16-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

16-1

COMPLETE COMPLETE BUSINESS BUSINESS

STATISTICSSTATISTICSbyby

AMIR D. ACZELAMIR D. ACZEL

&&

JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN

66thth edition (SIE) edition (SIE)

16-2

Chapter 16 Chapter 16

Sampling MethodsSampling Methods

16-3

• Using Statistics• Nonprobability Sampling and Bias• Stratified Random Sampling• Cluster Sampling• Systematic Sampling• Nonresponse

Sampling MethodsSampling Methods1616

16-4

• Apply nonprobability sampling methods• Decide when to conduct a stratified

sampling method• Compute estimates from stratified sample

results• Decide when to conduct a cluster sampling

method

LEARNING OUTCOMESLEARNING OUTCOMES1616

After studying this chapter you should be able to:After studying this chapter you should be able to:

16-5

• Compute estimates from cluster sampling results

• Decide when to conduct a systematic sampling method

• Compute estimates from systematic sample results

• Avoid nonresponse biases in estimates

LEARNING OUTCOMES (2)LEARNING OUTCOMES (2)1616

After studying this chapter you should be able to:After studying this chapter you should be able to:

16-6

• Sampling methods that do not use samples with known probabilities of selection are know as nonprobability sampling methodsnonprobability sampling methods.

• In nonprobability sampling methods, there is no objective way of evaluating how far away from the population parameter the estimate may be.

• FrameFrame - a list of people or things of interest from which a random sample can be chosen.

16-2 Nonprobability Sampling and 16-2 Nonprobability Sampling and BiasBias

16-7

In stratified random samplingstratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2 +...+ Nm =N.


16-3 Stratified Random Sampling16-3 Stratified Random Sampling

2 Stratum

1 Stratum

mStratum

1N

2N

mN

The m strata are non-overlapping.The m strata are non-overlapping.

NNm

i i

1NN

m

i i

1

PopulationPopulation

16-8



7654321 Group

NiNi

Population DistributionPopulation Distribution7654321 Group

nini

Sample DistributionSample Distribution

In proportional allocationproportional allocation, the relative frequencies in the sample (ni/n) are the same as those in the population (Ni/N) .

In proportional allocationproportional allocation, the relative frequencies in the sample (ni/n) are the same as those in the population (Ni/N) .

16-3 Stratified Random Sampling 16-3 Stratified Random Sampling (Continued)(Continued)

16-9

m

i iX

iWX

is

iX

i

ni

n

if

Ni

N

iW

1st

:sampling random stratifiedin The

2 :i stratumin varianceSample

:i stratumin mean Sample

2 :i stratum of varianceTrue

2 :population theof varianceTrue

i :i stratumin mean True

:population ofmean True

:i stratumin fraction Sampling

:i stratum of weight True

mean population the ofestimator

m

i iX

iWX

is

iX

i

ni

n

if

Ni

N

iW

1st

:sampling random stratifiedin The

2 :i stratumin varianceSample

:i stratumin mean Sample

2 :i stratum of varianceTrue

2 :population theof varianceTrue

i :i stratumin mean True

:population ofmean True

:i stratumin fraction Sampling

:i stratum of weight True

mean population the ofestimator

Relationship Between the Population Relationship Between the Population and a Stratified Random Sampleand a Stratified Random Sample

16-10

1. If the estimator of the mean in each stratum, Xi , is then the stratified

estimator of the mean, Xst is an estimator of the population mean, .

2. If the samples in the different strata are drawn independently of each other, then the variance of the stratified estimator of the population mean, Xst , is given by:

( ) = Xii=1

m

If sampling in all strata is random, then the variance of Xst is further equal to:

( ) =i=1

m

When the sampling fractions, , are small and may be ignored, we have:

unbiased

unbiased,

( )

.

( )

V Xst Wi V

V Xst Wii

ni

fi

fi

2

3

22

1

V Xst Wii

ni

( ) =i=1

m 22

1. If the estimator of the mean in each stratum, Xi , is then the stratified

estimator of the mean, Xst is an estimator of the population mean, .

2. If the samples in the different strata are drawn independently of each other, then the variance of the stratified estimator of the population mean, Xst , is given by:

( ) = Xii=1

m

If sampling in all strata is random, then the variance of Xst is further equal to:

( ) =i=1

m

When the sampling fractions, , are small and may be ignored, we have:

unbiased

unbiased,

( )

.

( )

V Xst Wi V

V Xst Wii

ni

fi

fi

2

3

22

1

V Xst Wii

ni

( ) =i=1

m 22

Properties of the Stratified Estimator Properties of the Stratified Estimator of the Sample Meanof the Sample Mean

16-11

4. If the sample allocation is proportional for all i , then

( ) =1 - f

n i=1

m

which reduces to

( ) =1

n i=1

m

when the sampling fraction is small. In addition, if the population variances in all strata are equal, then

( ) =2

n

when the sampling fraction is small.

ni

nN

iN

V X st Wi

V X st Wi

V X st

i

i

2

2

4. If the sample allocation is proportional for all i , then

( ) =1 - f

n i=1

m

which reduces to

( ) =1

n i=1

m

when the sampling fraction is small. In addition, if the population variances in all strata are equal, then

( ) =2

n

when the sampling fraction is small.

