122 101
126 101
0005 66 67 69 72 74 77 BO 83 85 8B 91 94 97 100 103 106 109 112 115
II 001 0025 005
66 67 68
68 70 72
71 73 75
74 76 79
76 80 83
79 83 86
82 86 90
85 90
89 93 98
91 97
101
95 100 105
9B 101 109
101 107 113
101 III 117
108 114 121
III liB 124
114 122 128
117 125 132
120 129 136
010 70 74 78 82 86 90 94 9B 103 107 III liS 119 124 128 132 136 HO 145
0001 78 79 79 81 e3 96 99 91 93 98 102 104 10 110 113 116 118 121 0005 78 80 81 95 88 91 94 97 100 103 106 110 113 116 120 123 116 130 133
I 001 001
78 90
91 93
84 86
97 90
90 93
93 97
96 101
100 105
103 lOB
107 112
110 r 16
114 120
117 124
121 126
125 132
128 136
132 1middot10
135 1--14
139 148
00 81 84 88 91 96 100 105 109 III 117 121 116 130 13~ 139 1J3 147 151 156 CW 83 S7 91 56 100 IDS 109 114 118 123 128 132 137 142 46 I~l 156 160 165
0031 91 ~I 93 95 97 100 103 106 109 112 115 liB 121 124 127 130 IH 137 140 0005 91 93 95 9 102 105 109 112 II 119 m 126 130 134 137 1lt1 1middot~5 149 152
I 001 Q015
92 93
94 96
97 100
101 104
104 108
108 III
III 116
115 120
119 125
123 129
117 133
131 137
135 142
139 146
143 lSI
147 ISS
lSI 159
ISS 164
159 168
005 9~ 98 102 107 III 116 110 IlS 129 134 139 143 149 153 157 162 67 172 176 010 96 101 105 110 115 120 125 130 135 140 145 150 ISS 160 166 171 176 181 IB6
00111 105 105 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 124 128 132 136 140 144 148 152 156 160 1M 19 173 001 106 108 112 116 119 123 1111 132 136 140 144 149 153 157 162 166 171 175 179 0015 107 III 115 119 123 1111 132 137 142 146 151 156 161 165 170 175 IBO 184 199 005 109 113 117 III 127 m 137 142 147 I5l 157 162 167 In 177 183 IB8 193 198 010 110 Jl6 121 126 131 137 112 147 153 158 164 169 175 180 186 191 197 203 208
0001 120 120 III 125 12B 133 135 138 142 145 149 153 157 161 164 16B 172 176 190 0005 120 123 126 129 133 137 141 1~5 ISO 154 158 163 167 172 176 181 185 190 191
15 001 0015
121 III
114 126
128 131
132 135
136 140
140 145
145 150
149 155
154 160
158 IloS
163 170
168 175
172 180
177 185
182 191
187 196
191 201
196 206
201 211
005 114 1111 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 115 221 010 126 131 137 143 148 154 160 16( In 178 184 189 195 201 207 213 219 225 231
0001 136 136 139 112 115 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO ISS 159 164 168 173 178 182 187 192 197 202 207 211 216
16 001 0015
137 13e
140 143
144 118
149 152
153 158
158 163
163 168
168 174
173 179
178 184
183 190
188 196
193 201
198 207
203 212
208 218
213 223
219 229
224 235
005 olD 115 151 156 162 167 173 179 185 191 197 202 208 214 220 ll6 232 238 241 010 142 148 151 160 166 173 179 185 191 198 ~ 211 217 113 230 236 243 2~9 256
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
20
TABLEA7 (Continued)
2 J 4 5 7 9 II 12 13 4 IS 16 17 18bull 0001 oW 154 156 159 163 161 111 175 179 183 188 192 197 201 206 211 215 0220 224 0l1li1 153 156 160 164 169 173 178 III 188 193 198 203 208 21~ 219 22~ 229 235 2~0
17 DOI DoB
154 156
158 160
162 165
167 171
172 176
171 182
182 188
187 193
192 199
198 205
203 211
209 217
21~ 223
220 229
225 235
231 2~1
236 2~7
2~2 253
2~7 259
LIS0
157 160
163 166
169 172
17~ 179
180 185
187 192
193 199
199 lD6
205 212
211 219
218 226
