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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Chap 7-1Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-1
Chapter 7
a!pling and a!pling "istributions
Basic Business Statistics for12th Edition
8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Chap 7-2Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-2
#earning $b%ecti&es
In this chapter, you learn:
'o distinguish bet(een di))erent sa!pling
!ethods
'he concept o) the sa!pling distribution
'o co!pute probabilities related to the sa!ple
!ean and the sa!ple proportion
'he i!portance o) the Central #i!it 'heore!
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Chap 7-*Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*
+hy a!ple
electing a sa!ple is less ti!e-consu!ing less costly than selecting e&ery ite! in
the population census/.
n analysis o) a sa!ple is lesscu!berso!e and !ore practical than ananalysis o) the entire population.
"C$
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Chap 7-Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-
a!pling Process 3egins +ith
a!pling 4ra!e
'he sa!pling )ra!e is a listing o) ite!s that
!a5e up the population
4ra!es are data sources such as populationlists, directories, or !aps
Inaccurate or biased results can result i) a
)ra!e e6cludes certain portions o) the
population sing di))erent )ra!es to generate data can
lead to dissi!ilar conclusions
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Chap 7-8Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-8
'ypes o) a!ples
Samples
Non-ProbabilitySamples
Judgment
Probability Samples
Simple
Random
Systematic
Stratified
Cluster
Convenience
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Chap 7-9Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-9
'ypes o) a!ples:
;onprobability a!ple
In a nonprobability sa!ple, ite!s included are
chosen (ithout regard to their probability o)
occurrence. In convenience sampling, ite!s are selected based
only on the )act that they are easy, ine6pensi&e, or
con&enient to sa!ple. In a judgment sample, you get the opinions o) pre-
selected e6perts in the sub%ect !atter.
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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'ypes o) a!ples:
Probability a!ple
In a probability sample, ite!s in the
sa!ple are chosen on the basis o) 5no(n
probabilities.
Probability Samples
SimpleRandom
Systematic Stratified Cluster
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Probability a!ple:
i!ple =ando! a!ple
E&ery indi&idual or ite! )ro! the )ra!e has ane>ual chance o) being selected.
election !ay be (ith replace!ent selectedindi&idual is returned to )ra!e )or possiblereselection/ or (ithout replace!ent selectedindi&idual isn?t returned to the )ra!e/.
a!ples obtained )ro! table o) rando!nu!bers or co!puter rando! nu!bergenerators.
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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electing a i!ple =ando! a!ple
sing =ando! ;u!ber 'able
a!pling 4ra!e 4or
Population +ith <80
Ite!s
Ite! ;a!e Ite! A3e& =. 001
lan B. 002
. .
. .
. .
. .
oann P. <@
Paul 4. <80
Portion Of A Random umber !able
@2<0 <<@2 *877@ 002<* <119* 07278
11100 02*0 12<90 79@7 @99 <@*@
0@<@* 2*@@7 200< @20 <<<72 0<01
!he "irst # Items in a simple
random sample
Ite! A @2
Ite! A <0<
Ite! A <@2 -- does not e6ist so ignore
Ite! A *8
Ite! A 77@
Ite! A 002
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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"ecide on sa!ple siDe: n
"i&ide )ra!e o) ; indi&iduals into groups o) 5
indi&iduals: 5;Fn
=ando!ly select one indi&idual )ro! the 1st
group
elect e&ery 5th indi&idual therea)ter
Probability a!ple:
yste!atic a!ple
$ %&
n $ %
' $ (&
"irst )roup
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Probability a!ple:
trati)ied a!ple
"i&ide population into t(o or !ore subgroups called strata/
according to so!e co!!on characteristic
si!ple rando! sa!ple is selected )ro! each subgroup, (ith sa!ple
siDes proportional to strata siDes
a!ples )ro! subgroups are co!bined into one
'his is a co!!on techni>ue (hen sa!pling population o) &oters,
strati)ying across racial or socio-econo!ic lines.
Population
*ivided
into %
strata
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Probability a!ple
Cluster a!ple
Population is di&ided into se&eral Gclusters, each representati&e o)
the population
si!ple rando! sa!ple o) clusters is selected
ll ite!s in the selected clusters can be used, or ite!s can be
chosen )ro! a cluster using another probability sa!pling techni>ue co!!on application o) cluster sa!pling in&ol&es election e6it polls,
(here certain election districts are selected and sa!pled.
