UNIVERSITY OF SWAZILAND

SUPPLEMENTARY EXAMINATION PAPER 2015

TITLE OF PAPER

COURSE CODE

TIME ALLOWED

REQUIRMENTS

INSTRUCTIONS

NON-PARAMETRIC ANALYSIS

ST409

2 (TWO) HOURS

CALCULATOR

ANSWER ANY THREE (3) QUESTIONS ALL QUESTIONS CARRY EQUAL MARKS

THIS PAPER IS NOT TO BE OPENED UNTIL PERMISSION HAS BEEN GRANTED BY THE INVIGILATOR

Page I

ANSWER ANY THREE QUESTIONS

For all questions clearly state the name ofthe test the null amp alternate hypotheses the test statistics the decision rule the level ofsignificance the decision amp the conclusions

QUESTION ONE [ 12+8 marks]

a Total annual precipitation is recorded yearly for 19 years This record is examined to see if the amount of precipitation is tending to increase or decrease The precipitation in inches was 4525 4583 4177 3626 4537 5225 3537 5716 3537 5832 4105 3372 457337904172360749833624 and 3990 Indicate the appropriate conclusion with a = 001 and also calculate the P-value

b An item A is manufactured using certain process Item B serves the same function as A but

manufactured using a new process The manufacturer wishes to detennine whether B is preferred to A by the consumer so she selects a random sample consisting of 10 consumers gives each of them oneA and one B and asks them to use the items for some period of time At the end of the allotted period of time the consumers report their preferences to the manufacturer Eight consumers preferred B to A 1 preferred A to B and 1 reported no preference Use an appropriate test to fmd out the manufacturers decision at 5 level of significance Also calculate the P-value

QUESTION TWO [20 marks]

Twelve MBA graduates are studied to measure the strength of the relationship between their score on the GMA T which they took prior to entering graduate school and their grade point average while they were in the MBA program Their GMA T scores and their GPAs are given below

Student 1 2 3 4 5 6 7 8 9 10 11 12 GMAT 710 610 640 580 545 560 610 530 560 540 570 560 GPA 40 40 39 38 37 36 35 35 35 33 32 32

Use either Spearmans p test or Kendalls T to test whether GMAT scores and GPAs are positively correlated Use a = 005

Page I

ANSWER ANY THREE QUESTIONS

For all questions clearly state the name ofthe test the null amp alternate hypotheses the test statistics the decision rule the level ofsignificance the decision amp the conclusions

QUESTION ONE [ 12+8 marks]

a Total annual precipitation is recorded yearly for 19 years This record is examined to see if the amount of precipitation is tending to increase or decrease The precipitation in inches was 4525 4583 4177 3626 4537 5225 3537 5716 3537 5832 4105 3372 457337904172360749833624 and 3990 Indicate the appropriate conclusion with a = 001 and also calculate the P-value

b An item A is manufactured using certain process Item B serves the same function as A but

manufactured using a new process The manufacturer wishes to detennine whether B is preferred to A by the consumer so she selects a random sample consisting of 10 consumers gives each of them oneA and one B and asks them to use the items for some period of time At the end of the allotted period of time the consumers report their preferences to the manufacturer Eight consumers preferred B to A 1 preferred A to B and 1 reported no preference Use an appropriate test to fmd out the manufacturers decision at 5 level of significance Also calculate the P-value

QUESTION TWO [20 marks]

Twelve MBA graduates are studied to measure the strength of the relationship between their score on the GMA T which they took prior to entering graduate school and their grade point average while they were in the MBA program Their GMA T scores and their GPAs are given below

Student 1 2 3 4 5 6 7 8 9 10 11 12 GMAT 710 610 640 580 545 560 610 530 560 540 570 560 GPA 40 40 39 38 37 36 35 35 35 33 32 32

Use either Spearmans p test or Kendalls T to test whether GMAT scores and GPAs are positively correlated Use a = 005

Page 2

QUESTION THREE [20 marks]

Each of the three aerospace companies has randomly selected a group of technical staff worker to participate in a training conference sponsored by a supplier finn Three companies have sent 6 5 and 7 employees respectively At the beginning of the session a preliminary test is given and the scores are shown in the following table At the 005 level use the Kruskal-Wallis test to determine whether evidence exists to conclude that the median scores for the three populations of technical staff workers could be the same

Technical Staff from Firm 1 Firm 2 Firm 3

67 64 75 57 73 61 62 72 76 59 68 71 70 65 78 67 74

79

QUESTION FOUR [20 marks]

Two computer software packages are being considered for use in the inventory control department )fa small manufacturing firm The firm has selected 12 different computing tasks that are typical of

the kinds ofjobs such a package would have to perform and then recorded the number of seconds each package required to complete each task given in the following table Do the data present sufficient evidence to indicate that both softwares performed the jobs at the same time Analyse the data by using the Wilcoxon Signed Rank Test with (l =010

Computing Time Required for Software Task Packages X and Y

A B C D E F G H I J K L

240 167 216 237 375 314 149 373 179 155 290 199

231 204 177 207 421 361 218 403 260 155 354 255

TABLE AI Nonna Distribution

Seleaed vUueI ~ -37190 zshy - -32905 Zoms = -1600 Zui = -16449 z- 37190 z_ = 3205 zm == 19600 ztS = 1644 0000 0001 0001 0003 0004 0001 0006 0001 0008 0009

000 -30902 -287B2 -27478 -26521 -25758 -25121 -2457l -24089 -23656

001 -23263 -22904 -22571 -22262 -21973 -21701 -21444 -2llDl -20969 -20749

001 -20537 -20335 -20141 -19954 -19n4 -19600 -19431 -19268 -19110 -IB957

OOl -18B08 -18663 -18522 -18384 -18250 -18119 -17991 -17866 -1n44 -17624

0D4 -17507 -17392 -17179 -17169 -17060 -16954 -1684 -16747 -16646 -16546

005 -1644 -16352 -16258 -16164 -16072 -15982 -15893 -15805 -157IB -15632

006 -15548 -15464 -15382 -15301 -15220 -15141 -15063 -14985 -1909 -I4B33 007 -14758 -14684 -14611 -14538 -14466 -1435 -143lS -14lS5 -14187 -14118

008 -14OSI -13984 -13917 -13852 -13787 -13722 -13658 -13595 -13532 -13469

Oot -13408 -13346 -13285 -1l225 -13165 -13106 -13047 -12988 -12lO -12873

010 -12816 -1275 -12702 -12646 -12591 -12536 -12481 -12426 -12372 -12319

011 -12165 12112 -12160 -12107 -12055 -IlOO4 -11952 -11901 -11850 -11800

012 -11750 -11700 -11650 -1middot1601 -11552 -11503 -11455 -11407 -11359 -11311

Oll -11264 -11217 -11i70 -11123 -lIon -1IOll -10985 -Io9l9 -10893 -10848

014 -1D803 -10758 -10714 -1D669 -Io6lS -10581 -10537 -1D494 -1D45O -1D407

015 -1D364 -10322 -1D279 -10237 -10194 -10152 -10110 -1D069 -10027 -09986

016 -0HS -09904 -09863 -09822 -0782 -09741 -09701 -09661 -09621 -09581

017 -0542 -0502 -09463 -0M24 -09385 -09346 -09307 -0926 -09230 -0912 018 -09154 -09116 -09078 -09040 -09002 -08965 -08927 -08B9O -08853 -08816

Olt -D8779 -08742 -08705 -08669 -086l3 -08596 -08560 -08524 -08488 -08452

020 -01416 -OM81 -08345 -oBlIO -08274 -08239 -08204 -08169 -08134 -010 021 -08064 -01030 -07995 -07961 -07926 -07892 -07858 -07824 -0n9O -0nS6

011 -07722 -07688 -07655 -07621 -07588 -07554 -07521 -07488 -07454 -07421

OU -07388 -07356 -07323 -07290 -07257 -07225 -07192 -07160 -07128 -07095

014 -07063 -07031 -0 -06967 -06935 -06903 -06871 -06840 -06808 -un6

TABLE AI (Continued)

0000 0001 0002 0003 0004 0005 0006 0007 0008 Ooot

015 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -06526 -06495 -06464 026 -06433 -06403 -06372 -06341 -06311 -06280 -06250 -06219 -0618 -06158 017 -0612B -0608 -06068 -06038 -06008 -05978 -85m -05918 -05888 -05858 028 -05828 -0579 -05769 -05740 -05710 -05681 -05651 -05622 -05592 -05563 019 -05534 -05505 -05476 -05446 -05417 -05388 -DS359 -05330 -05302 -05273 030 031 031

-05244 -0495 -046n

-05215 -04930 -04649

-05187 -04902 -04621

-05158 -04874 -04593

-0529 -04845 -04565

-05101 -04817 -04538

-05072 -04789 -04SIO

-05044 -04761 -04482

-05015 -04733 -04454

-04987 -04705 -04427

033 034 035

-04399 -D4I25 -03853

-01372 -04097 -03826

-04344 -04070 -03719

-04316 -04043 -03772

-04289 -04016 -03745

-04261 -03989 -03719

-04234 -03961 -03692

-04207 -0l934 -03665

-04179 -03907 -03638

-04152 -0l880 -03611

0bull3 -03585 -03558 -03531 -03505 -0H78 -OHSI -03-125 -03398 -03171 -03345 nn -0331 -onn -03266 -0327 -03113 -03186 -03160 -O3IH -03107 -03081 Il VH

-03055 -0273

-0302 -02767

-0002 -021i I

-O2~76

-027IS -02950 -Oi6aS

-02924 -016pound

-O1e -0267

-02871 -02611

-O2EmiddottS -O2~pound5

-02819 -025~~

Q4Q 1)([

-0252 -02175

-02508 -02250

-02482 -02224

-O2~St

-0219pound -020(30 -O2l7

-O24lK -021-7

-0237e -02121

-02m -02096

-0221 -02070

-0230 I -02045

1)(1shy

Ioil -02019 -01764

-019 -01738

-OIS -01713

-0112 -01687

-OI~17

-01662 -01891 -01637

-01866 -01611

-OIIMO -01586

-0IBI5 -01560

-0178S -01535

O4(

1145 -01510 -01257

-01484 -01231

-01459 -01206

-01434 -01181

-01408 -01156

-01383 -01130

-01358 -01105

-01332 -OIOBO

-01307 -01055

-01282 -01030

046 -01004 -00979 -00954 -00929 -00904 -00878 -00853 -00828 -00803 -oons 047 -00753 -00728 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -00476 -00451 -0D426 -00401 -00376 -00351 -00326 -00301 -00276 049 050

-OOlSl 00000

-00226 00025

-00201 00050

-00175 00075

-001 SO 00100

-00125 00125

-00100 00150

-00075 00175

-00050 00201

-00025 00226

051 OOlSl 00276 00301 00326 00351 00376 00401 00426 00451 00476 052 00502 00527 00552 00577 00602 00627 00652 O06n 00702 00728 053 00753 0On8 00803 00828 00853 00878 00904 00929 00954 00979

VI CI

054 01004 01030 01055 01080 01105 01130 01156 01181 01206 01231

Table AI (Continued)

bull 01 4 bull 007 8 bullbull 08 001 ooOl DOS bullbull09

055 01257 01212 01307 01332 01358 01383 01408 01-43-4 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738

57 01764 0178 01815 01840 01866 01891 01917 01942 0 bull 68 0193 058 02019 02045 02070 02096 02121 02147 02173 0218 02224 02250 59 02275 02301 02327 02353 02l78 02404 D243O 02456 02482 02SD8 060 02533 0255 02585 02611 D2637 02663 Dl689 02715 02741 02767 061 02793 02819 02845 02871 02898 0224 02950 02976 03002 03029 062 03055 O3OBI 03107 03134 03160 03186 03213 0323 03266 03292 D6l 03319 03345 il3372 03398 O34lS 03451 03478 Ol505 03531 03558 D64 03585 03611 O3UB 03665 03692 03719 037-45 urn 03799 03826 US 0l853 03880 Ol907 0334 03961 0398 04016 040-43 04070 04097 066 MIlS 04152 04179 04207 04234 04261 04289 04316 04344 0-4372 D67 04399 04427 04454 04482 04510 04538 04565 0-4593 04621 0-4649 G68

D 4677 D4959

04705 04987

04733 05015

04761 05044

04789 050n

04817 05101

048045 0512

04874 05158

04902 05187

0-430 05215

870 05244 05273 05302 05330 05359 05388 D5417 05446 05476 05505 D7I 05534 05563 05592 05622 05651 05681 os710 05740 05769 os799 D72 05828 05858 05888 0518 05948 05978 06008 D6038 06068 D6098 Dn 06128 06158 0618 06219 06lS0 06280 06311 06341 06372 06403 074 06433 06464 06495 06526 06557 06588 06620 06651 06682 06713 D75 06745 06776 0680B D6840 06871 06903 0935 0amp967 O6m 07031

76 07063 07095 07128 07160 07192 07225 07257 0720 07323 07356

77 07388 07421 07-454 07488 07521 07554 07588 07621 07655 01688 878 0772l 07756 07790 07124 07858 07892 07926 07961 07995 08030 D79 08064

08416 0809 08452

08134 08488

08169 08524

0B204 08560

08239 08596

8274 DB6l3

08310 08669

08)45 08705

08381 087-42

D81 o8rn 08816 08853 D889O 08927 08965 09002 0040 09078 09116 081 09154 0192 0230 09269 0307 09346 09385 0942-4 09463 09502

Table AI (Continued)

Dot bull 001 bull GOl l 084 5 bull 006 007 8 DD09

01l 09542 09581 09621 09661 09101 09741 0782 09822 09863 09904

D 09945 0986 10027 Ul069 10110 10152 10194 10237 1027 10322 US 10364 10407 10450 10494 10531 105BI 10625 1066 10714 10758 086 10803 IOH8 10893 10939 10985 11031 11077 11123 11170 11217 D87 11264 11311 11359 11407 11-455 11503 11552 11601 11650 11700 Oer 1[750 1180[) I85c) Ll901 11952 12004- 12055 12107 12160 12212 lUi 12265 12l1S 12372 12426 12481 12536 fml 12G46 12702 1275~

1gt0 12816 l28n 12gt30 12936 1gt047 UI06 I3IeS 13225 13285 Inlaquo [il IHoe U4E~ 1gt5J2 1355 136S( 12722 IlSY 13152 1l17 13SgtS ItS 11051 [AIlE IAlei 1(25 1ltl2r Ii$~f 1lt4gtlt ltSt 14middotE II 1468lt

IA75E I4S23 1lt0 IltN~[ ISGt 151~1 522( 153( 1 1~3E 154(lt

155ltpound IS631 IS7fB 15805 ISES3 160n Ltd IEmiddot25G 16352 I~ 16546 16646 16H7 l6lYi~middot 16954 17[)60 17169 ln7gt In~2

17507 1762lt1 17744 17866 179lt11 1amp1 IS 18250 IS3amp-lt 18522 18663 18808 IISS7 19110 1926S ~9431 19600 11774 199pound4 20141 l033S 20537 20749 20969 ll201 21444 21701 21973 22262 22571 22904 23263 23656 24089 24573 25121 25r5B 26521 27478 28782 30902

SOURCpound GlIIIlefllted by II L Iman Used with permlS$ion

bull The entries In this table are quantlles Z of the standard normal fllndom variable Z selected so P(Z s z) = p and P(Z gt z) = I - p Note thilt the value of p to two dedmal places determin whith row to use the third decimal place of p d rmines whidl column to us to fmd

510 APPENDIX

TABLE Al Chi-Squared Distribution

Uti 0999 p =0750 0900 usa G97S U90

1083(=1 132] 2706 3H1 502 66l5 71 1382

2 2773 605 5991 7378 9210 1060 93-48 12 1627

1 108 6251 7815 IIM 18-47 5385 7779 88 111 1318 116 2051

5 6626 9236 1107 1283 1509 1675

6 7H1 106-4 1259 15 1681 1855 12 7 9037 1202 107 1601 18bullf8 2UI 232

26ll8 1022 13]6 1551 175l 2009 2196

2788

10 1255 1599 1831 208 2UI 1519 1599 1139 1middot468 1692 1902 2167 235

2676 312II 1370 1728 1968 2192 273

1622 2830 l2tI1855 2103 23l12 185 2f7 2769 2982 35313 1598 1981 2236 1712 2106 2368 2612 2U ll32 3612

lU8 l280 37701825 2231 2500 279IS 2885 l200 3 27 3925

16 1937 235 2UO 3MI 35n 4079

17 2M9 2f77 2759 lO19 2311160 2599 2887 ll53 l81 37 bull 18

22n 2720 3014 l285 3619 lUI 38219 53120 2383 281 ll 1 3 17 3757 -4000 l89) 140 802962 3267 35821 293 4029 210 2711 26Df 3081 U92 3678

3201 3517 3808 16-4 18 73II 271 5118n20 36f2 39l7 298 552821 3U8 3765 065 ll 526215 293 3

30] 3556 )889 192 56-4 29 $f0516 55-4811 3153 367 4011 l19 96 18 3262 3792 13 6 828 SO 5619