22~ 233
231 239
237 24
2~3 253
250 260
256 261
263 27~
269 281
8l1li1 171 172 175 178 182 186 110 195 199 lD4 209 214 218 223 128 233 238 243 218 801 171 174 178 183 188 193 198 203 209 214 219 225 230 236 2~2 2~7 253 259 264
18 DOI DoB
172 174
176 179
181 184
186 110
191 116
116 2D2
202 208
208 214
213 220
219 227
225 233
231 239
237 216
242 252
H8 158
25~ 265
l6D 271
266 278
272 284
8os 176 181 188 194 lOll 1Ii1 213 230 227 233 240 247 254 260 267 274 281 288 295 018 178 185 192 199 lD6 213 230 227 234 241 249 256 263 270 278 28S 292 300 307 8MI 110 191 194 198 202 lD6 211 216 220 225 231 236 211 24 251 257 262 268 273 DD05 191 194 198 203 208 213 219 224 230 236 242 248 254 l6D 265 272 278 284 290
It DoI DD25
192 193
195 198
lOll lD4
206 210
211 216
211 223
223 229
229 236
23S 243
211 249
247 256
254 263
l6D 269
266 216
273 283
279 290
lB5 297
292 304
298 310
DoS 195 201 208 214 221 128 235 H2 24 256 263 271 278 lB5 292 300 307 31~ 321 DID 198 205 212 219 227 ~ 242 249 257 264 272 280 288 295 303 311 319 326 334 OMI 210 211 214 218 223 227 232 237 243 248 253 259 265 270 276 281 287 293 299 0l1li1 211 214 219 224 129 235 2~1 247 253 259 265 271 278 284 210 297 3D) 310 316
lD GDI DOB
1I2 213
216 219
221 225
227 231
233 238
239 245
2~S
251 251 259
258 266
264 273
271 280
278 287
284 294
291 301
298 309
304 ll6
311 323
318 330
325 338
005 010
215 218
222 226
229 233
236 HI
H1 249
250 257
158 265
265 273
273 281
280 189
288 297
295 305
301 III
III 111
318 330
326 338
l34 3~6
HI l5~
H 362
For norm grater than 20 the fIth quantile w of the Mam-WhiDley test statistic may be approximated by
w - n(N + 1)2 + Zvnm(N + 1)112
where Z Is the fIth quIIIltile of a standard nonnaI random variable obtained from Table AI and where N ~ m + IL
bull The entries In chiJ table quandies w of the Mam-Whitney test scatlstic T pen by Equation 511 for selected values of p Note that PIT lt w) S gt Upper quail-dies may be found from tha equation
w = n(n + m +1) - w
Critical regions correspond to values of T less than (or zruter than) but not equal to tha appropriate quantile
~________~_____bull - _ ____ _ _ ___ 0
middoti~i-ImiddotIR~ A
I f III -r if 13 3
ii Qt
i ~ I) tlS l
i ft 9090e fbull f~I Q
iA ~ 9shy r ~ ~ ~
SshyC
~~ ~ Ii ~ r
1 ~ i
SshyF iD
9 lt gt 3 L
Ii ~
lr ~ 0 3
~
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
APPENDIX s542 APPENDIX
TABLE AIO Quantiles of Spearmans p
n =0900 0950 0975 0990 0995 0999
4 08000 08000 5 07000 08000 09000 09000
6 06000 07714 08286 08857 09429
7 8
05357 05000
06786 06190
07500 07143
08571 08095
08929 08571
09643 09286
9 04667 05833 06833 07667 08167 09000
10 04424 05515 06364 07333 07818 08667
II 04182 05273 06091 07000 07455 08364
11 03986 04965 05804 06713 07203 08111
13 03791 04780 05549 06429 06978 07857 ~14 03626 04593 05341 06220 06747 07670
15 03500 04429 05179 06000 06500 07464
16 03382 04265 05000 05794 06324 0n65
17 03260 O4IIB 04853 05637 06152 07083
18 03148 03994 04696 05480 05975 06904
19 03070 03895 Qof579 05333 05825 06737
10 02977 03789 04451 05203 05684 06586
21 02909 03688 04351 05078 05545 06455
21 02829 03597 04241 04963 05426 06318
2l 02767 03518 04150 04852 05306 06186
24 02704 03435 04061 04748 05200 06070