Population
divided into
(+ clusters Randomly selected
clusters for sample
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Probability a!ple:
Co!paring a!pling ethods
i!ple rando! sa!ple and yste!atic sa!ple i!ple to use ay not be a good representation o) the population?s
underlying characteristics trati)ied sa!ple
Ensures representation o) indi&iduals across the entire
population
Cluster sa!ple ore cost e))ecti&e #ess e))icient need larger sa!ple to ac>uire the sa!e
le&el o) precision/
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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E&aluating ur&ey +orthiness
+hat is the purpose o) the sur&ey
Is the sur&ey based on a probability sa!ple
Co&erage error J appropriate )ra!e ;onresponse error J )ollo( up
easure!ent error J good >uestions elicit good
responses
a!pling error J al(ays e6ists
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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'ypes o) ur&ey Errors
Co&erage error or selection bias E6ists i) so!e groups are e6cluded )ro! the )ra!e and ha&e
no chance o) being selected
;onresponse error or bias People (ho do not respond !ay be di))erent )ro! those (ho
do respond
a!pling error
ariation )ro! sa!ple to sa!ple (ill al(ays e6ist easure!ent error
"ue to (ea5nesses in >uestion design, respondent error, and
inter&ie(er?s e))ects on the respondent GHa(thorne e))ect/
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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'ypes o) ur&ey Errors
Co&erage error
;on response error
a!pling error
easure!ent error
-.cluded from
frame
"ollo/ up on
nonresponses
Random
differences fromsample to sample
0ad or leading
1uestion
(continued)
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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a!pling "istributions
sa!pling distribution is a distribution o) all o) the
possible &alues o) a sa!ple statistic )or a gi&en siDe
sa!ple selected )ro! a population.
4or e6a!ple, suppose you sa!ple 80 students )ro! your
college regarding their !ean KP. I) you obtained !any
di))erent sa!ples o) 80, you (ill co!pute a di))erent
!ean )or each sa!ple. +e are interested in thedistribution o) the !ean KP )ro! all possible sa!ples o)
80 students.
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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"e&eloping aa!pling "istribution
Assume there is a population 2
Population siDe ;
=ando! &ariable, B,is age o) indi&iduals
alues o) B: 1<, 20,
22, 2 years/
A 0 C*
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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3
4
(
& (5 4& 44 4%
A 0 C *
ni)or! "istribution
P6/
6
(continued)
u!!ary easures )or the Population "istribution:
"e&eloping aa!pling "istribution
21
222201<;
BL i
=+++
=
=∑
2.2*9;
L/BM
2i
=−
= ∑
"C$
8/9/2019 Chapter 7:Sampling and Sampling Distributions
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19 possible sa!ples
sa!pling (ith
replace!ent/
;o( consider all possible sa!ples o) siDe n2
(st 4nd Observation
Obs (5 4& 44 4%
(5 1< 1@ 20 21
4& 1@ 20 21 22
44 20 21 22 2*
4% 21 22 2* 2
(continued)
"e&eloping aa!pling "istribution
19 a!ple
eans(st
Obs4nd Observation
1< 20 22 2
1< 1<,1< 1<,20 1<,22 1<,2
20 20,1< 20,20 20,22 20,2
22 22,1< 22,20 22,22 22,2
2 2,1< 2,20 2,22 2,2
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8/9/2019 Chapter 7:Sampling and Sampling Distributions
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a!pling "istribution o) ll a!ple eans
1< 1@ 20 21 22 2* 20
.1
.2
.*
P68
a!ple eans
"istribution19 a!ple eans
9
"e&eloping aa!pling "istribution
(continued)
no longer uni)or!/
9
"C$
8/9/2019 Chapter 7:Sampling and Sampling Distributions
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u!!ary easures o) this a!pling "istribution:
"e&eloping aa!pling "istribution
(continued)
21
16
24191918μ
X
1.5816
21)-(2421)-(1921)-(18σ
222
X ==
"C$
;ote: Here (e di&ide by 19 because there are 19
di))erent sa!ples o) siDe 2.