523 58303l71 390 156 572 9519 5367 597030 M80 4026 377 698 5019

6)69 6671 71-405181 5576 59l 563] 6317 6750 711 7615 79 86660 562

50 9195 996160 6698 740 7908 8l30 88l8

70 n58 8553 9053 1009502 1Df2 1123

1019 IOU ~112l IIU 12880 8813 9658 90 9865 1076 1131 1181 121 1213 m2

1296 1358 1-402100 1091 1185 12 ) I 0675 1282 16-45 1960 2)26 2576 l090

~

For k gt 100 uslbe approldmadon w - (t)(r + vlk - I) or lb more accurajI w

(I - ~+ r ~) where r Is be value from Ibe RlJldlrdbad normal distribution shown In Ibe b_

of Ik table SoUllCE Abrldamped ffom Table e Vol I of PelrlOft and Hanley (1976) with pennlulOll from Ibe 1IIomuilc1l

Trumes bull The entries In ibis table are qusndlu w of a chI-squared random varIIbl W wllk delrees of hedom s ad 10 P(W S w) - p and P(W gt w m I - p

APPENDIX 511

TABLE A3 Binomial Distribution

bullbull Wn y p =005 010 015 OW u J 030 0l5 OAO 05

1

1

5

6

7

o

o I 2

o 1 2 1

o

1 l 4 o

2bull l 4 5

o I 2 1 4 5 6 o I 2 1 4 5 6 1

09500 10000 09025 09975 10000

0857 09928 09999 10000

0815 09860 09995 10000 10000

017l8 0917 09988 10000 10000 10000

07351 O96n 09978 09999 10000 10000 10000

06983 09556 09962 09998 10000 10000 10000 10000

09000 10000 08100 09900 10000

07290 09720 09990 10000

06561 077 0996] 09999 10000

05905 09185 0991 09995 10000 10000

0531 08857 0842 09987 09999 10000 10000

0783 0850] 0973 09973 09998 10000 10000 10000

08500 10000 07225 09775 10000

06141 09392 09966 10000

05220 08905 09880 09995 10000

04437 08352 09734 09978 0999 10000

03771 07765 09527 09941

09996 10000 10000

0l206 07166 09262 09879 09988 09999 10000 10000

OBOOO 10000 06400 09600 10000

OSf2o

08960 09920 10000

010

011191 09720 09904

10000

0317 07373 09411 09933

09997

10000

02621 06554 09011 09930

09984

09999

10000

02097

057(7

08520 09667

09953

099

10000 10000

07500 10000 O565

09375

10000

OA219 OBIJ9

09114-1 10000 1))16middot1

Dnll) 11~)4n

091 10000

02373

06318 095 09844

09990 10000

017ilu ()5B9 01l30

09624 09954 09990 10000

01335 DA449 07564

092Y-1

09071 09987

09999

10000

07000 10000 04900 09100 10000

03430 07B40 09730 10000

02401 06517 09163 09919 10000

01681 05282 09369 09692 09976 10000

01176 OA202 07443 09295

09891 09993 10000

00821 03294 06471 08740 09712 09962 09998 10000

06500 06000 10000 10000 04225 03600 08775 08400 10000 10000

02746 02160 07182 06480 09571 09360 10000 10000

01785 01296 05630 004752 08735 08208 0850 09744 10000 10000

01160 00778 OA284 03370 07648 06826 09460 Q9130 09947 09898 10000 10000

00754 00467 03191 02333 06471 05443 08826 08208 09177 09590 09982 09959 10000 10000

00490 00280 02338 01586 05323 004199 08002 07102 09444 09037 09910 09812 09994 09984 10000 10000

05500 10000 03025 07975 10000

01664 05748 09089 10000

00915 03910 07585 09590 10000

00503 02562 05931 08688 09815 10000

00277 01636 04415 07447 09309 09917 10000

001S2 01024 03164 06083 08471 0963 09963 10000

w

o

lit

II III CO

coiD CO

CO

e III

bull

-------00000 ------oo=== ----gtgt~~~f ----===0bbbbS 2b~~~iD~8S b822~~~22S b-----v02SoS~8sectsectsectS~8~~i SSsectS88~iIS~ 8~S8sectsecti~ls sect8S8Siet

III rshym

t litII

Ii

o

E III

raquo o W mIII Z

D

-W

X

o

~

u o n~ o

i o

1 bullo p

p o-

bullo o

bullo o

n o

I a

~ mZ o ~

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

-lt -- 11

f c

s ~ Mbull -i 9 + It -

bull 1

0 ~ ~ 01 s I

D

UI W -O middot Wa-O UI W -O W - bull ---------- --------shy

iii~~~~eee~ee~~g~gg-DOIrDIID ~ CD-W J 0 00 sectbsecti~~~~~~~e~~ei~iii~2~~2~gw w- 1~~~~~~~~~~~wSgg o CD tJOCD~CJ~

~~pppppppppppppppppp--OOOOOOOOOOpOOOOOOOObull ~ ~~~~w-000o~8io~1~~~=~~~~~~w~a88sect118 00 0 g8~1~~ ~~~~~sectS 0 ~-~om~m iS-~- o m~ oww-o~-- m - g---~-o~~w w 0

f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

~pppppppPpP~pppppppp~~ampPpppppppppppppoopp

t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

i~~sect~iiiiiiiiiiiiiiiii~gi~iiiiiiiiiiiiiii

J

C

I

bulle D

e ebull e III

bull ~

bull

Cbull

bull bull ~

po III

~ IIf r-m

~

n 0 I It I I III CI

TABLE AI Nonna Distribution

Seleaed vUueI ~ -37190 zshy - -32905 Zoms = -1600 Zui = -16449 z- 37190 z_ = 3205 zm == 19600 ztS = 1644 0000 0001 0001 0003 0004 0001 0006 0001 0008 0009

000 -30902 -287B2 -27478 -26521 -25758 -25121 -2457l -24089 -23656

001 -23263 -22904 -22571 -22262 -21973 -21701 -21444 -2llDl -20969 -20749

001 -20537 -20335 -20141 -19954 -19n4 -19600 -19431 -19268 -19110 -IB957

OOl -18B08 -18663 -18522 -18384 -18250 -18119 -17991 -17866 -1n44 -17624

0D4 -17507 -17392 -17179 -17169 -17060 -16954 -1684 -16747 -16646 -16546

005 -1644 -16352 -16258 -16164 -16072 -15982 -15893 -15805 -157IB -15632

006 -15548 -15464 -15382 -15301 -15220 -15141 -15063 -14985 -1909 -I4B33 007 -14758 -14684 -14611 -14538 -14466 -1435 -143lS -14lS5 -14187 -14118

008 -14OSI -13984 -13917 -13852 -13787 -13722 -13658 -13595 -13532 -13469

Oot -13408 -13346 -13285 -1l225 -13165 -13106 -13047 -12988 -12lO -12873

010 -12816 -1275 -12702 -12646 -12591 -12536 -12481 -12426 -12372 -12319

011 -12165 12112 -12160 -12107 -12055 -IlOO4 -11952 -11901 -11850 -11800

012 -11750 -11700 -11650 -1middot1601 -11552 -11503 -11455 -11407 -11359 -11311

Oll -11264 -11217 -11i70 -11123 -lIon -1IOll -10985 -Io9l9 -10893 -10848

014 -1D803 -10758 -10714 -1D669 -Io6lS -10581 -10537 -1D494 -1D45O -1D407

015 -1D364 -10322 -1D279 -10237 -10194 -10152 -10110 -1D069 -10027 -09986

016 -0HS -09904 -09863 -09822 -0782 -09741 -09701 -09661 -09621 -09581

017 -0542 -0502 -09463 -0M24 -09385 -09346 -09307 -0926 -09230 -0912 018 -09154 -09116 -09078 -09040 -09002 -08965 -08927 -08B9O -08853 -08816

Olt -D8779 -08742 -08705 -08669 -086l3 -08596 -08560 -08524 -08488 -08452

020 -01416 -OM81 -08345 -oBlIO -08274 -08239 -08204 -08169 -08134 -010 021 -08064 -01030 -07995 -07961 -07926 -07892 -07858 -07824 -0n9O -0nS6

011 -07722 -07688 -07655 -07621 -07588 -07554 -07521 -07488 -07454 -07421

OU -07388 -07356 -07323 -07290 -07257 -07225 -07192 -07160 -07128 -07095

014 -07063 -07031 -0 -06967 -06935 -06903 -06871 -06840 -06808 -un6

TABLE AI (Continued)

0000 0001 0002 0003 0004 0005 0006 0007 0008 Ooot

015 -06745 -06713 -06682 -06651 -06620 -06588 -06557 -06526 -06495 -06464 026 -06433 -06403 -06372 -06341 -06311 -06280 -06250 -06219 -0618 -06158 017 -0612B -0608 -06068 -06038 -06008 -05978 -85m -05918 -05888 -05858 028 -05828 -0579 -05769 -05740 -05710 -05681 -05651 -05622 -05592 -05563 019 -05534 -05505 -05476 -05446 -05417 -05388 -DS359 -05330 -05302 -05273 030 031 031

-05244 -0495 -046n

-05215 -04930 -04649

-05187 -04902 -04621

-05158 -04874 -04593

-0529 -04845 -04565

-05101 -04817 -04538

-05072 -04789 -04SIO

-05044 -04761 -04482

-05015 -04733 -04454

-04987 -04705 -04427

033 034 035

-04399 -D4I25 -03853

-01372 -04097 -03826

-04344 -04070 -03719

-04316 -04043 -03772

-04289 -04016 -03745

-04261 -03989 -03719

-04234 -03961 -03692

-04207 -0l934 -03665

-04179 -03907 -03638

-04152 -0l880 -03611

0bull3 -03585 -03558 -03531 -03505 -0H78 -OHSI -03-125 -03398 -03171 -03345 nn -0331 -onn -03266 -0327 -03113 -03186 -03160 -O3IH -03107 -03081 Il VH

-03055 -0273

-0302 -02767

-0002 -021i I

-O2~76

-027IS -02950 -Oi6aS

-02924 -016pound

-O1e -0267

-02871 -02611

-O2EmiddottS -O2~pound5

-02819 -025~~

Q4Q 1)([

-0252 -02175

-02508 -02250

-02482 -02224

-O2~St

-0219pound -020(30 -O2l7

-O24lK -021-7

-0237e -02121

-02m -02096

-0221 -02070

-0230 I -02045

1)(1shy

Ioil -02019 -01764

-019 -01738

-OIS -01713

-0112 -01687

-OI~17

-01662 -01891 -01637

-01866 -01611

-OIIMO -01586

-0IBI5 -01560

-0178S -01535

O4(

1145 -01510 -01257

-01484 -01231

-01459 -01206

-01434 -01181

-01408 -01156

-01383 -01130

-01358 -01105

-01332 -OIOBO

-01307 -01055

-01282 -01030

046 -01004 -00979 -00954 -00929 -00904 -00878 -00853 -00828 -00803 -oons 047 -00753 -00728 -00702 -00677 -00652 -00627 -00602 -00577 -00552 -00527 048 -00502 -00476 -00451 -0D426 -00401 -00376 -00351 -00326 -00301 -00276 049 050

-OOlSl 00000

-00226 00025

-00201 00050

-00175 00075

-001 SO 00100

-00125 00125

-00100 00150

-00075 00175

-00050 00201

-00025 00226

051 OOlSl 00276 00301 00326 00351 00376 00401 00426 00451 00476 052 00502 00527 00552 00577 00602 00627 00652 O06n 00702 00728 053 00753 0On8 00803 00828 00853 00878 00904 00929 00954 00979

VI CI

054 01004 01030 01055 01080 01105 01130 01156 01181 01206 01231

Table AI (Continued)

bull 01 4 bull 007 8 bullbull 08 001 ooOl DOS bullbull09

055 01257 01212 01307 01332 01358 01383 01408 01-43-4 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738

57 01764 0178 01815 01840 01866 01891 01917 01942 0 bull 68 0193 058 02019 02045 02070 02096 02121 02147 02173 0218 02224 02250 59 02275 02301 02327 02353 02l78 02404 D243O 02456 02482 02SD8 060 02533 0255 02585 02611 D2637 02663 Dl689 02715 02741 02767 061 02793 02819 02845 02871 02898 0224 02950 02976 03002 03029 062 03055 O3OBI 03107 03134 03160 03186 03213 0323 03266 03292 D6l 03319 03345 il3372 03398 O34lS 03451 03478 Ol505 03531 03558 D64 03585 03611 O3UB 03665 03692 03719 037-45 urn 03799 03826 US 0l853 03880 Ol907 0334 03961 0398 04016 040-43 04070 04097 066 MIlS 04152 04179 04207 04234 04261 04289 04316 04344 0-4372 D67 04399 04427 04454 04482 04510 04538 04565 0-4593 04621 0-4649 G68

D 4677 D4959

04705 04987

04733 05015

04761 05044

04789 050n

04817 05101

048045 0512

04874 05158

04902 05187

0-430 05215

870 05244 05273 05302 05330 05359 05388 D5417 05446 05476 05505 D7I 05534 05563 05592 05622 05651 05681 os710 05740 05769 os799 D72 05828 05858 05888 0518 05948 05978 06008 D6038 06068 D6098 Dn 06128 06158 0618 06219 06lS0 06280 06311 06341 06372 06403 074 06433 06464 06495 06526 06557 06588 06620 06651 06682 06713 D75 06745 06776 0680B D6840 06871 06903 0935 0amp967 O6m 07031

76 07063 07095 07128 07160 07192 07225 07257 0720 07323 07356

77 07388 07421 07-454 07488 07521 07554 07588 07621 07655 01688 878 0772l 07756 07790 07124 07858 07892 07926 07961 07995 08030 D79 08064

08416 0809 08452

08134 08488

08169 08524

0B204 08560

08239 08596

8274 DB6l3

08310 08669

08)45 08705

08381 087-42

D81 o8rn 08816 08853 D889O 08927 08965 09002 0040 09078 09116 081 09154 0192 0230 09269 0307 09346 09385 0942-4 09463 09502

Table AI (Continued)

Dot bull 001 bull GOl l 084 5 bull 006 007 8 DD09

01l 09542 09581 09621 09661 09101 09741 0782 09822 09863 09904

D 09945 0986 10027 Ul069 10110 10152 10194 10237 1027 10322 US 10364 10407 10450 10494 10531 105BI 10625 1066 10714 10758 086 10803 IOH8 10893 10939 10985 11031 11077 11123 11170 11217 D87 11264 11311 11359 11407 11-455 11503 11552 11601 11650 11700 Oer 1[750 1180[) I85c) Ll901 11952 12004- 12055 12107 12160 12212 lUi 12265 12l1S 12372 12426 12481 12536 fml 12G46 12702 1275~

1gt0 12816 l28n 12gt30 12936 1gt047 UI06 I3IeS 13225 13285 Inlaquo [il IHoe U4E~ 1gt5J2 1355 136S( 12722 IlSY 13152 1l17 13SgtS ItS 11051 [AIlE IAlei 1(25 1ltl2r Ii$~f 1lt4gtlt ltSt 14middotE II 1468lt

IA75E I4S23 1lt0 IltN~[ ISGt 151~1 522( 153( 1 1~3E 154(lt

155ltpound IS631 IS7fB 15805 ISES3 160n Ltd IEmiddot25G 16352 I~ 16546 16646 16H7 l6lYi~middot 16954 17[)60 17169 ln7gt In~2

17507 1762lt1 17744 17866 179lt11 1amp1 IS 18250 IS3amp-lt 18522 18663 18808 IISS7 19110 1926S ~9431 19600 11774 199pound4 20141 l033S 20537 20749 20969 ll201 21444 21701 21973 22262 22571 22904 23263 23656 24089 24573 25121 25r5B 26521 27478 28782 30902

SOURCpound GlIIIlefllted by II L Iman Used with permlS$ion

bull The entries In this table are quantlles Z of the standard normal fllndom variable Z selected so P(Z s z) = p and P(Z gt z) = I - p Note thilt the value of p to two dedmal places determin whith row to use the third decimal place of p d rmines whidl column to us to fmd

510 APPENDIX

TABLE Al Chi-Squared Distribution

Uti 0999 p =0750 0900 usa G97S U90

1083(=1 132] 2706 3H1 502 66l5 71 1382

2 2773 605 5991 7378 9210 1060 93-48 12 1627

1 108 6251 7815 IIM 18-47 5385 7779 88 111 1318 116 2051

5 6626 9236 1107 1283 1509 1675

6 7H1 106-4 1259 15 1681 1855 12 7 9037 1202 107 1601 18bullf8 2UI 232

26ll8 1022 13]6 1551 175l 2009 2196

2788

10 1255 1599 1831 208 2UI 1519 1599 1139 1middot468 1692 1902 2167 235

2676 312II 1370 1728 1968 2192 273

1622 2830 l2tI1855 2103 23l12 185 2f7 2769 2982 35313 1598 1981 2236 1712 2106 2368 2612 2U ll32 3612

lU8 l280 37701825 2231 2500 279IS 2885 l200 3 27 3925

16 1937 235 2UO 3MI 35n 4079

17 2M9 2f77 2759 lO19 2311160 2599 2887 ll53 l81 37 bull 18

22n 2720 3014 l285 3619 lUI 38219 53120 2383 281 ll 1 3 17 3757 -4000 l89) 140 802962 3267 35821 293 4029 210 2711 26Df 3081 U92 3678

3201 3517 3808 16-4 18 73II 271 5118n20 36f2 39l7 298 552821 3U8 3765 065 ll 526215 293 3