25 02646 03362 03977 04654 05100 05962
16 02588 03299 03894 04564 05002 05856
27 02540 03236 03822 04481 04915 05757
18 02490 03175 03749 04401 04828 05660
19 02443 03113 03685 04320 04744 05567
lO 02400 03059 03620 04251 04665 05479
For n greater than 30 the approximate quantlles of p may be obtained from
Zwmiddot Vn _ 1
where z I the pth quantile of a standard normal random variable obtained from Table AI Souaa Adapted from Glasser and Winter (1161) with corrections with permlulon from the Blometrikd Trustees bull The entries In this table are selected quantUe w of the Spearman rank correlation coefficient p when used as a test stadstlc The lower quantlles may be obtalned from the equadon
w~ = WI_
The crltkal region corresponds to values of p smaller than (or greater than) but not including the approshypriate quantile Note that he median of p Is O
TABLE AI I Quantiles of the Kendall test statistic T = Nc - Nbullbull Quantiles of Kendalls 7 are given in parentheses Lower quantiles are the negative of the upper quantiles wp = -WI_p
n = 0900 0950 0975 0990 0995
4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000) 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 7 9 (04286) II (05238) 13 (06190) 15 (07143) 17 (0B095) 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)
10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) II 17 (03091) 21 (03818) 25 (04545) 29 (05273) 31 (05636) 11 18 (02n7) 24 (03636) 28 (04242) 34 (05152) 36 (05455) Il 22 (02821) 26 (03333) 32 (04103) 38 (048n) 42 (05285) 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857)
16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 17 32 (02353) 40 (02941) 48 (03529) 56 (04118) 62 (04559) 18 35 (02288) 43 (02810) 51 (03333) 61 (03987) 67 (04379) 19 37 (02164) 47 (02749) 55 (03216) 65 (03801) 73 (04269) 20 40 (02105) 50 (02632) 60 (03158) 70 (03684) 78 (04105)
21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600)
26 59 (01815) 75 (0230B) 89 (02738) 105 (03231) fl5 (03538) 17 61 (01738) 79 (02251) 93 (02650) III (03162) 123 (03504) 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386) 19 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) 30 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287)
II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204) II 80 (0613) 102 (02056) 120 (02419) 142 (02863) 158 (03185) l3 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) 34 87 (01551) III (01979) 131 (02335) ISS 02763) 173 (03084) 35 91 (01529) 115 (01933) 137 (02303) 163 (02739) 179 (03008)
36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) 37 98 (01-471) 126 (01892) ISO (02252) 176 (02643) 198 (02943)
I j i I
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
II
1
i APPENDIX
rABLE All (Continued)
or n_ dian 60 ~ quuKIIa 01 T IMY abaInu tom
jnln - I)(ln + 5)w Z 18
APPENDIX 545
TABLE A 11 QlWItile of the WilcOlmn ~iiiJI ~iBJli6l (it Statistic
(II + I) W WUI Wuu WOft5 lJilu -utu WOll) WI WUI
=lt1 o 5 o 6 o 7 o
OtoO 0910 0175 00 0995
183 (02603) 203 (02888) 8 I 191 (02578) 211 (028lt18)
2198 (02538) 220 (02821)
10 1206 (02512) 228 (02780)