8/9/2019 Chapter 7:Sampling and Sampling Distributions
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Co!paring the Population "istributionto the a!ple eans "istribution
1< 1@ 20 21 22 2* 2&
(
4
3PB/
(5 4& 44 4%
A 0 C *
&
(
4
3
Population;
PB/
B N
1.8<M 21LBB
==2.2*9M 21L ==
a!ple eans "istributionn 2
N
"C$
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Chap 7-2Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-2
a!pling "istribution o) 'he ean:
tandard Error o) the ean
"i))erent sa!ples o) the sa!e siDe )ro! the sa!e
population (ill yield di))erent sa!ple !eans
!easure o) the &ariability in the !ean )ro! sa!ple to
sa!ple is gi&en by the tandard Error o) the ean:'his assu!es that sa!pling is (ith replace!ent or
sa!pling is (ithout replace!ent )ro! an in)inite population/
;ote that the standard error o) the !ean decreases as
the sa!ple siDe increases
n
MMB =
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Chap 7-28Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-28
a!pling "istribution o) 'he ean:
I) the Population is ;or!al
I) a population is nor!al (ith !ean L and
standard de&iation M, the sa!pling distribution
o) is also nor!ally distributed (ith
and
B
LLB =
n
MM
B =
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Chap 7-29Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-29
O-&alue )or a!pling "istributiono) the ean
O-&alue )or the sa!pling distribution o) :
(here: sa!ple !ean
population !ean
population standard de&iation
n sa!ple siDe
BL
M
n
M
L/B
M
/LB
OB
B −=
−
=
B
"C$
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Chap 7-27Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-27
;or!al Population
"istribution
;or!al a!pling
"istribution
has the sa!e !ean/
a!pling "istribution Properties
i.e. is unbiased /66
6
LL6 =
L
6L
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Chap 7-2<Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-2<
a!pling "istribution Properties
s n increases,
decreasesarger
sample si;e
<maller
sample si;e
6
(continued)
6M
L
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Chap 7-2@Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-2@
"eter!ining n Inter&al Including
4i6ed Proportion o) the a!ple eans
4ind a sy!!etrically distributed inter&al around
that (ill include @8Q o) the sa!ple !eans (hen
*9<, M 18, and n 28.
ince the inter&al contains @8Q o) the sa!ple !eans
8Q o) the sa!ple !eans (ill be outside the inter&al. ince the inter&al is sy!!etric 2.8Q (ill be abo&e
the upper li!it and 2.8Q (ill be belo( the lo(er li!it. 4ro! the standardiDed nor!al table, the O score (ith
2.8Q 0.0280/ belo( it is -1.@9 and the O score (ith
2.8Q 0.0280/ abo&e it is 1.@9.
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Chap 7-*0Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*0
"eter!ining n Inter&al Including
4i6ed Proportion o) the a!ple eans
Calculating the lo(er li!it o) the inter&al
Calculating the upper li!it o) the inter&al
@8Q o) all sa!ple !eans o) sa!ple siDe 28 are
bet(een *92.12 and *7*.<<
12.36225
15)96.1(368 =−+=+=
n Z X L
σ μ
(continued)
88.373
25
15)96.1(368 =+=+=
n Z X U
σ μ
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Chap 7-*1Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*1
a!pling "istribution o) 'he ean:
I) the Population is not ;or!al
+e can apply the Central #i!it 'heore!:
E&en i) the population is not nor!al, Rsa!ple !eans )ro! the population (ill be
appro6i!ately nor!al as long as the sa!ple siDe islarge enough.
Properties o) the sa!pling distribution:
and
LL6 =n
MM6 =
"C$
l li "i t ib ti
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Chap 7-*2Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*2
Population "istribution
a!pling "istributionbeco!es nor!al as n increases/
Central 'endency
ariation
6
6
argersample
si;e
<maller
sample si;e
a!ple ean a!pling "istribution:
I) the Population is not ;or!al(continued)
a!pling distribution
properties:
LL6 =
n
M
M6 =
6L
L
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Chap 7-**Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-**
Ho( #arge is #arge Enough
4or !ost distributions, n S *0 (ill gi&e a
sa!pling distribution that is nearly nor!al
4or )airly sy!!etric distributions, n S 18
4or nor!al population distributions, the
sa!pling distribution o) the !ean is al(ays
nor!ally distributed
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Chap 7-*Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*
E6a!ple
uppose a population has !ean L < and
standard de&iation M *. uppose a rando!
sa!ple o) siDe n *9 is selected.