30] 3556 )889 192 56-4 29 $f0516 55-4811 3153 367 4011 l19 96 18 3262 3792 13 6 828 SO 5619

523 58303l71 390 156 572 9519 5367 597030 M80 4026 377 698 5019

6)69 6671 71-405181 5576 59l 563] 6317 6750 711 7615 79 86660 562

50 9195 996160 6698 740 7908 8l30 88l8

70 n58 8553 9053 1009502 1Df2 1123

1019 IOU ~112l IIU 12880 8813 9658 90 9865 1076 1131 1181 121 1213 m2

1296 1358 1-402100 1091 1185 12 ) I 0675 1282 16-45 1960 2)26 2576 l090

~

For k gt 100 uslbe approldmadon w - (t)(r + vlk - I) or lb more accurajI w

(I - ~+ r ~) where r Is be value from Ibe RlJldlrdbad normal distribution shown In Ibe b_

of Ik table SoUllCE Abrldamped ffom Table e Vol I of PelrlOft and Hanley (1976) with pennlulOll from Ibe 1IIomuilc1l

Trumes bull The entries In ibis table are qusndlu w of a chI-squared random varIIbl W wllk delrees of hedom s ad 10 P(W S w) - p and P(W gt w m I - p

APPENDIX 511

TABLE A3 Binomial Distribution

bullbull Wn y p =005 010 015 OW u J 030 0l5 OAO 05

1

1

5

6

7

o

o I 2

o 1 2 1

o

1 l 4 o

2bull l 4 5

o I 2 1 4 5 6 o I 2 1 4 5 6 1

09500 10000 09025 09975 10000

0857 09928 09999 10000

0815 09860 09995 10000 10000

017l8 0917 09988 10000 10000 10000

07351 O96n 09978 09999 10000 10000 10000

06983 09556 09962 09998 10000 10000 10000 10000

09000 10000 08100 09900 10000

07290 09720 09990 10000

06561 077 0996] 09999 10000

05905 09185 0991 09995 10000 10000

0531 08857 0842 09987 09999 10000 10000

0783 0850] 0973 09973 09998 10000 10000 10000

08500 10000 07225 09775 10000

06141 09392 09966 10000

05220 08905 09880 09995 10000

04437 08352 09734 09978 0999 10000

03771 07765 09527 09941

09996 10000 10000

0l206 07166 09262 09879 09988 09999 10000 10000

OBOOO 10000 06400 09600 10000

OSf2o

08960 09920 10000

010

011191 09720 09904

10000

0317 07373 09411 09933

09997

10000

02621 06554 09011 09930

09984

09999

10000

02097

057(7

08520 09667

09953

099

10000 10000

07500 10000 O565

09375

10000

OA219 OBIJ9

09114-1 10000 1))16middot1

Dnll) 11~)4n

091 10000

02373

06318 095 09844

09990 10000

017ilu ()5B9 01l30

09624 09954 09990 10000

01335 DA449 07564

092Y-1

09071 09987

09999

10000

07000 10000 04900 09100 10000

03430 07B40 09730 10000

02401 06517 09163 09919 10000

01681 05282 09369 09692 09976 10000

01176 OA202 07443 09295

09891 09993 10000

00821 03294 06471 08740 09712 09962 09998 10000

06500 06000 10000 10000 04225 03600 08775 08400 10000 10000

02746 02160 07182 06480 09571 09360 10000 10000

01785 01296 05630 004752 08735 08208 0850 09744 10000 10000

01160 00778 OA284 03370 07648 06826 09460 Q9130 09947 09898 10000 10000

00754 00467 03191 02333 06471 05443 08826 08208 09177 09590 09982 09959 10000 10000

00490 00280 02338 01586 05323 004199 08002 07102 09444 09037 09910 09812 09994 09984 10000 10000

05500 10000 03025 07975 10000

01664 05748 09089 10000

00915 03910 07585 09590 10000

00503 02562 05931 08688 09815 10000

00277 01636 04415 07447 09309 09917 10000

001S2 01024 03164 06083 08471 0963 09963 10000

w

o

lit

II III CO

coiD CO

CO

e III

bull

-------00000 ------oo=== ----gtgt~~~f ----===0bbbbS 2b~~~iD~8S b822~~~22S b-----v02SoS~8sectsectsectS~8~~i SSsectS88~iIS~ 8~S8sectsecti~ls sect8S8Siet

III rshym

t litII

Ii

o

E III

raquo o W mIII Z

D

-W

X

o

~

u o n~ o

i o

1 bullo p

p o-

bullo o

bullo o

n o

I a

~ mZ o ~

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

-lt -- 11

f c

s ~ Mbull -i 9 + It -

bull 1

0 ~ ~ 01 s I

D

UI W -O middot Wa-O UI W -O W - bull ---------- --------shy

iii~~~~eee~ee~~g~gg-DOIrDIID ~ CD-W J 0 00 sectbsecti~~~~~~~e~~ei~iii~2~~2~gw w- 1~~~~~~~~~~~wSgg o CD tJOCD~CJ~

~~pppppppppppppppppp--OOOOOOOOOOpOOOOOOOObull ~ ~~~~w-000o~8io~1~~~=~~~~~~w~a88sect118 00 0 g8~1~~ ~~~~~sectS 0 ~-~om~m iS-~- o m~ oww-o~-- m - g---~-o~~w w 0

f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

~pppppppPpP~pppppppp~~ampPpppppppppppppoopp

t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

i~~sect~iiiiiiiiiiiiiiiii~gi~iiiiiiiiiiiiiii

J

C

I

bulle D

e ebull e III

bull ~

bull

Cbull

bull bull ~

po III

~ IIf r-m

~

n 0 I It I I III CI

Table AI (Continued)

bull 01 4 bull 007 8 bullbull 08 001 ooOl DOS bullbull09

055 01257 01212 01307 01332 01358 01383 01408 01-43-4 0145 01484 056 01510 01535 01560 01586 01611 01637 01662 01687 01713 01738

57 01764 0178 01815 01840 01866 01891 01917 01942 0 bull 68 0193 058 02019 02045 02070 02096 02121 02147 02173 0218 02224 02250 59 02275 02301 02327 02353 02l78 02404 D243O 02456 02482 02SD8 060 02533 0255 02585 02611 D2637 02663 Dl689 02715 02741 02767 061 02793 02819 02845 02871 02898 0224 02950 02976 03002 03029 062 03055 O3OBI 03107 03134 03160 03186 03213 0323 03266 03292 D6l 03319 03345 il3372 03398 O34lS 03451 03478 Ol505 03531 03558 D64 03585 03611 O3UB 03665 03692 03719 037-45 urn 03799 03826 US 0l853 03880 Ol907 0334 03961 0398 04016 040-43 04070 04097 066 MIlS 04152 04179 04207 04234 04261 04289 04316 04344 0-4372 D67 04399 04427 04454 04482 04510 04538 04565 0-4593 04621 0-4649 G68

D 4677 D4959

04705 04987

04733 05015

04761 05044

04789 050n

04817 05101

048045 0512

04874 05158

04902 05187

0-430 05215

870 05244 05273 05302 05330 05359 05388 D5417 05446 05476 05505 D7I 05534 05563 05592 05622 05651 05681 os710 05740 05769 os799 D72 05828 05858 05888 0518 05948 05978 06008 D6038 06068 D6098 Dn 06128 06158 0618 06219 06lS0 06280 06311 06341 06372 06403 074 06433 06464 06495 06526 06557 06588 06620 06651 06682 06713 D75 06745 06776 0680B D6840 06871 06903 0935 0amp967 O6m 07031

76 07063 07095 07128 07160 07192 07225 07257 0720 07323 07356

77 07388 07421 07-454 07488 07521 07554 07588 07621 07655 01688 878 0772l 07756 07790 07124 07858 07892 07926 07961 07995 08030 D79 08064

08416 0809 08452

08134 08488

08169 08524

0B204 08560

08239 08596

8274 DB6l3

08310 08669

08)45 08705

08381 087-42

D81 o8rn 08816 08853 D889O 08927 08965 09002 0040 09078 09116 081 09154 0192 0230 09269 0307 09346 09385 0942-4 09463 09502

Table AI (Continued)

Dot bull 001 bull GOl l 084 5 bull 006 007 8 DD09

01l 09542 09581 09621 09661 09101 09741 0782 09822 09863 09904

D 09945 0986 10027 Ul069 10110 10152 10194 10237 1027 10322 US 10364 10407 10450 10494 10531 105BI 10625 1066 10714 10758 086 10803 IOH8 10893 10939 10985 11031 11077 11123 11170 11217 D87 11264 11311 11359 11407 11-455 11503 11552 11601 11650 11700 Oer 1[750 1180[) I85c) Ll901 11952 12004- 12055 12107 12160 12212 lUi 12265 12l1S 12372 12426 12481 12536 fml 12G46 12702 1275~

1gt0 12816 l28n 12gt30 12936 1gt047 UI06 I3IeS 13225 13285 Inlaquo [il IHoe U4E~ 1gt5J2 1355 136S( 12722 IlSY 13152 1l17 13SgtS ItS 11051 [AIlE IAlei 1(25 1ltl2r Ii$~f 1lt4gtlt ltSt 14middotE II 1468lt

IA75E I4S23 1lt0 IltN~[ ISGt 151~1 522( 153( 1 1~3E 154(lt

155ltpound IS631 IS7fB 15805 ISES3 160n Ltd IEmiddot25G 16352 I~ 16546 16646 16H7 l6lYi~middot 16954 17[)60 17169 ln7gt In~2

17507 1762lt1 17744 17866 179lt11 1amp1 IS 18250 IS3amp-lt 18522 18663 18808 IISS7 19110 1926S ~9431 19600 11774 199pound4 20141 l033S 20537 20749 20969 ll201 21444 21701 21973 22262 22571 22904 23263 23656 24089 24573 25121 25r5B 26521 27478 28782 30902

SOURCpound GlIIIlefllted by II L Iman Used with permlS$ion

bull The entries In this table are quantlles Z of the standard normal fllndom variable Z selected so P(Z s z) = p and P(Z gt z) = I - p Note thilt the value of p to two dedmal places determin whith row to use the third decimal place of p d rmines whidl column to us to fmd

510 APPENDIX

TABLE Al Chi-Squared Distribution

Uti 0999 p =0750 0900 usa G97S U90

1083(=1 132] 2706 3H1 502 66l5 71 1382

2 2773 605 5991 7378 9210 1060 93-48 12 1627

1 108 6251 7815 IIM 18-47 5385 7779 88 111 1318 116 2051

5 6626 9236 1107 1283 1509 1675

6 7H1 106-4 1259 15 1681 1855 12 7 9037 1202 107 1601 18bullf8 2UI 232

26ll8 1022 13]6 1551 175l 2009 2196

2788

10 1255 1599 1831 208 2UI 1519 1599 1139 1middot468 1692 1902 2167 235

2676 312II 1370 1728 1968 2192 273

1622 2830 l2tI1855 2103 23l12 185 2f7 2769 2982 35313 1598 1981 2236 1712 2106 2368 2612 2U ll32 3612

lU8 l280 37701825 2231 2500 279IS 2885 l200 3 27 3925

16 1937 235 2UO 3MI 35n 4079

17 2M9 2f77 2759 lO19 2311160 2599 2887 ll53 l81 37 bull 18

22n 2720 3014 l285 3619 lUI 38219 53120 2383 281 ll 1 3 17 3757 -4000 l89) 140 802962 3267 35821 293 4029 210 2711 26Df 3081 U92 3678

3201 3517 3808 16-4 18 73II 271 5118n20 36f2 39l7 298 552821 3U8 3765 065 ll 526215 293 3

30] 3556 )889 192 56-4 29 $f0516 55-4811 3153 367 4011 l19 96 18 3262 3792 13 6 828 SO 5619

523 58303l71 390 156 572 9519 5367 597030 M80 4026 377 698 5019

6)69 6671 71-405181 5576 59l 563] 6317 6750 711 7615 79 86660 562

50 9195 996160 6698 740 7908 8l30 88l8

70 n58 8553 9053 1009502 1Df2 1123

1019 IOU ~112l IIU 12880 8813 9658 90 9865 1076 1131 1181 121 1213 m2

1296 1358 1-402100 1091 1185 12 ) I 0675 1282 16-45 1960 2)26 2576 l090

~

For k gt 100 uslbe approldmadon w - (t)(r + vlk - I) or lb more accurajI w

(I - ~+ r ~) where r Is be value from Ibe RlJldlrdbad normal distribution shown In Ibe b_

of Ik table SoUllCE Abrldamped ffom Table e Vol I of PelrlOft and Hanley (1976) with pennlulOll from Ibe 1IIomuilc1l

Trumes bull The entries In ibis table are qusndlu w of a chI-squared random varIIbl W wllk delrees of hedom s ad 10 P(W S w) - p and P(W gt w m I - p

APPENDIX 511

TABLE A3 Binomial Distribution

bullbull Wn y p =005 010 015 OW u J 030 0l5 OAO 05

1

1

5

6

7

o

o I 2

o 1 2 1

o

1 l 4 o

2bull l 4 5

o I 2 1 4 5 6 o I 2 1 4 5 6 1

09500 10000 09025 09975 10000

0857 09928 09999 10000

0815 09860 09995 10000 10000

017l8 0917 09988 10000 10000 10000

07351 O96n 09978 09999 10000 10000 10000

06983 09556 09962 09998 10000 10000 10000 10000

09000 10000 08100 09900 10000

07290 09720 09990 10000

06561 077 0996] 09999 10000

05905 09185 0991 09995 10000 10000

0531 08857 0842 09987 09999 10000 10000

0783 0850] 0973 09973 09998 10000 10000 10000

08500 10000 07225 09775 10000

06141 09392 09966 10000

05220 08905 09880 09995 10000

04437 08352 09734 09978 0999 10000

03771 07765 09527 09941

09996 10000 10000

0l206 07166 09262 09879 09988 09999 10000 10000

OBOOO 10000 06400 09600 10000

OSf2o

08960 09920 10000

010

011191 09720 09904

10000

0317 07373 09411 09933

09997

10000

02621 06554 09011 09930

09984

09999

10000

02097

057(7

08520 09667

09953

099

10000 10000

07500 10000 O565

09375

10000

OA219 OBIJ9

09114-1 10000 1))16middot1

Dnll) 11~)4n

091 10000

02373

06318 095 09844

09990 10000

017ilu ()5B9 01l30

09624 09954 09990 10000

01335 DA449 07564

092Y-1

09071 09987

09999

10000

07000 10000 04900 09100 10000

03430 07B40 09730 10000

02401 06517 09163 09919 10000

01681 05282 09369 09692 09976 10000

01176 OA202 07443 09295

09891 09993 10000

00821 03294 06471 08740 09712 09962 09998 10000

06500 06000 10000 10000 04225 03600 08775 08400 10000 10000

02746 02160 07182 06480 09571 09360 10000 10000

01785 01296 05630 004752 08735 08208 0850 09744 10000 10000

01160 00778 OA284 03370 07648 06826 09460 Q9130 09947 09898 10000 10000

00754 00467 03191 02333 06471 05443 08826 08208 09177 09590 09982 09959 10000 10000

00490 00280 02338 01586 05323 004199 08002 07102 09444 09037 09910 09812 09994 09984 10000 10000

05500 10000 03025 07975 10000

01664 05748 09089 10000

00915 03910 07585 09590 10000

00503 02562 05931 08688 09815 10000

00277 01636 04415 07447 09309 09917 10000

001S2 01024 03164 06083 08471 0963 09963 10000

w

o

lit

II III CO

coiD CO

CO

e III

bull

-------00000 ------oo=== ----gtgt~~~f ----===0bbbbS 2b~~~iD~8S b822~~~22S b-----v02SoS~8sectsectsectS~8~~i SSsectS88~iIS~ 8~S8sectsecti~ls sect8S8Siet

III rshym

t litII

Ii

o

E III

raquo o W mIII Z

D

-W

X

o

~

u o n~ o

i o

1 bullo p

p o-

bullo o

bullo o

n o

I a

~ mZ o ~

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

-lt -- 11

f c

s ~ Mbull -i 9 + It -

bull 1

0 ~ ~ 01 s I

D

UI W -O middot Wa-O UI W -O W - bull ---------- --------shy

iii~~~~eee~ee~~g~gg-DOIrDIID ~ CD-W J 0 00 sectbsecti~~~~~~~e~~ei~iii~2~~2~gw w- 1~~~~~~~~~~~wSgg o CD tJOCD~CJ~