6211 (02lt171) 235 (02729) III (0lff7) 215 (02713) 12 8 228 (02lt110) 252 (0266lt1) 11 10 236 (02383) 262 (026lt16) 14 11 215 (02367) 271 (02618) 15 16 151 (02310) 279 (02581)
16 20260 (02305) 288 (02551) 17 24268 (02279) 296 (02517)
277 (02261) lOS (02190) 18 28 28S (02235) 315 (02lt171) 19 II 291 (02217) 321 (023) 10 38 102 (02192) ]34 (02124) 11 111 (02173) 313 (02397)
22 19319 (02118) 353 (02377) 2l 55128 (02130) 362 (02151) 21 62336 (02105) 3n (02m)
315 (02087) 181 (02305) 15 69 155 (02075) ]91 (02285) 26 76 16lt1 (02056) 402 (02271) 27 81
18 92 29 101 30 110 31 119
tw tom rha nandltd normal dlllCllludaft aNM by Tabla 1 ~ra qWUldIu of IMY II 129
lnadfrom 31 119
w-~ 5 l lI(n - I) J4 15
149 160
rtclcal raatons corrupond to vaJuu 01 T1_dian (or Ius dian) IIuI Me IIdIIdInJ rha Ippropnara 36 172 iUIIldl Nora chac rha mulan 01 T II O Qulndlu for are obtained by dhlldlnl the qlWltllu of T by (II shy 1)12
n 38
181 196
QUIICI Adapcad tom Tabla I hit (1971) wkh parmIoIon from the author It lOB 10 221 41 235 11 218
0 o o j 3 1 5 0 o -~ S 6 75 0 3 - v 9 9 105
3 4 - II 12 11 2 1 6 I 14 16 18 4 6 ) Ii i 18 20 225 6 9 II IS IJ n 25 275 8 I 14 12 J 27 30 13
10 11 Ie 1 21 32 36 39 11 18 21 j 311 42 455
16 22 26 J -J~I 44 48 525 20 26 31 jT ~~_I 51 5S 60 2lt1 30 )( j Si 58 63 68 28 35 42 -~I ~H 65 71 765 11 11 48 ~ 73 80 855 38 17 5lt1 d 1- 82 09 95 53 61middot 1) L 91 98 105 50 59 68 Ie toO 108 1155 56 67 16 t- IOU 110 119 1265 63 71 04 -)$ 110 10 30 138 70 82 91 lOS Ilu 31 1lt11 ISO 77 90 101 II-l 3i 143 IS) 1625 85 9 II t I~ 1- ISS 165 1755 1lt1 108 120 135 ~-l 167 178 189
102 117 131 1- h~ 100 192 103 III 127 141 Isa IIIl 193 206 2175 121 138 152 110 I~I 207 220 2325 131 118 161 un 205 221 235 248 111 160 m 1 Ilt) 13amp 250 261 152 171 188 108 3j lSI 266 2805 161 183 201 in -iiJ 266 282 2975 175 196 214 1Jl 20) 293 299 liS 187 209 228 251 hI 299 317 333 199 m 242 2gt 195 316 335 3515 212 236 257 lin 3 I 334 353 3705 ns 250 272 198 31lt) 352 372 390 239 265 287 31--1 341 371 391 lt110 253 280 30] m liS 390 111 lt1305 267 295 320 349 3pound1-1 -109 131 4515
10 15 21 28 36 45 55 66 78 91
105 120 136 153 171 190 210 231 2S) 276 300 125 351 378 106 lt135 165
4 528 561 595 630 666 703 711 780 820 861 903
~ w m II ~ B ~ ~
~ ~ ~ W ~i n ~ ~ ~
~ ~ ~ ~ ~
103 (01-465) 107 (011) 110 (01 3n) 11lt1 (01390) 119 (01382) 123 (01362) 128 (01353) III (olm) IlS (OllOf) 111 (0130-4) I (01277) ISO (01276) 153 (01219) 159 (01217) 162 (01222) 168 (01219) 173 (01209) In (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (01111)
Ill (01861) 117 (01819) 1lt12 (01821) 6 (01780) 151 (017Sf) 157 (01739) 62 (01712) 168 (01 697) 173 (01671) 179 (01656) 186 (016lt19) 190 (01616) 197 (01608) 203 (01592) 208 (0156 2 (01553) 221 (015 ) ll7 (01529) 232 (01506) 2040 (0150-4) 2lt15 (01182) 251 (01167) 258 (01lt158)
ISS (0ll05) 161 (02173) 168 (02151) 17lt1 (02Ill) 181 (02101) 187 (02071) 191 (02051) 200 (02020) 207 (02000) 213 (01970) 210 (01950) 228 (01939) 2ll (01902) 211 (01890) 218 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792)
281 (01779) 291 (01760) 299 (01718) 306 (01729)
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
5-4 APPENDIX If APPENDIX 5-47
TABLE All (Continued) TABLE All Quantile of the Kolmogoi T Si1itistk-n(n + I) OnemiddotSlded Tat