+hat is the probability that the sa!ple !ean is
bet(een 7.< and <.2
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Chap 7-*8Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*8
E6a!ple
olution:
E&en i) the population is not nor!ally
distributed, the central li!it theore! can be
used n S *0/
R so the sa!pling distribution o) is
appro6i!ately nor!al
R (ith !ean <
Rand standard de&iation
(continued)
6
6L
0.8*9
*
n
MM 6 ===
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Chap 7-*9Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*9
E6a!ple
olution continued/:
(continued)
0.3108 0.3446-0.65540.4)ZP(-0.4
363
8-8.2
nσ
μ-X
363
8-7.8P8.2)XP(7.8
=
O7.< <.2 -0. 0.
a!pling
"istribution
tandard ;or!al
"istribution
Population
"istribution
a!ple tandardiDe
<L = <LB = 0LD =6B
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Chap 7-*7Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*7
Population Proportions
π the proportion o) the population ha&inga characteristic o) interest
a!ple proportion p/ pro&ides an esti!ate o) π:
0 T p T 1
p is appro6i!ately distributed as a nor!al distribution(hen n is large
assu!ing sa!pling (ith replace!ent )ro! a )inite population or(ithout replace!ent )ro! an in)inite population/
siDesa!ple
interesto) sticcharacteritheha&ingsa!pletheinite!so) nu!ber
n
Bp ==
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Chap 7-*<Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*<
a!pling "istribution o) p
ppro6i!ated by a
nor!al distribution i):
(here and
(here U population proportion/
a!pling "istributionP ps/
3
4(
& & 4 % + 5 ( p
π =pLn
/1Mp
π π −=
5)n(1
5nand
≥−
≥
π
π
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Chap 7-*@Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*@
O-alue )or Proportions
n/1
p
M
pO
p π π
π π
−
−=
−=
tandardiDe p to a O &alue (ith the )or!ula:
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Chap 7-0Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-0
E6a!ple
I) the true proportion o) &oters (ho support
Proposition is π 0., (hat is the probability
that a sa!ple o) siDe 200 yields a sa!ple
proportion bet(een 0.0 and 0.8
i.e.: if π $ &% and n $ 4&&, /hat is
P6&%& = p = &%#8 >
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Chap 7-1Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-1
E6a!ple
if π $ &% and n $ 4&&, /hat is
P6&%& = p = &%#8 >
(continued)
0.0*9200
0./0.1n
/1Mp =−=−= π π
1./OP0
0.0*90.00.8O
0.0*90.00.0P0.8/pP0.0
≤≤=
−≤≤−=≤≤
4ind :
Con&ert tostandardiDed
nor!al:
pM
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Chap 7-2Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-2
E6a!ple
O0.8 1.
0.281
tandardiDe
a!pling "istributiontandardiDed
;or!al "istribution
if π $ &% and n $ 4&&, /hat is
P6&%& = p = &%#8 >
(continued)
tiliDe the cu!ulati&e nor!al table:
P0 T O T 1./ 0.@281 J 0.8000 0.281
0.0 0p
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Chap 7-*Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall Chap 7-*
Chapter u!!ary
"iscussed probability and nonprobability sa!ples
"escribed )our co!!on probability sa!ples
E6a!ined sur&ey (orthiness and types o) sur&ey
errors Introduced sa!pling distributions
"escribed the sa!pling distribution o) the !ean 4or nor!al populations
sing the Central #i!it 'heore! "escribed the sa!pling distribution o) a proportion
Calculated probabilities using sa!pling distributions
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Online !opic
a!pling 4ro! 4inite Populations
Basic Business Statistics for12th Edition
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#earning $b%ecti&es
In this section, you learn:
'o 5no( (hen )inite population corrections are
needed 'o 5no( ho( to utiliDe )inite population
correction )actors in calculating standard errors
4inite Population Correction
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4inite Population Correction
4actors
sed to calculate the standard error o) both the
sa!ple !ean and the sa!ple proportion
;eeded (hen the sa!ple siDe, n, is !ore than
8Q o) the population siDe ; i.e. n F ; V 0.08/
'he 4inite Population Correction 4actor Is:
"C$
1 fpc
−
−=
N
n N
sing 'he )pc In Calculating
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sing 'he )pc In Calculating
tandard Errors"C$
<tandard -rror of the ?ean for "inite Populations
1−
−=
N
n N
n X
σ σ
<tandard -rror of the Proportion for "inite Populations
1
)1(
−
−−=
N
n N
n p
π π σ
sing 'he )pc =educes 'he
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sing 'he )pc =educes 'he
tandard Error
'he )pc is al(ays less than 1
o (hen it is used it reduces the standard error
=esulting in !ore precise esti!ates o)
population para!eters
"C$
sing )pc +ith 'he ean
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sing )pc +ith 'he ean -
E6a!ple"C$
uppose a rando! sa!ple o) siDe 100 is dra(n )ro! a
population o) siDe 1,000 (ith a standard de&iation o) 0.
Here n100, ;1,000 and 100F1,000 0.10 V 0.08.
o using the )pc )or the standard error o) the !ean (e get:
8.311000
1001000
100
40=
−
−= X
σ