~~pppppppppppppppppp--OOOOOOOOOOpOOOOOOOObull ~ ~~~~w-000o~8io~1~~~=~~~~~~w~a88sect118 00 0 g8~1~~ ~~~~~sectS 0 ~-~om~m iS-~- o m~ oww-o~-- m - g---~-o~~w w 0

f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

~pppppppPpP~pppppppp~~ampPpppppppppppppoopp

t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

i~~sect~iiiiiiiiiiiiiiiii~gi~iiiiiiiiiiiiiii

J

C

I

bulle D

e ebull e III

bull ~

bull

Cbull

bull bull ~

po III

~ IIf r-m

~

n 0 I It I I III CI

510 APPENDIX

TABLE Al Chi-Squared Distribution

Uti 0999 p =0750 0900 usa G97S U90

1083(=1 132] 2706 3H1 502 66l5 71 1382

2 2773 605 5991 7378 9210 1060 93-48 12 1627

1 108 6251 7815 IIM 18-47 5385 7779 88 111 1318 116 2051

5 6626 9236 1107 1283 1509 1675

6 7H1 106-4 1259 15 1681 1855 12 7 9037 1202 107 1601 18bullf8 2UI 232

26ll8 1022 13]6 1551 175l 2009 2196

2788

10 1255 1599 1831 208 2UI 1519 1599 1139 1middot468 1692 1902 2167 235

2676 312II 1370 1728 1968 2192 273

1622 2830 l2tI1855 2103 23l12 185 2f7 2769 2982 35313 1598 1981 2236 1712 2106 2368 2612 2U ll32 3612

lU8 l280 37701825 2231 2500 279IS 2885 l200 3 27 3925

16 1937 235 2UO 3MI 35n 4079

17 2M9 2f77 2759 lO19 2311160 2599 2887 ll53 l81 37 bull 18

22n 2720 3014 l285 3619 lUI 38219 53120 2383 281 ll 1 3 17 3757 -4000 l89) 140 802962 3267 35821 293 4029 210 2711 26Df 3081 U92 3678

3201 3517 3808 16-4 18 73II 271 5118n20 36f2 39l7 298 552821 3U8 3765 065 ll 526215 293 3

30] 3556 )889 192 56-4 29 $f0516 55-4811 3153 367 4011 l19 96 18 3262 3792 13 6 828 SO 5619

523 58303l71 390 156 572 9519 5367 597030 M80 4026 377 698 5019

6)69 6671 71-405181 5576 59l 563] 6317 6750 711 7615 79 86660 562

50 9195 996160 6698 740 7908 8l30 88l8

70 n58 8553 9053 1009502 1Df2 1123

1019 IOU ~112l IIU 12880 8813 9658 90 9865 1076 1131 1181 121 1213 m2

1296 1358 1-402100 1091 1185 12 ) I 0675 1282 16-45 1960 2)26 2576 l090

~

For k gt 100 uslbe approldmadon w - (t)(r + vlk - I) or lb more accurajI w

(I - ~+ r ~) where r Is be value from Ibe RlJldlrdbad normal distribution shown In Ibe b_

of Ik table SoUllCE Abrldamped ffom Table e Vol I of PelrlOft and Hanley (1976) with pennlulOll from Ibe 1IIomuilc1l

Trumes bull The entries In ibis table are qusndlu w of a chI-squared random varIIbl W wllk delrees of hedom s ad 10 P(W S w) - p and P(W gt w m I - p

APPENDIX 511

TABLE A3 Binomial Distribution

bullbull Wn y p =005 010 015 OW u J 030 0l5 OAO 05

1

1

5

6

7

o

o I 2

o 1 2 1

o

1 l 4 o

2bull l 4 5

o I 2 1 4 5 6 o I 2 1 4 5 6 1

09500 10000 09025 09975 10000

0857 09928 09999 10000

0815 09860 09995 10000 10000

017l8 0917 09988 10000 10000 10000

07351 O96n 09978 09999 10000 10000 10000

06983 09556 09962 09998 10000 10000 10000 10000

09000 10000 08100 09900 10000

07290 09720 09990 10000

06561 077 0996] 09999 10000

05905 09185 0991 09995 10000 10000

0531 08857 0842 09987 09999 10000 10000

0783 0850] 0973 09973 09998 10000 10000 10000

08500 10000 07225 09775 10000

06141 09392 09966 10000

05220 08905 09880 09995 10000

04437 08352 09734 09978 0999 10000

03771 07765 09527 09941

09996 10000 10000

0l206 07166 09262 09879 09988 09999 10000 10000

OBOOO 10000 06400 09600 10000

OSf2o

08960 09920 10000

010

011191 09720 09904

10000

0317 07373 09411 09933

09997

10000

02621 06554 09011 09930

09984

09999

10000

02097

057(7

08520 09667

09953

099

10000 10000

07500 10000 O565

09375

10000

OA219 OBIJ9

09114-1 10000 1))16middot1

Dnll) 11~)4n

091 10000

02373

06318 095 09844

09990 10000

017ilu ()5B9 01l30

09624 09954 09990 10000

01335 DA449 07564

092Y-1

09071 09987

09999

10000

07000 10000 04900 09100 10000

03430 07B40 09730 10000

02401 06517 09163 09919 10000

01681 05282 09369 09692 09976 10000

01176 OA202 07443 09295

09891 09993 10000

00821 03294 06471 08740 09712 09962 09998 10000

06500 06000 10000 10000 04225 03600 08775 08400 10000 10000

02746 02160 07182 06480 09571 09360 10000 10000

01785 01296 05630 004752 08735 08208 0850 09744 10000 10000

01160 00778 OA284 03370 07648 06826 09460 Q9130 09947 09898 10000 10000

00754 00467 03191 02333 06471 05443 08826 08208 09177 09590 09982 09959 10000 10000

00490 00280 02338 01586 05323 004199 08002 07102 09444 09037 09910 09812 09994 09984 10000 10000

05500 10000 03025 07975 10000

01664 05748 09089 10000

00915 03910 07585 09590 10000

00503 02562 05931 08688 09815 10000

00277 01636 04415 07447 09309 09917 10000

001S2 01024 03164 06083 08471 0963 09963 10000

w

o

lit

II III CO

coiD CO

CO

e III

bull

-------00000 ------oo=== ----gtgt~~~f ----===0bbbbS 2b~~~iD~8S b822~~~22S b-----v02SoS~8sectsectsectS~8~~i SSsectS88~iIS~ 8~S8sectsecti~ls sect8S8Siet

III rshym

t litII

Ii

o

E III

raquo o W mIII Z

D

-W

X

o

~

u o n~ o

i o

1 bullo p

p o-

bullo o

bullo o

n o

I a

~ mZ o ~

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

-lt -- 11

f c

s ~ Mbull -i 9 + It -

bull 1

0 ~ ~ 01 s I

D

UI W -O middot Wa-O UI W -O W - bull ---------- --------shy

iii~~~~eee~ee~~g~gg-DOIrDIID ~ CD-W J 0 00 sectbsecti~~~~~~~e~~ei~iii~2~~2~gw w- 1~~~~~~~~~~~wSgg o CD tJOCD~CJ~

~~pppppppppppppppppp--OOOOOOOOOOpOOOOOOOObull ~ ~~~~w-000o~8io~1~~~=~~~~~~w~a88sect118 00 0 g8~1~~ ~~~~~sectS 0 ~-~om~m iS-~- o m~ oww-o~-- m - g---~-o~~w w 0

f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

~pppppppPpP~pppppppp~~ampPpppppppppppppoopp

t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

i~~sect~iiiiiiiiiiiiiiiii~gi~iiiiiiiiiiiiiii

J

C

I

bulle D

e ebull e III

bull ~

bull

Cbull

bull bull ~

po III

~ IIf r-m

~

n 0 I It I I III CI

w

o

lit

II III CO

coiD CO

CO

e III

bull

-------00000 ------oo=== ----gtgt~~~f ----===0bbbbS 2b~~~iD~8S b822~~~22S b-----v02SoS~8sectsectsectS~8~~i SSsectS88~iIS~ 8~S8sectsecti~ls sect8S8Siet

III rshym

t litII

Ii

o

E III

raquo o W mIII Z

D

-W

X

o

~

u o n~ o

i o

1 bullo p

p o-

bullo o

bullo o

n o

I a

~ mZ o ~

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

-lt -- 11

f c

s ~ Mbull -i 9 + It -

bull 1

0 ~ ~ 01 s I

D

UI W -O middot Wa-O UI W -O W - bull ---------- --------shy

iii~~~~eee~ee~~g~gg-DOIrDIID ~ CD-W J 0 00 sectbsecti~~~~~~~e~~ei~iii~2~~2~gw w- 1~~~~~~~~~~~wSgg o CD tJOCD~CJ~

~~pppppppppppppppppp--OOOOOOOOOOpOOOOOOOObull ~ ~~~~w-000o~8io~1~~~=~~~~~~w~a88sect118 00 0 g8~1~~ ~~~~~sectS 0 ~-~om~m iS-~- o m~ oww-o~-- m - g---~-o~~w w 0

f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

~pppppppPpP~pppppppp~~ampPpppppppppppppoopp

t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

i~~sect~iiiiiiiiiiiiiiiii~gi~iiiiiiiiiiiiiii

J

C

I

bulle D

e ebull e III

bull ~

bull

Cbull

bull bull ~

po III

~ IIf r-m

~

n 0 I It I I III CI

o

~

u o n~ o

i o

1 bullo p

p o-

bullo o

bullo o

n o

I a

~ mZ o ~

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

-lt -- 11

f c

s ~ Mbull -i 9 + It -

bull 1

0 ~ ~ 01 s I

D

UI W -O middot Wa-O UI W -O W - bull ---------- --------shy

iii~~~~eee~ee~~g~gg-DOIrDIID ~ CD-W J 0 00 sectbsecti~~~~~~~e~~ei~iii~2~~2~gw w- 1~~~~~~~~~~~wSgg o CD tJOCD~CJ~

~~pppppppppppppppppp--OOOOOOOOOOpOOOOOOOObull ~ ~~~~w-000o~8io~1~~~=~~~~~~w~a88sect118 00 0 g8~1~~ ~~~~~sectS 0 ~-~om~m iS-~- o m~ oww-o~-- m - g---~-o~~w w 0

f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

~pppppppPpP~pppppppp~~ampPpppppppppppppoopp

t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

i~~sect~iiiiiiiiiiiiiiiii~gi~iiiiiiiiiiiiiii

J

C

I

bulle D

e ebull e III

bull ~

bull

Cbull

bull bull ~

po III

~ IIf r-m

~

n 0 I It I I III CI

-lt

III I m

e o

c w _o

= In

Cgt cgt

co mIn Z o

2 0-28shy

co III

X

X

III

c ww o w -o

II o in o raquo m

Z g

o o

o

= o

O Wlaol_ bull

1

obull 1ft

= 1ft

~ o mW 1ft Z

g

= 1ft

X

bull O o- w 0 w -0

II o is

F

F 0shyo

F 0shy

o

o ia III

F

O oshy w _o

o W on

o o

I I m Z o X

~ 5 bull51 -lt$11middotLf T~fflqSo bullbulld iii JIP~a ~ So shyl If middot ~ ri1 ga ~ il i ii

iif Ii c01 c~i

~~- j rbull S

i3 I III8-

lI -So o -l shyt

c31 I

1 l i

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f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

~pppppppppppppppppppp ~ppppppppppppppppppp

i~ m~~w -00~g8~~88~ 8~~I~~~~ie~g8sect8secti~ st~ ~~ shym mbullbull ~m ~o~w 00 gS~~ 8~=~t~~~~~-00 ~pppppppppppppppppppp ~ppppppppppppppppppp -iooooosect0811~~~~e~~~~~~~sect~lolo~ 8 m~~

1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

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t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

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f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

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1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

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t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

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f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

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1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

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t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

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f~w - ~=w-0~8 2~-~ 0~_wiiiii~~~~iisisecti~iiiisect i~~~~~~~eeegggggg~ii0- 0 0 ~ 0 ~~~o~ ~o -~ ~-~CJ- -0

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1g~It~~woti~g sect~~ ~o~ o--ww sect sect 8 mm-~0 1t~ -0 g ~pppppppppppppppppppp

i~~eeeegiiggg~gsect~i1~~I~Eti~~~~~8sect11~~88~ ~~~~~~~ 88 0 8 0~~ m w 88 mo~ ~ ~~ ~-88000 ~~~Pipppppppppppppppp

g~~~~~~~g~~~~~~~~ppgI=w~ ~ie~8888i~11118 8~~ ~w~~~-08888811sect~i~~2~~~8~~~8 8 --~ -w w

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t~~ ~t_15~~~~~~~~~1111111111 sect~~~~i~~isectIIsect01881sect - osect 88 0 ~pppp~pppppppppppppp~pooooooooooooooooooo

1sectE~~~~sect88sect11111181 8~~~~sect~sect111888~8111~~o -~ w -880 8 8 ~~w 888 8

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10

TABLEA7 (Continued)

n mampl 1 ~ 5 6 9 10 II 11 11 I~ IS 16 17 18 19bull 0001 153 1$4 156 159 163 167 171 175 179 183 188 192 197 201 206 211 215 220 22~ 0005 153 156 160 I~ 169 173 178 183 188 193 198 203 208 214 219 224 229 235 2~

17 001 Ul5

1$4 156

158 160

1pound2 165

1pound7 171

In 176

177 182

182 188

187 193

192 199

198 20S

203 211

209 217

214 2ll

220 229

225 235

231 241

236 217

242 253

2~7 259

005 157 163 169 174 180 187 193 199 205 211 218 224 231 237 243 250 256 263 269 010 160 16pound In 179 185 192 9 206 212 219 216 233 239 2 253 260 267 274 281 eo1 171 In 175 178 182 186 190 195 199 204 209 214 218 m ll8 233 238 243 248

18

5 OolS

171 In 174

174 176 119

178 181 I~

183 185 190

188 191 196

193 196 202

198 202 208

203 208 214

209 213 220

214 219 221

219 225 233

225 231 239

230 237 2

216 242 252

2~2

248 258

247 254 265

253 260 271

259 266 278

2 272 284

0 bullbull 5 116 181 188 194 200 201 213 220 221 233 240 247 25~ 260 267 274 281 188 295 bullbull10 178 185 192 199 206 213 220 227 234 241 249 256 263 270 278 285 292 300 307 0001 190 191 194 198 202 206 211 216 220 225 231 236 241 246 251 257 1pound2 268 213 bullbull05 191 194 198 203 208 213 21t 224 230 236 242 248 251 260 265 272 278 lO4 290

19 001 15

19l 193

I9S 198

200 204

206 210

211 216

217 2ll

2ll 229

229 236

235 241

241 24

247 256

251 263

260 269

2 276

213 283

279 290

285 27

292 304

298 310

005 195 201 208 214 221 ll8 23S 242 249 256 263 271 278 285 29l 300 307 314 321 010 198 205 212 219 227 114 142 249 257 2~ 272 lao 188 295 303 311 319 326 334 0001 210 211 214 118 2ll 227 232 237 243 248 253 25 165 270 276 281 287 193 29 0005 211 214 219 22~ 229 235 2~1 2~7 253 259 165 271 278 204 290 297 303 310 316

10 001 DolS

212 213

21 219

221 225

227 231

233 118

119 24S

2~5

251 251 259

258 266

~ 213

271 lO0

278 281

204 294

291 301

298 309

304 316

311 323

318 330

325 33S

005 215 122 229 236 ro 250 258 265 213 280 288 295 303 311 318 326 334 341 ~9 010 218 226 233 241 2~ 251 26S 273 281 18 297 305 313 321 330 338 3 354 362

For n Dr m 1 than 20 the ptll quande w of the Mann-Whi1ney test tistic may be approximated by

w - nN + 1)2 + Vnm(N + 1)12

wIlere Z Is tile pth quantile of I standard normal rondom variable obtained from Table AI and where N m+ n bull The entries in this cable are quantiles w of the Mann-Whltney leSt statistic T given by Equation 511 for selected values Df p Note that P(T lt wpl S p Upper qua

tiles may be found from the equation

w = n(n + m + I) - Wi Critical regions correspond to values of T less than (or greater than) but not equal to the appropriate quantile

t-------- ------- shy

~ m Z CI

gtlt

TABLE A7 Quantiles ofthe Mann-Whitney Test Statistic I~ 15 16 17 18 19 20

10 II 12 m = 2fgt

3 3 3 3 3 3 0001 3 3 3 3 3 4 43 3 33 1 13 3 30005 3 3 5 53 ~ 43 3 3 001 3 3 3 3 3 6 65 5 5 52 4 43 30025 3 3 3 8 86 6 74 5 5 5

6 7005 3 3 3 4 4 10 10 II II7 8 8 8 3 4 4 5 5 5010 7 7 7 76 6 6 66 660001 6 6 6 9 10 109 98 8 87 7 7

0005 6 6 6 6 6 II II II 128 9 9 9 10 108 8001 6 6 6 6 6 13 14 14 15II II 12 12 139 10 10 0025 6 6 6 7 8 16 16 17 1812 13 14 14 15

9 9 10 II II 126 7 7 8005 19 20 21 2215 16 17 17 1812 12 13 14

010 7 8 8 9 10 II 13 14 14 14II II 12 12 12 1310 10 II10 10 10 100001 10 10 16 16 17 17 18 1914 14 1512 13 13

10 10 10 10 II II 120005 19 20 20 2114 15 16 16 17 18 1813 1410 II 12 12 15 16 17001 10 10 22 23 24 2518 19 20 21 22

0025 10 10 II 12 13 14 15 26 27 28 2919 20 21 22 23 2516 17 18005 10 II 12 13 14 15 29 31 32 3323 14 26 27 2820 21 2216 17 18010 II 12 14 15

21 22 23 2318 19 19 20 2117 17 1815 15 15 15 15 160001 15 26 27 28 2921 22 23 23 24 2517 18 19 2015 16 170005 15 15 29 30 31 3214 25 26 27 2818 19 20 21 22 23

22 23 24 25001 15 15 16 17 33 34 35 3627 28 29 30 31

17 18 19 ~I0015 15 16 36 38 39 4129 31 32 34 3527 2822 24 2517 18 20 21005 16 41 43 44 4633 34 36 38 39 010 17 18 20 21 23 24 26 28 29 31