Igt w W bullbullbull w WI w Wue Wuo W WU 2 = 090 095 0975 099 09 P =090 D95 0975 099 0995
Two-Sided TelC 41 163 282 311 ]]7 366 403 29 452 473 946 = 080 090 095 098 U9J = 080 090 095 098 099 4 277 297 328 3S4 385 4ll 50 473 495 990
n I 0900 0950 0975 0990 O)J~ 21 0226 0259 0287 0321 0345 291 31l 3044 371 40l 2 471 495 5175 1035 1 061H 0776 0H2 0900 0929 ~ 0221 0253 0281 031 0337l08 329 361 390 41) 463 517 5405 10814 2 3 0565 0636 0708 0785 0029 ~ 0216 0247 0275 0307 0330
47 32 3046 379 08 2 H 514 SiO 5604 1128 4 093 0565 0624 0689 0134 i 0212 0242 0269 0301 0323 8 liO l6l 397 28 463 505 536 563 SS8 1176 5 OM7 0509 0563 0627 066~i 2 0208 0238 0264 0295 031749 357 381 416 7 483 527 559 587 6125 1225 0410 068 0519 0577 O6Ijmiddot [ 0204 0233 0259 0290 01(1 50 37 398 35 67 so 550 583 611 6375 1275 7 0381 036 OA83 0518 OS j 0200 0229 02504 0284 0305
8 0358 MID 04504 0507 05middotl ttl 0117 0225 0250 0279 0300 For n laJr than SO dI Ith quantll w or dI Wilcoxon lipad ranks _ ltadldc rnay b approldmatad by w - [n(n + I)H] + 9 0339 0387 OA30 0180 051j ly 0193 0221 0246 0275 0295 rvn(n + I)(ln + 1)1204 whara z Is da Idl quanaJ of a IWIdard normal random _1aII1 obtlllnad from Table AI 10 0323 0369 0409 OA57 oAIl~ ~t 0190 021B 0242 0270 0190 SoIJllCl Adaptod from Hamr lind Owen (170) wkh parrnlulon from da Amorlan Mathematical Society II 03OB 0152 0191 OA37 OA6J 1 0187 0214 0ll9 0266 0285 Th anrrtu In dill abla IUII qwmdIu W of dI Wilcoxon d nnka cut natlRlc T IIWn by Equation 573 101 bullbullcted wi- II 0296 0338 0375 0419 O44i J 0184 D211 0234 0262 02BI u of s Oso QlWltllbullbull w rar I gt oSO may b computed from dI llqUadan 3 0285 0325 0361 0404 OAn ~ 0182 0208 0231 0258 0217
1-4 0275 011 0349 0390 0418 Jj 0179 0205 0227 0254 0273w ~ 11(11 +1)11- WI_ t15 0266 0304 0318 0177 OA04 0177 0202 0224 0151 0269J
whr n(n + 1)12 II ampWan In the amphE hand column In dI ab Note diu JIr lt w) I lind JIr gt w ) s I - II H Is 16 Dl5B 0295 0317 0366 0392 ~ 0174 0199 0221 0147 0265 true Critical loIIs corrupand to vatuu of T leu dian (or Irauer than) but not Inctudlnl the apprltgtpriara qlWldIa 17 Dl50 02B6 0118 0355 0381 1 0172 0196 021B 02 0262
18 02 0279 0309 0346middot 0371 3ltj 0170 0194 0215 0141 0258 19 0ll7 0271 OlOI 0337 0361 J~ 0169 0191 021l 0218 0255 20 0232 0265 0294 Oll 03SL ~~o 0165 0189 0210 0ll5 0252
Applo)(lmailu 107 122 136 152 163 fol n gt 40 Yo Yo Yo Yo Yo
SoUAeE Adapaod from Tabla 1of Miller (1956) Ued with permission or the American Statistical Auadadon Th anulalln rhIs cabII_d quantll w of the Kolmogorov rest Statistics T P and T- as donned by Equation 611101 cwo-llded tutI lind by Equadonl 612 and 613 for one-sided t Ieject H at the Ilvol If Teceed the I - qIWItUe In rhIs tibia Tha qlWldlu IUII enct for n S 40 in the cwo-tailed test The other quantUe are approximations dlat ara aquaI torhe lUa quantll In IIIOIt cuobullbull A blttlr approximation fal 11 gt -10 results If (n + YniiO Is used Instead of Yn In dI denominator
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
556 APPENDIX APPENDIX
TABLE A10 Quantiles of the Smit-nov Test Statistic for Two Samples of Different Size II and m
TABLE AI9 Quantlles of the Smirnov Test Statistic for Two Samples of Equal Size II
