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




#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!



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$



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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'ypes o) a!ples

Samples

Non-ProbabilitySamples

Judgment

Probability Samples

Simple

Random

Systematic

Stratified

Cluster

Convenience

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'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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'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

"C$


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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.

"C$


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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

"C$



"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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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

"C$



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

"C$


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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/

"C$



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

"C$



'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/

"C$



'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)

"C$



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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"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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3

4

(

& (5 4& 44 4%

A 0 C *

ni)or! "istribution

P6/

6

(continued)

u!!ary easures )or the Population "istribution:


21

222201<;

BL i

=+++

=

=∑

2.2*9;

L/BM

2i

=−

= ∑

"C$



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)


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

"C$



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


(continued)

no longer uni)or!/

9

"C$



u!!ary easures o) this a!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.


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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$




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 =

"C$





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 =

"C$




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$




;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

"C$



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

"C$



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

"C$





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




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

"C$



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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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

"C$




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 ===

"C$




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

"C$




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 ==

"C$



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

"C$




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 >

"C$




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

"C$




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

"C$




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



Chap 7-8Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

#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




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




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



Chap 7-<Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

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



Chap 7-@Copyright ©2012 Pearson Education, Inc. publishing as Prentice Hall

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

Chapter 7:Sampling and Sampling Distributions

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Transcript of Chapter 7:Sampling and Sampling Distributions