31 32 33 3426 26 27 28 29 3023 24 2521 21 21 210001 21 21 37 38 39 4031 32 33 34 35 21 22 23 24 25 26 27 28 29

0005 21 37 38 40 41 42 4431 33 H 3528 29 30001 21 21 23 24 25 26 44 46 47 4936 38 39 41 4333 3527 28 30 32

32 34 360015 21 23 24 25 48 50 52 5438 39 41 43 45 47

25 27 29 30 39 41005 22 24 53 56 58 6043 45 47 49 51

010 23 25 27 29 31 33 35 37 42 43 44 4536 37 38 39 40

28 28 28 29 30 31 32 34 35 0001 28 48 50 51 5339 41 42 44 45 47

29 30 32 33 35 36 380005 28 28 52 53 55 5743 45 46 48 5040 4135 36 38001 28 29 30 32 33 53 55 57 59 61 6345 47 49 5141 4334 35 37 390015 28 30 32 62 64 66 6850 53 55 57 59

31 33 35 37 40 42 44 46 48 005 29 67 70 72 7552 55 57 60 62 6547 5040 42 45010 30 33 35 37

54 55 57 5846 48 49 51 52 00 36 36 36 61 63 65 67

42 43 4537 38 39 41 48 50 52 54 55 57 59

36 38 39 41 43 44 460805 36 65 67 69 7152 54 56 59 61 6348 5043 44 46001 36 37 39 41 71 73 75 7856 59 61 63 66 68 39 41 43 45 ~7 50 52 54

0015 37 76 78 81 8463 65 68 70 7347 50 52 55 57 60

US 38 40 42 45 (4 67 70 73 76 79 B2 85 88 91

59 6144 47 50 53010 39 42 ~

TABLE A7 (Continued)

m =2 6 7 9 10 II 12 13 I~ 15 16 17 18 19fgt

0001 45 45 45 47 48 ~9 51 53 54 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 49 51 53 55 57 59 62 64 66 68 70 73 75 77 79 B2 001 45 47 49 51 53 55 57 60 62 64 67 69 72 74 77 79 82 84 86 0025 46 48 50 53 56 58 61 63 66 69 72 74 n 80 83 85 88 91 94 005 47 50 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 010 48 51 55 58 61 64 68 71 H n 81 84 87 91 94 98 101 104 108

0001 55 55 56 57 59 61 62 64 66 68 70 73 75 77 79 81 83 85 88 0005 55 56 58 60 62 65 67 69 72 74 n 80 82 85 87 90 93 95 98

10 001 0025

55 56

57 59

59 61

62 64

(4

67 67 70

69 73

72 76

75 79

78 82

80 85

83 B9

86 92

89 95

92 98

94 101

97 104

100 108

103 III

005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III 114 118 010 59 62 66 69 73 n 80 84 88 92 95 99 103 107 110 114 118 122 126

0001 66 66 67 69 71 73 75 77 79 B2 84 87 89 91 94 96 99 101 104 0005 66 67 69 72 H 77 80 83 85 88 91 94 97 100 103 106 109 112 115

II 001 0025

66 67

68 70

71 73

74 76

76 80

79 83

82 86

85 90

89 93

92 97

95 100

98 104

101 107

104 III

108 114

III 118

114 122

117 125

120 129

005 68 72 75 79 83 86 90 94 98 101 105 109 113 117 121 124 128 132 136 010 70 74 78 82 86 90 94 98 103 107 III 115 119 124 128 132 136 140 145

0001 78 78 79 81 83 86 88 91 93 96 98 102 104 106 110 113 116 liB 121 0005 78 80 82 85 88 91 94 97 100 103 106 110 113 116 120 1]3 126 130 133

12 001 0025

78 80

81 83

84 86

87 90

90 93

93 97

96 101

100 IDS

103 108

107 112

110 116

114 120

117 124

121 128

125 III

128 136

132 140

135 1-14

139 148

005 81 84 88 92 96 100 IDS 109 III 117 121 126 130 134 139 H3 H7 IS I 156 010 8l 87 91 96 100 105 109 114 118 123 128 132 137 142 146 151 156 160 165

0001 91 91 9l 95 97 100 103 106 109 112 115 118 121 124 127 130 134 137 140 0005 91 9l 95 99 102 IDS 109 112 116 119 III 126 130 134 137 141 145 149 152

13 001 0025

92 93

94 96

97 100

101 104

104 108

108 112

112 116

115 120

119 125

123 129

127 133

131 137

135 142

139 146

143 151

147 ISS

151 159

ISS 164

159 168

005 94 98 102 107 III 116 120 125 129 13~ 139 143 148 153 157 162 167 172 176 010 96 101 105 110 115 120 125 130 135 140 145 ISO ISS 160 166 171 176 IBI IB6

0001 105 IDS 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 114 128 132 136 140 144 148 152 156 160 164 169 173

14 001 0025

106 107

108 III

112 115

116 119

119 123

123 128

128 132

132 137

136 142

140 146

144 151

149 156

153 161

157 165

162 170

166 175

171 180

175 184

179 189

005 109 113 117 122 127 132 137 142 147 152 157 162 167 172 177 183 18B 193 198 010 110 116 121 126 131 137 142 147 153 158 164 169 175 180 186 191 197 203 208

0001 120 120 122 125 128 133 135 138 1~2 145 149 ISl 157 161 164 168 172 176 180 0005 120 12l 126 129 133 137 141 145 150 154 158 163 167 172 176 181 185 190 194

15 001 0025

121 122

124 126

128 III

132 135

136 140

140 145

145 150

149 ISS

154 160

158 165

163 170

168 175

172 180

177 185

182 191

187 196

191 201

196 206

201 211

005 124 128 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 215 221 010 126 III Il7 143 148 154 160 166 172 178 184 189 195 201 207 213 219 225 231

0001 136 136 139 142 145 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO 155 159 164 168 173 178 182 187 192 197 202 207 211 216

16 001 0025

137 138

140 143

144 148

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

005 140 145 151 156 162 167 173 179 185 191 197 202 208 214 220 226 232 238 244 010 142 148 154 160 166 173 179 185 191 198 204 211 217 223 230 236 243 249 256

20

541 APPENDIX APPENDIX 543 ~

V 1 j

TABLE Ala Quantiles of Spearmans f

TABLE All Quantiles of the Kendall test statistic T = N - Nd bull Quantiles of Kendalls T are given in parentheses Lower quantiles are the negative of

4 08000 08000

n p 0900 0950 0915 0990 0995 0999

the upper quantiles wp = -WI_p5 07000 08000 09000 09000 6 06000 07714 08286 08857 09429 n p 0900 0950 0915 0990 0995 7 05357 06786 07500 08571 08929 09643 8 05000 06190 07143 08095 08571 09286 4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 9 04667 05833 06833 07667 08167 09000 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000)

10 04424 05515 06364 07333 07818 08667 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 1 04182 05273 06091 07000 07455 08364 7 9 (04286) 11 (05238) 13 (06190) 15 (07143) 17 (08095) 12 03986 04965 05804 06713 07203 08112 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 13 03791 04780 05549 06429 06978 07857 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)

-14 03626 04593 05341 06220 06747 07670 10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 15 03500 04429 05179 06000 06500 07464 II 17 (03091) 21 (03818) 25 (0A545) 29 (05273) 31 (05636) 16 03382 04265 05000 05794 06324 07265 12 18 (02727) 24 (03636) 28 (0A242) H (05152) 36 (05455) 11 03260 MII8 04853 05637 06152 07083 Il 22 (02821) 26 (03333) 32 (0A103) 38 (04872) 42 (05285) 18 03148 03994 04696 05480 05975 06904 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 19 03070 03895 04579 05333 05825 06737 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 20 02977 03789 04451 05203 05684 06586 16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 21 02909 03688 04351 05078 05545 06455 17 32 (02353) 40 (01941) 48 (03529) 56 (04118) 62 (04559) 22 02829 03597 04241 04963 05426 06318 18 35 (02288) 43 (02810) 5 I (03333) 61 (03987) 67 (04379) 23 02767 03518 04150 04852 05306 06186 19 37 (02164) 47 (02749f 55 (03216) 65 (03801) 73 (04269) 24 02704 03435 OA061 04748 05200 06070 20 40 (02105) SO (02632) 60 (03158) 70 (03684) 78 (04105) 25 02646 03362 03977 04654 05100 05962 21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 26 02588 03299 03894 04564 05002 05856 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 27 02540 03236 03822 04481 04915 05757 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 28 02490 03175 03749 04401 04828 05660 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 29 02443 03113 03685 04320 04744 05567 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600) lO 02400 03059 03620 OA25 I 04665 05479 26 59 (01815) 75 (02308) 89 (02738) 105 (03231) 115 (03538)

27 61 (01138) 79 (02251) 93 (01650) III (03162) 123 (03504) For n greater than 30 the approximate quandles of p may b obtained from 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386)

29 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) shy4 lO 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287) II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204)

w vn _I

where z Is the pth quantile of a standard normal random variable obtained from Table A I l2 80 (01613) I02 (02056) 120 (02419) 142 (02863) 158 (03185)Sou~CE Adapted from Glasser and Winter (1961) with correctlons with permission from the Biometrika r 33 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) Trustees 34 87 (01551) III (01979) 131 (01335) 155 (02763) 173 (03084)bull The entries In this table are selected quantlles w of the Spmn rtnk correladon coefl1clent p when used I lS 91 (01529) I 15 (01933) 137 (02303) 163 (02739) 179 (03008)as a test statisdc The lower quandles may be obtained from the equation

36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) Iw = -w_) l1 98 (01471) 126 (01892) 150 (02252) 176 (02643) 198 (02943)

The critical region corresponds to values of p smaller than (or greater thn) but not Including the appro- I priate quantile Note that the median of p is O )

44 APPENDIX

rABLE All (Continued) oII 0950 0915 0990 0995 0900 1 o

Ior n IC chon 60 IfIprmdmauo 1111 of T may ba obCllned from

n(n - 1)(lI +5)w 18

APPENDIX 545

TABLE AI2 Quantllas Githa Wilcoxon jigrlltlfjd IfilI~poundgt Yilt Statistic

nn + I) W W WbullbullIU Wu iV11J middotiuJIJ UlIIbull30 W 2

n=-4 o 5 o

183 (01603) 203 (02888) I 1 I (02578) 211 (02848)

f 2198 (02538) 220 (02821) 10 206 (02512) 228 (02780)

21] (027 ) 235 (02729) II 6 11 B

228 (0210) 252 (0266-4) ~ 221 (027) 2 5 (02713)

13 10 236 (02383) 262 (O26-46) -4 13 245 (02367) 271 (02618) 15 16253 (02340) 279 (02581)

20160 (02305) 288 (02553) 268 (02279) 296 (02517) 11 2 277 (01161) 305 (0290) II 2B 285 (02235) 315 (0271) 19 33 29-4 (02217) 32 (02 3) 20 3B 302 (02192) 3l-4 (02 2 ) 11 1II (02173) l-43 (02397)

21 319 (02148) 353 (02377)

13 55328 (02IlO) 362 (02351) 2 61ll6 (02105) 17l (02lll)

3 5 (02087) 381 (02305) 15 69 355 (02075) 391 (0ll85) 16 76 364 (02056) -402 (oll71) 17 amp4

~ 19 W II n ~ w ~

~ ~ W 19 W i I ~ ~ ~ ~

~ ~

W 19 W

10l (0165) 107 (0) 110 (01372)

II114 (0ll90) (0 I l82)

123 (Oll62) 128 (0135l) Il2 (01333) 135 (0130lt4) 1-4 (0130lt4) I (Ol277) ISO (01276) 153 (0129) 15 (012-47) 162 (01222) 168 (01219) 173 (01209) 177 (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (0111)

III (01863) 137 (01amp49) 2 (01821) 146 (01780) 151 (0175) 157 (01739) 162 (01712) 168 (01697) 17] (01671) 179 (01656) 186 (01 6-4 190 (01616) I97 (01608) 20] (01592) 208 (01569) 21 (0155]) 221 (015) 227 (01529) 232 (01506) 240 (0150lt4) 2-45 (01482) 251 (0167) 258 (0158)

155 (02205) 161 (02173) 168 (0215lt4) 17 (02122) 181 (02102) 187 (02071) 19-4 (02OSI) 200 (02020) 207 (02000) 213 (01970) 220 (01 ISO) 228 (01939) 233 (01902) 2 1 (01890) 18 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792) 284 (0171) 291 (01760) 299 (0178) 306 (0ln9)

11 92 2f 101 30 110 31 119

lit I from ch sandard nonnal dlJUlbutlon lIVen by Tabl AI ApprmdmaUO dll of TlIIlIy ba II 129obtalnld from

II 139 w iii 1IVljli(1n==+5) lt 19

lVn(n-l) 15 160 rIdCII reston CO~llpond co will of T ur chut (or Ion thin) INC not Includlnc dIo IfIPropnaUO 172 JllIINIIo N_ chu chi mldlan of T II 0 Quandli for T Ire obcablad by dlvldll dIo quanlllH of T by 3

n 184(n - 1)11 31 196

0UIlCL AltIapced from Tabl I alrr (1974) wlch pnnlillon from dIo audlor 1f 108 0 221 135 tl 148

o o o I 2 4

6 8

10 13

16 10 2 18 33 38 50 56 63 70 77 B5 9-4

102 III III 131 141 152 163 175 187 199 211 115 139 253 167

o o I 3

4 6 9

14 18 22 26 30 35 41 7 53 59 67 7 82 90 99

108 117 127 138 18 160 171 183 196 209 m 136 150 265 180 295

o

3 6 9

II 14 III 11 26 31 36 41 48 51 61 611 76 81 92

101 III 120 131 1 1 152 164 176 188 201 214 219 2-42 257 272 287 303 320

3

)

II I~

J8

~

1i

I (

-fj

- ~ d Iil

It Gf ~J~

10$

II-l 11~

DS 1 6

156 170 182 195

208

211 1J(

251

166 1al 199 31-4

331 349

j

c J

I

i~

b J~

u

) ~

)t)

~~I~

~ I ~t~

r~

u)

1

lOll Illl

DJ 131 I ~ I~ IL_I

I(~

171l

IJI 205 ]I~

m 2-10

liS)

imiddotJ

19$

311

329 )4

)5

3iW

3 5 8

If 14 III 11 21 32 39 44 51 S85 13 112 91

100 110 120 131 143 155 167 100 193 207 22

136 lSI 166 21l) 299 316 334 352 311 390 409

6 9

12 16 20 25 30 36 42

-48 55 63 71 80 09 98

108 119 130 1-41 153 165 179 192 206 220 235 250 266 282 299 ll7 l35 l53 372 391 411

-431

5 75

105 14 18 225 275 33 39 455 525 60 68 765 955 95

105 1155 1265 138 ISO 1625 1755 IB9 203 2175 2325 HB 26-4 2805 2975 315 333 l515 3705 390 410 305 4515

10 15 21 28 l6 -45 55 66 78 91

105 120 136 153 171 190 210 211 253 276 300 325 l51 378 406 35 465 496 528 561 595 610 666 703

7 1 780 820 B61 903

--

bullbull

546 APPENDIX If APPENDIX 547

TAB~EAll (Continued) TABLEAU Quantile of the Kolmogoiov I1 s~tistkmiddot

1(1 + I) OnemiddotSlded Tot P 090 095 0975 099 09)$ fgt = 090 095 0975 099 0995

TwO-SIded Tesl u 263 282 311 337 366 403 29 52 73 916

UIIS WUI WoOII WU Wa11 WUI WLlI W Wau 1

P = 080 090 US 090 II) fi = 080 090 095 098 099 271 297 328 351 38S 50 73 990U 5 1=1 0900 0950 U75 0990 0995 lo = 11 0226 0259 0281 0321 Ol+l 5 292 313 l+f 3n 403 +f2 71 95 5175 1035

1 06804 0716 082 0900 0919 t 0221 025l 0281 0l14 03l7 308 329 362 190 2] 463 92 511 50405 1081 1 0565 0636 0108 0785 0019 r 0216 0247 0275 0307 0330 7 324 36 379 0408 +12 804 51 5040 564 1128 093 0565 062 0689 07)4 ~ 0212 02-42 0269 0301 0323 3-40 363 397 28 46] 505 536 563 5118 1176 5 0+17 0509 0563 0627 0669 ~ 0208 0238 0264 0295 0l17

9 351 381 416 +11 483 521 559 581 6125 IUS 6 010 0-468 0519 0577 0617 ~ 0204 0233 0259 0290 0]11

50 ll 398 435 461 504 550 58l 611 6315 1215 7 0381 0436 0-483 0538 056 j 0100 0229 025-4 0284 0l05 8 0358 010 M5-4 0507 OS4 ttl 0197 0225 0250 0219 0300