One-sided Test One-Sided Test = 090 095 0975 099 0995 = 090 095 0975 099 0995 Two-Sided Test Two-Sided Test = 080 090 095 098 099 = 080 090 095 098 099
One-Sided T e5t Two-SIded Test
HI = 1 Ht = 9 10
HI == 2 Ht == 3
p =090 =080
1718 9110 56 34
095 090
0975 095
099 099
0995 099
11=3 213 213 n = 22 7122 8122 8122 10122 10122 5 45 115
bull 4 5 6 7
311 35 36 417
34 35 416 417
311 415 416 517
415 56 517
415 56 57
23 24 25 26
7123 7124 7125 726
8123 8124 8125 8126
9123 9124 9125 9126
10123 10124 1025 10126
1023 11124 11125 11126
6 7 8 9
10
56 517 314 719 710
56 617 78 819 45
78 819 9110
8 48 48 58 518 618 27 7127 8127 9127 11127 11127 9 49 519 519 619 619 28 8128 9128 10128 11128 12128
10 4110 5110 6110 6110 7110 29 8129 9129 10129 11129 12129 II 5111 5111 6111 7111 711 30 8130 930 10130 1130 12130 12 5112 512 6112 712 7112 31 8131 931 1031 11131 12131 Il 5113 6113 613 713 8113 32 832 932 1032 12132 12132
14 5114 6114 714 7114 8114 3l 8133 9133 1133 1233 131ll 15 515 6115 7115 8115 8115 34 8131 1034 11134 12134 1334 16 6116 6116 716 8116 916 35 8135 1035 11135 12135 1335 17 617 717 717 8117 9117 36 9136 1036 11136 12136 1336 18 618 718 8118 918 9118 37 9137 1037 1137 1337 1337 19 6119 719 8119 919 9119 38 9138 1038 1138 1338 1438 20 6120 7120 8120 9120 10120 39 939 1039 1139 1339 11139 21 6121 7121 8121 9121 10121 40 9140 10140 12140 1340 14140
Approximation 152 173 192 215 230 for n gt 40 Yo Yo Yo Yo Yo
SOURCE Adapted from Blmbaum and Hall (l960l with permission from the Institute of Mathematlcal Statistics
bull Th entries In this table aruelected quantles w of the Smlrnoy two-sample teSt statistic T defined II) Equations 632 and 633 for th one-tailed test and defined by Equation 631 for the cwo-taIled tesc Reject Ha at the level a If T exceeds the I - a quanshytil of T as glyen In this table The telt statistic II a discrete random variable so the exact level of Ilgnificance may be less than the apparent tI uId In thll table
HI = 3
H =4
HI =5
H =6
Ht == 4 5 6 7 8 9
10 12
Ht = 5 6 7 8 9
10 12 16
Ht = 6 7 8 9
10 15 20
Nt = 7 8 9
10 12 18
34 213 213 213 518 213 315 712 35 7112
1728 518 519
11120 7112 916 35 417
11120 519 112 8115 112
23142 112 12 12 112 119
34 45 213 517 311 213 7110 213 311 213 517 518 213
13120 213 58 213
2335 518 35 35 35
11120 417 7112 519
1730 7112 519
45 56 617 34 719 45 314 45 311 311 314 314 710 213
11116 213 517
27140 31145 710 213 35
29142 213 213
1930 7112
11118
617 718 819 9110 56 415 516 617 78 79 15 311 34 56
2935 415 719 710
1115 710 517 34
Illl8 710 213 213
819 910
11112
56 617 78 819 45 56
13116 56 617 45 415 45
11115 34 56 34 719
11115 ll1
13118 24 11124 112 7112 58 213
~)
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
APPENDIX APPENDIX 559
TABLE A21 The t Distribution
Degrees of Freedom p = 06 075 09 095 0975 099 0995 09975 0999 09995
0325 1000 3078 6314 11706 31821 63657 12732 31831 63662 2 0289 0816 IB86 2120 4303 6965 9925 14089 22327 11598ILE A20 (Continued)
0277 0765 1638 2353 3182 4541 5841 7453 10214 12924
ISlded Test P 090 095 0915 099 0995 4 0271 07~1 1533 2132 2776 3747 4604 5598 7173 8610 090 095 099 099)middotSided Test P = 080 5 0267 0727 IA76 2015 2571 3365 4032 4773 5893 6869