For n larpr than SO the pth quantile w of the WUcoxon IIod ranks _ mdltlc may be approlClmlted by w - [n(1I + 1)4] + 9 0]39 0381 0430 0180 0513 1) 019] 0221 0246 0275 0295 zVn(n + I)(ln + 1I24 whero 11 the prh quamUe 01 slandaid normal random rlabla obtained amp-om TaIqAI 10 0323 0369 0409 00457 040) ]JI 0190 0218 0H2 0270 0210 SOURCE Adapted rom Harter and Owen (1910) wIth PIIrmlssl~n from the American Mathematical Society II 0308 0lS2 0391 0137 0460 I 0187 0214 02]8 0266 0285 bull lb entries In this labl ara quandles Wi of the WilcOlllOR slcn8d ranks ten nadltlc 1 at- by equation 513 for eted valshy 12 0296 0338 0315 0-419 OA4~ J 0184 0211 0234 0262 0281 u of s 050 Quanlle w for I gt 050 may b computad frOm die equation Il 0285 0325 0361 0401 OAn 0182 0208 02ll US8 0277

I 0275 031-4 0l9 0]90 OAII3 ~-~ 0179 0205 0221 0254 0273Wi - IIII + I )12 - WI_

15 0266 030-4 0338 0377 00404 J~ 0171 0202 0221 0251 0269 where n(n + 112 Is elven In the nihc hand column In th tablbull Nora rhac 1(1 lt w) s I and 1(1 gt w )s I - If H Is 16 0258 0295 0327 0]66 0311 ~o 0171 099 022 02-41 0265 true Critical regions c~rrpond CO Iues of 1 leu than (or limtill dwt) but not Indudln the appropriate quantile 11 0250 0286 0318 0355 0381 0172 0196 0218 02+1 0262

02+1 0219 0]09 0346 0111 jllJ 0170 0194 0215 0241 0258 19 0231 0211 OlOI 03]7 0161 lY 0168 0191 0213 0238 0255 10 02ll 0265 0291 0]29 0352 10 0165 0189 0210 0235 0252

ApproldmatiOi 107 122 136 152 163 forogt40 Yo Yo Yo Yo Yo

SoURCE Adopad from Table 1 01 Millar (1956) Used wlch peomitsion of he American Statitlcal Association The antlies In this tabla are selceld andl w of lthe I(olmogoroy to taddcs r r+ and r- as dellned by Equation 611 for cwoslded - and by Equations 612 and 613 for onemiddotslded e$[S noeet H It tho loel a if r excood the I - a quantile liven In thIII labIe lb quantlles ace for n s ~o In he twomiddot tailed lt bullbull~ The ocher anII bullbull Ir approximations that are aquallOthce quantilbullbull In most cu A bonor approximation for n gt 10 results If (n + VnIiO)n Is usod Instead of V In the denomlnacor

TABLE A7 Quantiles ofthe Mann-Whitney Test Statistic I~ 15 16 17 18 19 20

10 II 12 m = 2fgt

3 3 3 3 3 3 0001 3 3 3 3 3 4 43 3 33 1 13 3 30005 3 3 5 53 ~ 43 3 3 001 3 3 3 3 3 6 65 5 5 52 4 43 30025 3 3 3 8 86 6 74 5 5 5

6 7005 3 3 3 4 4 10 10 II II7 8 8 8 3 4 4 5 5 5010 7 7 7 76 6 6 66 660001 6 6 6 9 10 109 98 8 87 7 7

0005 6 6 6 6 6 II II II 128 9 9 9 10 108 8001 6 6 6 6 6 13 14 14 15II II 12 12 139 10 10 0025 6 6 6 7 8 16 16 17 1812 13 14 14 15

9 9 10 II II 126 7 7 8005 19 20 21 2215 16 17 17 1812 12 13 14

010 7 8 8 9 10 II 13 14 14 14II II 12 12 12 1310 10 II10 10 10 100001 10 10 16 16 17 17 18 1914 14 1512 13 13

10 10 10 10 II II 120005 19 20 20 2114 15 16 16 17 18 1813 1410 II 12 12 15 16 17001 10 10 22 23 24 2518 19 20 21 22

0025 10 10 II 12 13 14 15 26 27 28 2919 20 21 22 23 2516 17 18005 10 II 12 13 14 15 29 31 32 3323 14 26 27 2820 21 2216 17 18010 II 12 14 15

21 22 23 2318 19 19 20 2117 17 1815 15 15 15 15 160001 15 26 27 28 2921 22 23 23 24 2517 18 19 2015 16 170005 15 15 29 30 31 3214 25 26 27 2818 19 20 21 22 23

22 23 24 25001 15 15 16 17 33 34 35 3627 28 29 30 31

17 18 19 ~I0015 15 16 36 38 39 4129 31 32 34 3527 2822 24 2517 18 20 21005 16 41 43 44 4633 34 36 38 39 010 17 18 20 21 23 24 26 28 29 31

31 32 33 3426 26 27 28 29 3023 24 2521 21 21 210001 21 21 37 38 39 4031 32 33 34 35 21 22 23 24 25 26 27 28 29

0005 21 37 38 40 41 42 4431 33 H 3528 29 30001 21 21 23 24 25 26 44 46 47 4936 38 39 41 4333 3527 28 30 32

32 34 360015 21 23 24 25 48 50 52 5438 39 41 43 45 47

25 27 29 30 39 41005 22 24 53 56 58 6043 45 47 49 51

010 23 25 27 29 31 33 35 37 42 43 44 4536 37 38 39 40

28 28 28 29 30 31 32 34 35 0001 28 48 50 51 5339 41 42 44 45 47

29 30 32 33 35 36 380005 28 28 52 53 55 5743 45 46 48 5040 4135 36 38001 28 29 30 32 33 53 55 57 59 61 6345 47 49 5141 4334 35 37 390015 28 30 32 62 64 66 6850 53 55 57 59

31 33 35 37 40 42 44 46 48 005 29 67 70 72 7552 55 57 60 62 6547 5040 42 45010 30 33 35 37

54 55 57 5846 48 49 51 52 00 36 36 36 61 63 65 67

42 43 4537 38 39 41 48 50 52 54 55 57 59

36 38 39 41 43 44 460805 36 65 67 69 7152 54 56 59 61 6348 5043 44 46001 36 37 39 41 71 73 75 7856 59 61 63 66 68 39 41 43 45 ~7 50 52 54

0015 37 76 78 81 8463 65 68 70 7347 50 52 55 57 60

US 38 40 42 45 (4 67 70 73 76 79 B2 85 88 91

59 6144 47 50 53010 39 42 ~

TABLE A7 (Continued)

m =2 6 7 9 10 II 12 13 I~ 15 16 17 18 19fgt

0001 45 45 45 47 48 ~9 51 53 54 56 58 60 61 63 65 67 69 71 72 0005 45 46 47 49 51 53 55 57 59 62 64 66 68 70 73 75 77 79 B2 001 45 47 49 51 53 55 57 60 62 64 67 69 72 74 77 79 82 84 86 0025 46 48 50 53 56 58 61 63 66 69 72 74 n 80 83 85 88 91 94 005 47 50 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 100 010 48 51 55 58 61 64 68 71 H n 81 84 87 91 94 98 101 104 108

0001 55 55 56 57 59 61 62 64 66 68 70 73 75 77 79 81 83 85 88 0005 55 56 58 60 62 65 67 69 72 74 n 80 82 85 87 90 93 95 98

10 001 0025

55 56

57 59

59 61

62 64

(4

67 67 70

69 73

72 76

75 79

78 82

80 85

83 B9

86 92

89 95

92 98

94 101

97 104

100 108

103 III

005 57 60 63 67 70 73 76 80 83 87 90 93 97 100 104 107 III 114 118 010 59 62 66 69 73 n 80 84 88 92 95 99 103 107 110 114 118 122 126

0001 66 66 67 69 71 73 75 77 79 B2 84 87 89 91 94 96 99 101 104 0005 66 67 69 72 H 77 80 83 85 88 91 94 97 100 103 106 109 112 115

II 001 0025

66 67

68 70

71 73

74 76

76 80

79 83

82 86

85 90

89 93

92 97

95 100

98 104

101 107

104 III

108 114

III 118

114 122

117 125

120 129

005 68 72 75 79 83 86 90 94 98 101 105 109 113 117 121 124 128 132 136 010 70 74 78 82 86 90 94 98 103 107 III 115 119 124 128 132 136 140 145

0001 78 78 79 81 83 86 88 91 93 96 98 102 104 106 110 113 116 liB 121 0005 78 80 82 85 88 91 94 97 100 103 106 110 113 116 120 1]3 126 130 133

12 001 0025

78 80

81 83

84 86

87 90

90 93

93 97

96 101

100 IDS

103 108

107 112

110 116

114 120

117 124

121 128

125 III

128 136

132 140

135 1-14

139 148

005 81 84 88 92 96 100 IDS 109 III 117 121 126 130 134 139 H3 H7 IS I 156 010 8l 87 91 96 100 105 109 114 118 123 128 132 137 142 146 151 156 160 165

0001 91 91 9l 95 97 100 103 106 109 112 115 118 121 124 127 130 134 137 140 0005 91 9l 95 99 102 IDS 109 112 116 119 III 126 130 134 137 141 145 149 152

13 001 0025

92 93

94 96

97 100

101 104

104 108

108 112

112 116

115 120

119 125

123 129

127 133

131 137

135 142

139 146

143 151

147 ISS

151 159

ISS 164

159 168

005 94 98 102 107 III 116 120 125 129 13~ 139 143 148 153 157 162 167 172 176 010 96 101 105 110 115 120 125 130 135 140 145 ISO ISS 160 166 171 176 IBI IB6

0001 105 IDS 107 109 112 115 118 121 125 128 131 135 138 142 145 149 152 156 160 0005 105 107 110 113 117 121 114 128 132 136 140 144 148 152 156 160 164 169 173

14 001 0025

106 107

108 III

112 115

116 119

119 123

123 128

128 132

132 137

136 142

140 146

144 151

149 156

153 161

157 165

162 170

166 175

171 180

175 184

179 189

005 109 113 117 122 127 132 137 142 147 152 157 162 167 172 177 183 18B 193 198 010 110 116 121 126 131 137 142 147 153 158 164 169 175 180 186 191 197 203 208

0001 120 120 122 125 128 133 135 138 1~2 145 149 ISl 157 161 164 168 172 176 180 0005 120 12l 126 129 133 137 141 145 150 154 158 163 167 172 176 181 185 190 194

15 001 0025

121 122

124 126

128 III

132 135

136 140

140 145

145 150

149 ISS

154 160

158 165

163 170

168 175

172 180

177 185

182 191

187 196

191 201

196 206

201 211

005 124 128 III 139 144 149 154 160 165 171 176 182 187 193 198 204 209 215 221 010 126 III Il7 143 148 154 160 166 172 178 184 189 195 201 207 213 219 225 231

0001 136 136 139 142 145 148 152 156 160 164 168 172 176 180 185 189 193 197 202 0005 136 139 142 146 ISO 155 159 164 168 173 178 182 187 192 197 202 207 211 216

16 001 0025

137 138

140 143

144 148

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

005 140 145 151 156 162 167 173 179 185 191 197 202 208 214 220 226 232 238 244 010 142 148 154 160 166 173 179 185 191 198 204 211 217 223 230 236 243 249 256

20

541 APPENDIX APPENDIX 543 ~

V 1 j

TABLE Ala Quantiles of Spearmans f

TABLE All Quantiles of the Kendall test statistic T = N - Nd bull Quantiles of Kendalls T are given in parentheses Lower quantiles are the negative of

4 08000 08000

n p 0900 0950 0915 0990 0995 0999

the upper quantiles wp = -WI_p5 07000 08000 09000 09000 6 06000 07714 08286 08857 09429 n p 0900 0950 0915 0990 0995 7 05357 06786 07500 08571 08929 09643 8 05000 06190 07143 08095 08571 09286 4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 9 04667 05833 06833 07667 08167 09000 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000)

10 04424 05515 06364 07333 07818 08667 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 1 04182 05273 06091 07000 07455 08364 7 9 (04286) 11 (05238) 13 (06190) 15 (07143) 17 (08095) 12 03986 04965 05804 06713 07203 08112 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 13 03791 04780 05549 06429 06978 07857 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)

-14 03626 04593 05341 06220 06747 07670 10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 15 03500 04429 05179 06000 06500 07464 II 17 (03091) 21 (03818) 25 (0A545) 29 (05273) 31 (05636) 16 03382 04265 05000 05794 06324 07265 12 18 (02727) 24 (03636) 28 (0A242) H (05152) 36 (05455) 11 03260 MII8 04853 05637 06152 07083 Il 22 (02821) 26 (03333) 32 (0A103) 38 (04872) 42 (05285) 18 03148 03994 04696 05480 05975 06904 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 19 03070 03895 04579 05333 05825 06737 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 20 02977 03789 04451 05203 05684 06586 16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 21 02909 03688 04351 05078 05545 06455 17 32 (02353) 40 (01941) 48 (03529) 56 (04118) 62 (04559) 22 02829 03597 04241 04963 05426 06318 18 35 (02288) 43 (02810) 5 I (03333) 61 (03987) 67 (04379) 23 02767 03518 04150 04852 05306 06186 19 37 (02164) 47 (02749f 55 (03216) 65 (03801) 73 (04269) 24 02704 03435 OA061 04748 05200 06070 20 40 (02105) SO (02632) 60 (03158) 70 (03684) 78 (04105) 25 02646 03362 03977 04654 05100 05962 21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 26 02588 03299 03894 04564 05002 05856 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 27 02540 03236 03822 04481 04915 05757 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 28 02490 03175 03749 04401 04828 05660 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 29 02443 03113 03685 04320 04744 05567 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600) lO 02400 03059 03620 OA25 I 04665 05479 26 59 (01815) 75 (02308) 89 (02738) 105 (03231) 115 (03538)

27 61 (01138) 79 (02251) 93 (01650) III (03162) 123 (03504) For n greater than 30 the approximate quandles of p may b obtained from 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386)

29 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) shy4 lO 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287) II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204)

w vn _I

where z Is the pth quantile of a standard normal random variable obtained from Table A I l2 80 (01613) I02 (02056) 120 (02419) 142 (02863) 158 (03185)Sou~CE Adapted from Glasser and Winter (1961) with correctlons with permission from the Biometrika r 33 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) Trustees 34 87 (01551) III (01979) 131 (01335) 155 (02763) 173 (03084)bull The entries In this table are selected quantlles w of the Spmn rtnk correladon coefl1clent p when used I lS 91 (01529) I 15 (01933) 137 (02303) 163 (02739) 179 (03008)as a test statisdc The lower quandles may be obtained from the equation

36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) Iw = -w_) l1 98 (01471) 126 (01892) 150 (02252) 176 (02643) 198 (02943)

The critical region corresponds to values of p smaller than (or greater thn) but not Including the appro- I priate quantile Note that the median of p is O )

44 APPENDIX

rABLE All (Continued) oII 0950 0915 0990 0995 0900 1 o

Ior n IC chon 60 IfIprmdmauo 1111 of T may ba obCllned from

n(n - 1)(lI +5)w 18

APPENDIX 545

TABLE AI2 Quantllas Githa Wilcoxon jigrlltlfjd IfilI~poundgt Yilt Statistic

nn + I) W W WbullbullIU Wu iV11J middotiuJIJ UlIIbull30 W 2

n=-4 o 5 o

183 (01603) 203 (02888) I 1 I (02578) 211 (02848)

f 2198 (02538) 220 (02821) 10 206 (02512) 228 (02780)

21] (027 ) 235 (02729) II 6 11 B

228 (0210) 252 (0266-4) ~ 221 (027) 2 5 (02713)

13 10 236 (02383) 262 (O26-46) -4 13 245 (02367) 271 (02618) 15 16253 (02340) 279 (02581)

20160 (02305) 288 (02553) 268 (02279) 296 (02517) 11 2 277 (01161) 305 (0290) II 2B 285 (02235) 315 (0271) 19 33 29-4 (02217) 32 (02 3) 20 3B 302 (02192) 3l-4 (02 2 ) 11 1II (02173) l-43 (02397)

21 319 (02148) 353 (02377)

13 55328 (02IlO) 362 (02351) 2 61ll6 (02105) 17l (02lll)

3 5 (02087) 381 (02305) 15 69 355 (02075) 391 (0ll85) 16 76 364 (02056) -402 (oll71) 17 amp4

~ 19 W II n ~ w ~

~ ~ W 19 W i I ~ ~ ~ ~

~ ~

W 19 W

10l (0165) 107 (0) 110 (01372)

II114 (0ll90) (0 I l82)

123 (Oll62) 128 (0135l) Il2 (01333) 135 (0130lt4) 1-4 (0130lt4) I (Ol277) ISO (01276) 153 (0129) 15 (012-47) 162 (01222) 168 (01219) 173 (01209) 177 (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (0111)

III (01863) 137 (01amp49) 2 (01821) 146 (01780) 151 (0175) 157 (01739) 162 (01712) 168 (01697) 17] (01671) 179 (01656) 186 (01 6-4 190 (01616) I97 (01608) 20] (01592) 208 (01569) 21 (0155]) 221 (015) 227 (01529) 232 (01506) 240 (0150lt4) 2-45 (01482) 251 (0167) 258 (0158)

155 (02205) 161 (02173) 168 (0215lt4) 17 (02122) 181 (02102) 187 (02071) 19-4 (02OSI) 200 (02020) 207 (02000) 213 (01970) 220 (01 ISO) 228 (01939) 233 (01902) 2 1 (01890) 18 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792) 284 (0171) 291 (01760) 299 (0178) 306 (0ln9)

11 92 2f 101 30 110 31 119

lit I from ch sandard nonnal dlJUlbutlon lIVen by Tabl AI ApprmdmaUO dll of TlIIlIy ba II 129obtalnld from

II 139 w iii 1IVljli(1n==+5) lt 19

lVn(n-l) 15 160 rIdCII reston CO~llpond co will of T ur chut (or Ion thin) INC not Includlnc dIo IfIPropnaUO 172 JllIINIIo N_ chu chi mldlan of T II 0 Quandli for T Ire obcablad by dlvldll dIo quanlllH of T by 3

n 184(n - 1)11 31 196

0UIlCL AltIapced from Tabl I alrr (1974) wlch pnnlillon from dIo audlor 1f 108 0 221 135 tl 148

o o o I 2 4

6 8

10 13

16 10 2 18 33 38 50 56 63 70 77 B5 9-4

102 III III 131 141 152 163 175 187 199 211 115 139 253 167

o o I 3

4 6 9

14 18 22 26 30 35 41 7 53 59 67 7 82 90 99

108 117 127 138 18 160 171 183 196 209 m 136 150 265 180 295

o

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II 14 III 11 26 31 36 41 48 51 61 611 76 81 92

101 III 120 131 1 1 152 164 176 188 201 214 219 2-42 257 272 287 303 320

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136 lSI 166 21l) 299 316 334 352 311 390 409

6 9

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-48 55 63 71 80 09 98

108 119 130 1-41 153 165 179 192 206 220 235 250 266 282 299 ll7 l35 l53 372 391 411

-431

5 75

105 14 18 225 275 33 39 455 525 60 68 765 955 95

105 1155 1265 138 ISO 1625 1755 IB9 203 2175 2325 HB 26-4 2805 2975 315 333 l515 3705 390 410 305 4515

10 15 21 28 l6 -45 55 66 78 91

105 120 136 153 171 190 210 211 253 276 300 325 l51 378 406 35 465 496 528 561 595 610 666 703

7 1 780 820 B61 903

--

bullbull

546 APPENDIX If APPENDIX 547

TAB~EAll (Continued) TABLEAU Quantile of the Kolmogoiov I1 s~tistkmiddot

1(1 + I) OnemiddotSlded Tot P 090 095 0975 099 09)$ fgt = 090 095 0975 099 0995

TwO-SIded Tesl u 263 282 311 337 366 403 29 52 73 916

UIIS WUI WoOII WU Wa11 WUI WLlI W Wau 1

P = 080 090 US 090 II) fi = 080 090 095 098 099 271 297 328 351 38S 50 73 990U 5 1=1 0900 0950 U75 0990 0995 lo = 11 0226 0259 0281 0321 Ol+l 5 292 313 l+f 3n 403 +f2 71 95 5175 1035

1 06804 0716 082 0900 0919 t 0221 025l 0281 0l14 03l7 308 329 362 190 2] 463 92 511 50405 1081 1 0565 0636 0108 0785 0019 r 0216 0247 0275 0307 0330 7 324 36 379 0408 +12 804 51 5040 564 1128 093 0565 062 0689 07)4 ~ 0212 02-42 0269 0301 0323 3-40 363 397 28 46] 505 536 563 5118 1176 5 0+17 0509 0563 0627 0669 ~ 0208 0238 0264 0295 0l17

9 351 381 416 +11 483 521 559 581 6125 IUS 6 010 0-468 0519 0577 0617 ~ 0204 0233 0259 0290 0]11

50 ll 398 435 461 504 550 58l 611 6315 1215 7 0381 0436 0-483 0538 056 j 0100 0229 025-4 0284 0l05 8 0358 010 M5-4 0507 OS4 ttl 0197 0225 0250 0219 0300

For n larpr than SO the pth quantile w of the WUcoxon IIod ranks _ mdltlc may be approlClmlted by w - [n(1I + 1)4] + 9 0]39 0381 0430 0180 0513 1) 019] 0221 0246 0275 0295 zVn(n + I)(ln + 1I24 whero 11 the prh quamUe 01 slandaid normal random rlabla obtained amp-om TaIqAI 10 0323 0369 0409 00457 040) ]JI 0190 0218 0H2 0270 0210 SOURCE Adapted rom Harter and Owen (1910) wIth PIIrmlssl~n from the American Mathematical Society II 0308 0lS2 0391 0137 0460 I 0187 0214 02]8 0266 0285 bull lb entries In this labl ara quandles Wi of the WilcOlllOR slcn8d ranks ten nadltlc 1 at- by equation 513 for eted valshy 12 0296 0338 0315 0-419 OA4~ J 0184 0211 0234 0262 0281 u of s 050 Quanlle w for I gt 050 may b computad frOm die equation Il 0285 0325 0361 0401 OAn 0182 0208 02ll US8 0277

I 0275 031-4 0l9 0]90 OAII3 ~-~ 0179 0205 0221 0254 0273Wi - IIII + I )12 - WI_

15 0266 030-4 0338 0377 00404 J~ 0171 0202 0221 0251 0269 where n(n + 112 Is elven In the nihc hand column In th tablbull Nora rhac 1(1 lt w) s I and 1(1 gt w )s I - If H Is 16 0258 0295 0327 0]66 0311 ~o 0171 099 022 02-41 0265 true Critical regions c~rrpond CO Iues of 1 leu than (or limtill dwt) but not Indudln the appropriate quantile 11 0250 0286 0318 0355 0381 0172 0196 0218 02+1 0262

02+1 0219 0]09 0346 0111 jllJ 0170 0194 0215 0241 0258 19 0231 0211 OlOI 03]7 0161 lY 0168 0191 0213 0238 0255 10 02ll 0265 0291 0]29 0352 10 0165 0189 0210 0235 0252

ApproldmatiOi 107 122 136 152 163 forogt40 Yo Yo Yo Yo Yo

SoURCE Adopad from Table 1 01 Millar (1956) Used wlch peomitsion of he American Statitlcal Association The antlies In this tabla are selceld andl w of lthe I(olmogoroy to taddcs r r+ and r- as dellned by Equation 611 for cwoslded - and by Equations 612 and 613 for onemiddotslded e$[S noeet H It tho loel a if r excood the I - a quantile liven In thIII labIe lb quantlles ace for n s ~o In he twomiddot tailed lt bullbull~ The ocher anII bullbull Ir approximations that are aquallOthce quantilbullbull In most cu A bonor approximation for n gt 10 results If (n + VnIiO)n Is usod Instead of V In the denomlnacor

541 APPENDIX APPENDIX 543 ~

V 1 j

TABLE Ala Quantiles of Spearmans f

TABLE All Quantiles of the Kendall test statistic T = N - Nd bull Quantiles of Kendalls T are given in parentheses Lower quantiles are the negative of

4 08000 08000

n p 0900 0950 0915 0990 0995 0999

the upper quantiles wp = -WI_p5 07000 08000 09000 09000 6 06000 07714 08286 08857 09429 n p 0900 0950 0915 0990 0995 7 05357 06786 07500 08571 08929 09643 8 05000 06190 07143 08095 08571 09286 4 4 (06667) 4 (06667) 6 (10000) 6 (10000) 6 (10000) 9 04667 05833 06833 07667 08167 09000 5 6 (06000) 6 (06000) 8 (08000) 8 (08000) 10 (10000)

10 04424 05515 06364 07333 07818 08667 6 7 (04667) 9 (06000) II (07333) II (07333) 13 (08667) 1 04182 05273 06091 07000 07455 08364 7 9 (04286) 11 (05238) 13 (06190) 15 (07143) 17 (08095) 12 03986 04965 05804 06713 07203 08112 8 10 (03571) 14 (05000) 16 (05714) 18 (06429) 20 (07143) 13 03791 04780 05549 06429 06978 07857 9 12 (03333) 16 (04444) 18 (05000) 22 (06111) 24 (06667)

-14 03626 04593 05341 06220 06747 07670 10 IS (03333) 19 (04222) 21 (04667) 25 (05556) 27 (06000) 15 03500 04429 05179 06000 06500 07464 II 17 (03091) 21 (03818) 25 (0A545) 29 (05273) 31 (05636) 16 03382 04265 05000 05794 06324 07265 12 18 (02727) 24 (03636) 28 (0A242) H (05152) 36 (05455) 11 03260 MII8 04853 05637 06152 07083 Il 22 (02821) 26 (03333) 32 (0A103) 38 (04872) 42 (05285) 18 03148 03994 04696 05480 05975 06904 14 23 (02527) 31 (03407) 35 (03846) 41 (04505) 45 (04945) 19 03070 03895 04579 05333 05825 06737 15 27 (02571) 33 (03143) 39 (03714) 47 (04476) 51 (04857) 20 02977 03789 04451 05203 05684 06586 16 28 (02333) 36 (03000) 44 (03667) 50 (04167) 56 (04667) 21 02909 03688 04351 05078 05545 06455 17 32 (02353) 40 (01941) 48 (03529) 56 (04118) 62 (04559) 22 02829 03597 04241 04963 05426 06318 18 35 (02288) 43 (02810) 5 I (03333) 61 (03987) 67 (04379) 23 02767 03518 04150 04852 05306 06186 19 37 (02164) 47 (02749f 55 (03216) 65 (03801) 73 (04269) 24 02704 03435 OA061 04748 05200 06070 20 40 (02105) SO (02632) 60 (03158) 70 (03684) 78 (04105) 25 02646 03362 03977 04654 05100 05962 21 42 (02000) 54 (02571) 64 (03048) 76 (03619) 84 (04000) 26 02588 03299 03894 04564 05002 05856 22 45 (01948) 59 (02554) 69 (02987) 81 (03506) 89 (03853) 27 02540 03236 03822 04481 04915 05757 2l 49 (01937) 63 (02490) 73 (02885) 87 (03439) 97 (03834) 28 02490 03175 03749 04401 04828 05660 24 52 (01884) 66 (02391) 78 (02826) 92 (03333) 102 (03696) 29 02443 03113 03685 04320 04744 05567 25 56 (01867) 70 (02333) 84 (02800) 98 (03267) 108 (03600) lO 02400 03059 03620 OA25 I 04665 05479 26 59 (01815) 75 (02308) 89 (02738) 105 (03231) 115 (03538)

27 61 (01138) 79 (02251) 93 (01650) III (03162) 123 (03504) For n greater than 30 the approximate quandles of p may b obtained from 28 66 (01746) 84 (02222) 98 (02593) 116 (03069) 128 (03386)

29 68 (01675) 88 (02167) 104 (02562) 124 (03054) 136 (03350) shy4 lO 73 (01678) 93 (02138) 109 (02506) 129 (02966) 143 (03287) II 75 (01613) 97 (02086) 115 (02473) 135 (02903) 149 (03204)

w vn _I

where z Is the pth quantile of a standard normal random variable obtained from Table A I l2 80 (01613) I02 (02056) 120 (02419) 142 (02863) 158 (03185)Sou~CE Adapted from Glasser and Winter (1961) with correctlons with permission from the Biometrika r 33 84 (01591) 106 (02008) 126 (02386) 150 (02841) 164 (03106) Trustees 34 87 (01551) III (01979) 131 (01335) 155 (02763) 173 (03084)bull The entries In this table are selected quantlles w of the Spmn rtnk correladon coefl1clent p when used I lS 91 (01529) I 15 (01933) 137 (02303) 163 (02739) 179 (03008)as a test statisdc The lower quandles may be obtained from the equation

36 94 (01492) 120 (01905) 144 (02286) 170 (02698) 188 (02984) Iw = -w_) l1 98 (01471) 126 (01892) 150 (02252) 176 (02643) 198 (02943)

The critical region corresponds to values of p smaller than (or greater thn) but not Including the appro- I priate quantile Note that the median of p is O )

44 APPENDIX

rABLE All (Continued) oII 0950 0915 0990 0995 0900 1 o

Ior n IC chon 60 IfIprmdmauo 1111 of T may ba obCllned from

n(n - 1)(lI +5)w 18

APPENDIX 545

TABLE AI2 Quantllas Githa Wilcoxon jigrlltlfjd IfilI~poundgt Yilt Statistic

nn + I) W W WbullbullIU Wu iV11J middotiuJIJ UlIIbull30 W 2

n=-4 o 5 o

183 (01603) 203 (02888) I 1 I (02578) 211 (02848)

f 2198 (02538) 220 (02821) 10 206 (02512) 228 (02780)

21] (027 ) 235 (02729) II 6 11 B

228 (0210) 252 (0266-4) ~ 221 (027) 2 5 (02713)

13 10 236 (02383) 262 (O26-46) -4 13 245 (02367) 271 (02618) 15 16253 (02340) 279 (02581)

20160 (02305) 288 (02553) 268 (02279) 296 (02517) 11 2 277 (01161) 305 (0290) II 2B 285 (02235) 315 (0271) 19 33 29-4 (02217) 32 (02 3) 20 3B 302 (02192) 3l-4 (02 2 ) 11 1II (02173) l-43 (02397)

21 319 (02148) 353 (02377)

13 55328 (02IlO) 362 (02351) 2 61ll6 (02105) 17l (02lll)

3 5 (02087) 381 (02305) 15 69 355 (02075) 391 (0ll85) 16 76 364 (02056) -402 (oll71) 17 amp4

~ 19 W II n ~ w ~

~ ~ W 19 W i I ~ ~ ~ ~

~ ~

W 19 W

10l (0165) 107 (0) 110 (01372)

II114 (0ll90) (0 I l82)

123 (Oll62) 128 (0135l) Il2 (01333) 135 (0130lt4) 1-4 (0130lt4) I (Ol277) ISO (01276) 153 (0129) 15 (012-47) 162 (01222) 168 (01219) 173 (01209) 177 (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (0111)

III (01863) 137 (01amp49) 2 (01821) 146 (01780) 151 (0175) 157 (01739) 162 (01712) 168 (01697) 17] (01671) 179 (01656) 186 (01 6-4 190 (01616) I97 (01608) 20] (01592) 208 (01569) 21 (0155]) 221 (015) 227 (01529) 232 (01506) 240 (0150lt4) 2-45 (01482) 251 (0167) 258 (0158)

155 (02205) 161 (02173) 168 (0215lt4) 17 (02122) 181 (02102) 187 (02071) 19-4 (02OSI) 200 (02020) 207 (02000) 213 (01970) 220 (01 ISO) 228 (01939) 233 (01902) 2 1 (01890) 18 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792) 284 (0171) 291 (01760) 299 (0178) 306 (0ln9)

11 92 2f 101 30 110 31 119

lit I from ch sandard nonnal dlJUlbutlon lIVen by Tabl AI ApprmdmaUO dll of TlIIlIy ba II 129obtalnld from

II 139 w iii 1IVljli(1n==+5) lt 19

lVn(n-l) 15 160 rIdCII reston CO~llpond co will of T ur chut (or Ion thin) INC not Includlnc dIo IfIPropnaUO 172 JllIINIIo N_ chu chi mldlan of T II 0 Quandli for T Ire obcablad by dlvldll dIo quanlllH of T by 3

n 184(n - 1)11 31 196

0UIlCL AltIapced from Tabl I alrr (1974) wlch pnnlillon from dIo audlor 1f 108 0 221 135 tl 148

o o o I 2 4

6 8

10 13

16 10 2 18 33 38 50 56 63 70 77 B5 9-4

102 III III 131 141 152 163 175 187 199 211 115 139 253 167

o o I 3

4 6 9

14 18 22 26 30 35 41 7 53 59 67 7 82 90 99

108 117 127 138 18 160 171 183 196 209 m 136 150 265 180 295

o

3 6 9

II 14 III 11 26 31 36 41 48 51 61 611 76 81 92

101 III 120 131 1 1 152 164 176 188 201 214 219 2-42 257 272 287 303 320

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If 14 III 11 21 32 39 44 51 S85 13 112 91

100 110 120 131 143 155 167 100 193 207 22

136 lSI 166 21l) 299 316 334 352 311 390 409

6 9

12 16 20 25 30 36 42

-48 55 63 71 80 09 98

108 119 130 1-41 153 165 179 192 206 220 235 250 266 282 299 ll7 l35 l53 372 391 411

-431

5 75

105 14 18 225 275 33 39 455 525 60 68 765 955 95

105 1155 1265 138 ISO 1625 1755 IB9 203 2175 2325 HB 26-4 2805 2975 315 333 l515 3705 390 410 305 4515

10 15 21 28 l6 -45 55 66 78 91

105 120 136 153 171 190 210 211 253 276 300 325 l51 378 406 35 465 496 528 561 595 610 666 703

7 1 780 820 B61 903

--

bullbull

546 APPENDIX If APPENDIX 547

TAB~EAll (Continued) TABLEAU Quantile of the Kolmogoiov I1 s~tistkmiddot

1(1 + I) OnemiddotSlded Tot P 090 095 0975 099 09)$ fgt = 090 095 0975 099 0995

TwO-SIded Tesl u 263 282 311 337 366 403 29 52 73 916

UIIS WUI WoOII WU Wa11 WUI WLlI W Wau 1

P = 080 090 US 090 II) fi = 080 090 095 098 099 271 297 328 351 38S 50 73 990U 5 1=1 0900 0950 U75 0990 0995 lo = 11 0226 0259 0281 0321 Ol+l 5 292 313 l+f 3n 403 +f2 71 95 5175 1035