6 0265 071B 1440 1943 M47 3143 3707 4317 5208 59591 N2 = 8 2756 3356 58 41156 314
7 0263 0711 1415 1895 2365 2998 3499 4029 4785 54089 31163 519 4063 517 47163 1860 8 0262 0706 1397 2306 2896 3355 3833 450 I 504110 33170 39170 4300 7110 517
911 517 9 0261 0703 1383 1833 2262 2821 3250 3690 4297 478114 30 112 417 28 317 13128 15128 1728 9114 10 0260 0700 1372 1812 2228 2764 3169 3581 4IH 4587
8 N2 =9 49 13124 58 213 34 II 0260 0697 1363 1796 2201 2718 3106 3497 4025 4437 10 1940 21140 2340 2740 7110 11 0259 0695 1356 1782 2179 2681 3055 3428 3930 4318
12 11124 12 7112 518 213 13 0259 069~ 1350 1771 2160 2650 3012 3372 3852 4221 16 7116 112 9116 5111 58
14 0258 0692 1345 1761 2145 2624 2977 3326 3787 4140 32 1332 716 112 9116 1932
15 0258 0691 1341 1753 2131 2602 29~7 3286 3733 4073 9 N2 =10 7115 112 26145 213 31145
16 025B 0690 1377 1746 2120 2583 2921 3252 3686 401512 419 112 519 11118 213
17 0257 06B9 1333 1740 2110 2567 2898 3222 3646 396515 19145 22145 8115 35 29145 18 0257 06BB 1330 1734 2101 2552 2878 3197 3610 392218 7118 49 112 59 11118
19136 519 19 0257 06BB 1328 1729 2093 2539 2861 3174 3579 )88336 1336 512 1736 7115 112 1730 9130 10 0257 0687 1325 1725 2086 2528 28-45 3153 3552 385010 N2 =IS 215
20 215 9120 112 11120 35 11 0257 0686 1323 1721 2080 2518 2831 3135 3527 3819
40 7120 215 9120 112 21 0256 0686 1321 1717 2074 2508 2819 3119 3505 3792
12 Nt =15 2360 9120 112 11120 712 II 0256 0685 1319 1714 2069 2500 2807 3104 3485 3767 16 318 7116 23148 13124 712 14 0256 0685 1318 1711 2064 2492 2797 3091 3467 3745 18 1336 5112 1736 19136 519
25 0256 068~ 1316 1708 2060 2485 2787 3078 3450 3725 20 1130 512 7115 31160 1730
26 0256 068~ 1315 1706 2056 2479 2779 3067 3435 3707 15 N2 =20 7120 215 1330 29160 31160
27 0256 068~ 1314 1703 2052 2473 2771 3057 3421 3690 16 Nt =20 2780 31180 17140 19140 41180 28 0256 0683 1311 1701 2048 2467 2763 3047 3408 3674
29 0256 0683 131 I 1699 2045 2462 2756 3038 3396 3659 e sample
107Jm +11 122Jm+11 136Jm + 11 152Jm + n 163Jm+11 30 0256 0683 1310 1697 2042 2457 2750 3030 3385 3646Oldmadon mil mnmil mil mil 40 0255 0681 1303 1684 2021 2423 2704 2971 3307 3551
60 025-4 0679 1296 167 I 2000 2390 2660 2915 3232 3460e Adapted from Massey (1952) with permission from the Institute of Mathematial Slatildes entries In thl lable are selected quantile w of the Smlmov test statlldc r for two samples defined no 025-4 0677 1289 1658 1980 2358 2617 2860 3160 3)7)
lations 631632 and 633 To enter the table let N be the smaller sample size and lat N be the 0253 067~ 1282 16~5 1960 2326 2576 2801 3090 3291 ample 51e Reject H at the level Dt if r exceeds w_ as given In thl lable If n and m are not COyshy
by this lable use the large sample approximation given at the end of the table or consult exact tables SoURCE Reprinted from Vol I of Pearson and Hardey (I976) with permission from the llIometriko Trustees nand Jennrich which appear In Harter and Owen I 970) lOr n m s 100 bullThe entries in this table are quantlles w of tho t distribUtion for various degrees of freedom Quantiles w for p lt 05 may be
computed from the equation
Wp = -WI_II
Note that w 0 for all degre of freedom bull
bull lt~~~ bull