1 06804 0716 082 0900 0919 t 0221 025l 0281 0l14 03l7 308 329 362 190 2] 463 92 511 50405 1081 1 0565 0636 0108 0785 0019 r 0216 0247 0275 0307 0330 7 324 36 379 0408 +12 804 51 5040 564 1128 093 0565 062 0689 07)4 ~ 0212 02-42 0269 0301 0323 3-40 363 397 28 46] 505 536 563 5118 1176 5 0+17 0509 0563 0627 0669 ~ 0208 0238 0264 0295 0l17

9 351 381 416 +11 483 521 559 581 6125 IUS 6 010 0-468 0519 0577 0617 ~ 0204 0233 0259 0290 0]11

50 ll 398 435 461 504 550 58l 611 6315 1215 7 0381 0436 0-483 0538 056 j 0100 0229 025-4 0284 0l05 8 0358 010 M5-4 0507 OS4 ttl 0197 0225 0250 0219 0300

For n larpr than SO the pth quantile w of the WUcoxon IIod ranks _ mdltlc may be approlClmlted by w - [n(1I + 1)4] + 9 0]39 0381 0430 0180 0513 1) 019] 0221 0246 0275 0295 zVn(n + I)(ln + 1I24 whero 11 the prh quamUe 01 slandaid normal random rlabla obtained amp-om TaIqAI 10 0323 0369 0409 00457 040) ]JI 0190 0218 0H2 0270 0210 SOURCE Adapted rom Harter and Owen (1910) wIth PIIrmlssl~n from the American Mathematical Society II 0308 0lS2 0391 0137 0460 I 0187 0214 02]8 0266 0285 bull lb entries In this labl ara quandles Wi of the WilcOlllOR slcn8d ranks ten nadltlc 1 at- by equation 513 for eted valshy 12 0296 0338 0315 0-419 OA4~ J 0184 0211 0234 0262 0281 u of s 050 Quanlle w for I gt 050 may b computad frOm die equation Il 0285 0325 0361 0401 OAn 0182 0208 02ll US8 0277

I 0275 031-4 0l9 0]90 OAII3 ~-~ 0179 0205 0221 0254 0273Wi - IIII + I )12 - WI_

15 0266 030-4 0338 0377 00404 J~ 0171 0202 0221 0251 0269 where n(n + 112 Is elven In the nihc hand column In th tablbull Nora rhac 1(1 lt w) s I and 1(1 gt w )s I - If H Is 16 0258 0295 0327 0]66 0311 ~o 0171 099 022 02-41 0265 true Critical regions c~rrpond CO Iues of 1 leu than (or limtill dwt) but not Indudln the appropriate quantile 11 0250 0286 0318 0355 0381 0172 0196 0218 02+1 0262

02+1 0219 0]09 0346 0111 jllJ 0170 0194 0215 0241 0258 19 0231 0211 OlOI 03]7 0161 lY 0168 0191 0213 0238 0255 10 02ll 0265 0291 0]29 0352 10 0165 0189 0210 0235 0252

ApproldmatiOi 107 122 136 152 163 forogt40 Yo Yo Yo Yo Yo

SoURCE Adopad from Table 1 01 Millar (1956) Used wlch peomitsion of he American Statitlcal Association The antlies In this tabla are selceld andl w of lthe I(olmogoroy to taddcs r r+ and r- as dellned by Equation 611 for cwoslded - and by Equations 612 and 613 for onemiddotslded e$[S noeet H It tho loel a if r excood the I - a quantile liven In thIII labIe lb quantlles ace for n s ~o In he twomiddot tailed lt bullbull~ The ocher anII bullbull Ir approximations that are aquallOthce quantilbullbull In most cu A bonor approximation for n gt 10 results If (n + VnIiO)n Is usod Instead of V In the denomlnacor

44 APPENDIX

rABLE All (Continued) oII 0950 0915 0990 0995 0900 1 o

Ior n IC chon 60 IfIprmdmauo 1111 of T may ba obCllned from

n(n - 1)(lI +5)w 18

APPENDIX 545

TABLE AI2 Quantllas Githa Wilcoxon jigrlltlfjd IfilI~poundgt Yilt Statistic

nn + I) W W WbullbullIU Wu iV11J middotiuJIJ UlIIbull30 W 2

n=-4 o 5 o

183 (01603) 203 (02888) I 1 I (02578) 211 (02848)

f 2198 (02538) 220 (02821) 10 206 (02512) 228 (02780)

21] (027 ) 235 (02729) II 6 11 B

228 (0210) 252 (0266-4) ~ 221 (027) 2 5 (02713)

13 10 236 (02383) 262 (O26-46) -4 13 245 (02367) 271 (02618) 15 16253 (02340) 279 (02581)

20160 (02305) 288 (02553) 268 (02279) 296 (02517) 11 2 277 (01161) 305 (0290) II 2B 285 (02235) 315 (0271) 19 33 29-4 (02217) 32 (02 3) 20 3B 302 (02192) 3l-4 (02 2 ) 11 1II (02173) l-43 (02397)

21 319 (02148) 353 (02377)

13 55328 (02IlO) 362 (02351) 2 61ll6 (02105) 17l (02lll)

3 5 (02087) 381 (02305) 15 69 355 (02075) 391 (0ll85) 16 76 364 (02056) -402 (oll71) 17 amp4

~ 19 W II n ~ w ~

~ ~ W 19 W i I ~ ~ ~ ~

~ ~

W 19 W

10l (0165) 107 (0) 110 (01372)

II114 (0ll90) (0 I l82)

123 (Oll62) 128 (0135l) Il2 (01333) 135 (0130lt4) 1-4 (0130lt4) I (Ol277) ISO (01276) 153 (0129) 15 (012-47) 162 (01222) 168 (01219) 173 (01209) 177 (01192) 182 (01182) 186 (01165) 191 (01155) 197 (01151) 202 (0111)

III (01863) 137 (01amp49) 2 (01821) 146 (01780) 151 (0175) 157 (01739) 162 (01712) 168 (01697) 17] (01671) 179 (01656) 186 (01 6-4 190 (01616) I97 (01608) 20] (01592) 208 (01569) 21 (0155]) 221 (015) 227 (01529) 232 (01506) 240 (0150lt4) 2-45 (01482) 251 (0167) 258 (0158)

155 (02205) 161 (02173) 168 (0215lt4) 17 (02122) 181 (02102) 187 (02071) 19-4 (02OSI) 200 (02020) 207 (02000) 213 (01970) 220 (01 ISO) 228 (01939) 233 (01902) 2 1 (01890) 18 (01870) 256 (01858) 263 (01838) 269 (01811) 276 (01792) 284 (0171) 291 (01760) 299 (0178) 306 (0ln9)

11 92 2f 101 30 110 31 119

lit I from ch sandard nonnal dlJUlbutlon lIVen by Tabl AI ApprmdmaUO dll of TlIIlIy ba II 129obtalnld from

II 139 w iii 1IVljli(1n==+5) lt 19

lVn(n-l) 15 160 rIdCII reston CO~llpond co will of T ur chut (or Ion thin) INC not Includlnc dIo IfIPropnaUO 172 JllIINIIo N_ chu chi mldlan of T II 0 Quandli for T Ire obcablad by dlvldll dIo quanlllH of T by 3

n 184(n - 1)11 31 196

0UIlCL AltIapced from Tabl I alrr (1974) wlch pnnlillon from dIo audlor 1f 108 0 221 135 tl 148

o o o I 2 4

6 8

10 13

16 10 2 18 33 38 50 56 63 70 77 B5 9-4

102 III III 131 141 152 163 175 187 199 211 115 139 253 167

o o I 3

4 6 9

14 18 22 26 30 35 41 7 53 59 67 7 82 90 99

108 117 127 138 18 160 171 183 196 209 m 136 150 265 180 295

o

3 6 9

II 14 III 11 26 31 36 41 48 51 61 611 76 81 92

101 III 120 131 1 1 152 164 176 188 201 214 219 2-42 257 272 287 303 320

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If 14 III 11 21 32 39 44 51 S85 13 112 91

100 110 120 131 143 155 167 100 193 207 22

136 lSI 166 21l) 299 316 334 352 311 390 409

6 9

12 16 20 25 30 36 42

-48 55 63 71 80 09 98

108 119 130 1-41 153 165 179 192 206 220 235 250 266 282 299 ll7 l35 l53 372 391 411

-431

5 75

105 14 18 225 275 33 39 455 525 60 68 765 955 95

105 1155 1265 138 ISO 1625 1755 IB9 203 2175 2325 HB 26-4 2805 2975 315 333 l515 3705 390 410 305 4515

10 15 21 28 l6 -45 55 66 78 91

105 120 136 153 171 190 210 211 253 276 300 325 l51 378 406 35 465 496 528 561 595 610 666 703

7 1 780 820 B61 903

--

bullbull

546 APPENDIX If APPENDIX 547

TAB~EAll (Continued) TABLEAU Quantile of the Kolmogoiov I1 s~tistkmiddot

1(1 + I) OnemiddotSlded Tot P 090 095 0975 099 09)$ fgt = 090 095 0975 099 0995

TwO-SIded Tesl u 263 282 311 337 366 403 29 52 73 916

UIIS WUI WoOII WU Wa11 WUI WLlI W Wau 1

P = 080 090 US 090 II) fi = 080 090 095 098 099 271 297 328 351 38S 50 73 990U 5 1=1 0900 0950 U75 0990 0995 lo = 11 0226 0259 0281 0321 Ol+l 5 292 313 l+f 3n 403 +f2 71 95 5175 1035

1 06804 0716 082 0900 0919 t 0221 025l 0281 0l14 03l7 308 329 362 190 2] 463 92 511 50405 1081 1 0565 0636 0108 0785 0019 r 0216 0247 0275 0307 0330 7 324 36 379 0408 +12 804 51 5040 564 1128 093 0565 062 0689 07)4 ~ 0212 02-42 0269 0301 0323 3-40 363 397 28 46] 505 536 563 5118 1176 5 0+17 0509 0563 0627 0669 ~ 0208 0238 0264 0295 0l17

9 351 381 416 +11 483 521 559 581 6125 IUS 6 010 0-468 0519 0577 0617 ~ 0204 0233 0259 0290 0]11

50 ll 398 435 461 504 550 58l 611 6315 1215 7 0381 0436 0-483 0538 056 j 0100 0229 025-4 0284 0l05 8 0358 010 M5-4 0507 OS4 ttl 0197 0225 0250 0219 0300

For n larpr than SO the pth quantile w of the WUcoxon IIod ranks _ mdltlc may be approlClmlted by w - [n(1I + 1)4] + 9 0]39 0381 0430 0180 0513 1) 019] 0221 0246 0275 0295 zVn(n + I)(ln + 1I24 whero 11 the prh quamUe 01 slandaid normal random rlabla obtained amp-om TaIqAI 10 0323 0369 0409 00457 040) ]JI 0190 0218 0H2 0270 0210 SOURCE Adapted rom Harter and Owen (1910) wIth PIIrmlssl~n from the American Mathematical Society II 0308 0lS2 0391 0137 0460 I 0187 0214 02]8 0266 0285 bull lb entries In this labl ara quandles Wi of the WilcOlllOR slcn8d ranks ten nadltlc 1 at- by equation 513 for eted valshy 12 0296 0338 0315 0-419 OA4~ J 0184 0211 0234 0262 0281 u of s 050 Quanlle w for I gt 050 may b computad frOm die equation Il 0285 0325 0361 0401 OAn 0182 0208 02ll US8 0277

I 0275 031-4 0l9 0]90 OAII3 ~-~ 0179 0205 0221 0254 0273Wi - IIII + I )12 - WI_

15 0266 030-4 0338 0377 00404 J~ 0171 0202 0221 0251 0269 where n(n + 112 Is elven In the nihc hand column In th tablbull Nora rhac 1(1 lt w) s I and 1(1 gt w )s I - If H Is 16 0258 0295 0327 0]66 0311 ~o 0171 099 022 02-41 0265 true Critical regions c~rrpond CO Iues of 1 leu than (or limtill dwt) but not Indudln the appropriate quantile 11 0250 0286 0318 0355 0381 0172 0196 0218 02+1 0262

02+1 0219 0]09 0346 0111 jllJ 0170 0194 0215 0241 0258 19 0231 0211 OlOI 03]7 0161 lY 0168 0191 0213 0238 0255 10 02ll 0265 0291 0]29 0352 10 0165 0189 0210 0235 0252

ApproldmatiOi 107 122 136 152 163 forogt40 Yo Yo Yo Yo Yo

SoURCE Adopad from Table 1 01 Millar (1956) Used wlch peomitsion of he American Statitlcal Association The antlies In this tabla are selceld andl w of lthe I(olmogoroy to taddcs r r+ and r- as dellned by Equation 611 for cwoslded - and by Equations 612 and 613 for onemiddotslded e$[S noeet H It tho loel a if r excood the I - a quantile liven In thIII labIe lb quantlles ace for n s ~o In he twomiddot tailed lt bullbull~ The ocher anII bullbull Ir approximations that are aquallOthce quantilbullbull In most cu A bonor approximation for n gt 10 results If (n + VnIiO)n Is usod Instead of V In the denomlnacor

--

bullbull

546 APPENDIX If APPENDIX 547

TAB~EAll (Continued) TABLEAU Quantile of the Kolmogoiov I1 s~tistkmiddot

1(1 + I) OnemiddotSlded Tot P 090 095 0975 099 09)$ fgt = 090 095 0975 099 0995

TwO-SIded Tesl u 263 282 311 337 366 403 29 52 73 916

UIIS WUI WoOII WU Wa11 WUI WLlI W Wau 1

P = 080 090 US 090 II) fi = 080 090 095 098 099 271 297 328 351 38S 50 73 990U 5 1=1 0900 0950 U75 0990 0995 lo = 11 0226 0259 0281 0321 Ol+l 5 292 313 l+f 3n 403 +f2 71 95 5175 1035

1 06804 0716 082 0900 0919 t 0221 025l 0281 0l14 03l7 308 329 362 190 2] 463 92 511 50405 1081 1 0565 0636 0108 0785 0019 r 0216 0247 0275 0307 0330 7 324 36 379 0408 +12 804 51 5040 564 1128 093 0565 062 0689 07)4 ~ 0212 02-42 0269 0301 0323 3-40 363 397 28 46] 505 536 563 5118 1176 5 0+17 0509 0563 0627 0669 ~ 0208 0238 0264 0295 0l17

9 351 381 416 +11 483 521 559 581 6125 IUS 6 010 0-468 0519 0577 0617 ~ 0204 0233 0259 0290 0]11

50 ll 398 435 461 504 550 58l 611 6315 1215 7 0381 0436 0-483 0538 056 j 0100 0229 025-4 0284 0l05 8 0358 010 M5-4 0507 OS4 ttl 0197 0225 0250 0219 0300

For n larpr than SO the pth quantile w of the WUcoxon IIod ranks _ mdltlc may be approlClmlted by w - [n(1I + 1)4] + 9 0]39 0381 0430 0180 0513 1) 019] 0221 0246 0275 0295 zVn(n + I)(ln + 1I24 whero 11 the prh quamUe 01 slandaid normal random rlabla obtained amp-om TaIqAI 10 0323 0369 0409 00457 040) ]JI 0190 0218 0H2 0270 0210 SOURCE Adapted rom Harter and Owen (1910) wIth PIIrmlssl~n from the American Mathematical Society II 0308 0lS2 0391 0137 0460 I 0187 0214 02]8 0266 0285 bull lb entries In this labl ara quandles Wi of the WilcOlllOR slcn8d ranks ten nadltlc 1 at- by equation 513 for eted valshy 12 0296 0338 0315 0-419 OA4~ J 0184 0211 0234 0262 0281 u of s 050 Quanlle w for I gt 050 may b computad frOm die equation Il 0285 0325 0361 0401 OAn 0182 0208 02ll US8 0277

I 0275 031-4 0l9 0]90 OAII3 ~-~ 0179 0205 0221 0254 0273Wi - IIII + I )12 - WI_

15 0266 030-4 0338 0377 00404 J~ 0171 0202 0221 0251 0269 where n(n + 112 Is elven In the nihc hand column In th tablbull Nora rhac 1(1 lt w) s I and 1(1 gt w )s I - If H Is 16 0258 0295 0327 0]66 0311 ~o 0171 099 022 02-41 0265 true Critical regions c~rrpond CO Iues of 1 leu than (or limtill dwt) but not Indudln the appropriate quantile 11 0250 0286 0318 0355 0381 0172 0196 0218 02+1 0262

02+1 0219 0]09 0346 0111 jllJ 0170 0194 0215 0241 0258 19 0231 0211 OlOI 03]7 0161 lY 0168 0191 0213 0238 0255 10 02ll 0265 0291 0]29 0352 10 0165 0189 0210 0235 0252

ApproldmatiOi 107 122 136 152 163 forogt40 Yo Yo Yo Yo Yo

SoURCE Adopad from Table 1 01 Millar (1956) Used wlch peomitsion of he American Statitlcal Association The antlies In this tabla are selceld andl w of lthe I(olmogoroy to taddcs r r+ and r- as dellned by Equation 611 for cwoslded - and by Equations 612 and 613 for onemiddotslded e$[S noeet H It tho loel a if r excood the I - a quantile liven In thIII labIe lb quantlles ace for n s ~o In he twomiddot tailed lt bullbull~ The ocher anII bullbull Ir approximations that are aquallOthce quantilbullbull In most cu A bonor approximation for n gt 10 results If (n + VnIiO)n Is usod Instead of V In the denomlnacor