ni

nN

iN

V X st Wi

V X st Wi

V X st

i

i

2

2

Properties of the Stratified Estimator Properties of the Stratified Estimator of the Sample Mean (continued)of the Sample Mean (continued)

16-12

)1(m

1=i

2i=)(2

:random is stratumeach in sampling If

iin data 1

2)(2i

:is , 2i

i, stratum of variancepopulation theofestimator unbiasedAn

2

i

i

fn

Si

W

stXS

in

iXX

S

)1(m

1=i

2i=)(2

:random is stratumeach in sampling If

iin data 1

2)(2i

:is , 2i

i, stratum of variancepopulation theofestimator unbiasedAn

2

i

i

fn

Si

W

stXS

in

iXX

S

When the Population Variance is When the Population Variance is UnknownUnknown

16-13

m

i in

is

in

in

iN

iN

m

i ini

s

in

iN

iN

stXsz

1 )1(

42

)/(

2

1=

2

)(

= df Effective

:freedom of degrees effective The

formula. above in the value- t theuse

unknown, are variancespopulation theand small, are sizes sample When the

)(st

x

:sampling

stratified using , mean, population for the interval confidence )100%-(1A

2

m

i in

is

in

in

iN

iN

m

i ini

s

in

iN

iN

stXsz

1 )1(

42

)/(

2

1=

2

)(

= df Effective

:freedom of degrees effective The

formula. above in the value- t theuse

unknown, are variancespopulation theand small, are sizes sample When the

)(st

x

:sampling

stratified using , mean, population for the interval confidence )100%-(1A

2

Confidence Interval for the Population Confidence Interval for the Population Mean in Stratified SamplingMean in Stratified Sampling

16-14

Population True SamplingNumber Weights Sample Fraction

Group of Firms (Wi) Sizes (fi) 1. Diversified service companies 100 0.20 20 0.202. Commercial banking companies 100 0.20 20 0.203. Financial service companies 150 0.30 30 0.304. Retailing companies 50 0.10 10 0.105. Transportation companies 50 0.10 10 0.106. Utilities 50 0.10 10 0.10

N = 500 n = 100

Population True SamplingNumber Weights Sample Fraction

Group of Firms (Wi) Sizes (fi) 1. Diversified service companies 100 0.20 20 0.202. Commercial banking companies 100 0.20 20 0.203. Financial service companies 150 0.30 30 0.304. Retailing companies 50 0.10 10 0.105. Transportation companies 50 0.10 10 0.106. Utilities 50 0.10 10 0.10

N = 500 n = 100

Stratum Mean Variance ni Wi Wixi

1 52.7 97650 20 0.2 10.54 156.240 2 112.6 64300 20 0.2 22.52 102.880 3 85.6 76990 30 0.3 25.68 184.776 4 12.6 18320 10 0.1 1.26 14.656 5 8.9 9037 10 0.1 0.89 7.230 6 52.3 83500 10 0.1 5.23 66.800

Estimated Mean: 66.12 532.582Estimated standard error of mean: 23.08

1 f

nWi si

2 95% Confdence Interval: x

st

66

z s Xst

2

12 1 96 23 0866 12 45 2420 88 111 36

( )

. ( . )( . )

. .[ . , . ]

95% Confdence Interval: x

st

66

z s Xst

2

12 1 96 23 0866 12 45 2420 88 111 36

( )

. ( . )( . )

. .[ . , . ]

Example 16-2Example 16-2

16-15

Example 16-2 Using the templateExample 16-2 Using the template

Observe that the computer gives a slightly more precise interval than the hand computation on the previous slide.

Observe that the computer gives a slightly more precise interval than the hand computation on the previous slide.

16-16

Stratified estimator of the population proportion, ,

The approximate variance of

V(

When the finite - population correction factors, must be considered:

V(

When proportional allocation is used:

V(

p

Pst Wi Pii

m

Pst

Pst WiPi Qinii

m

f

PstN

Ni

Ni

ni

Pi QiN

inii

m

Pstf

nWi Pi Qii

m

i

,

)

,

) ( )

( )

)

1

2

1

1

2

2

11

1

1

Stratified estimator of the population proportion, ,

The approximate variance of

V(

When the finite - population correction factors, must be considered:

V(

When proportional allocation is used:

V(

p

Pst Wi Pii

m

Pst

Pst WiPi Qinii

m

f

PstN

Ni

Ni

ni

Pi QiN

inii

m

Pstf

nWi Pi Qii

m

i

,

)

,

) ( )

( )

)

1

2

1

1

2

2

11

1

1

Stratified Sampling for the Population Stratified Sampling for the Population ProportionProportion

16-17

NumberGroup Wi ni fi InterestedMetropolitan 0.65 130 0.65 28 0.14 0.0005756Nonmetropolitan 0.35 70 0.35 18 0.09 0.0003099

Estimated proportion: 0.23 0.0008855Estimated standard error: 0.0297574

90% confidence interval:[0.181,0.279]

Wi pi

Wi pi qi

n

90% Confdence Interval: p

st ( )

. ( . )( . )

. .[ . , . ]

z s Pst

2

0 23 1 645 0 2970 23 0 0490 181 0 279

Stratified Sampling for the Population Stratified Sampling for the Population Proportion: Example 16-1 (Continued)Proportion: Example 16-1 (Continued)

16-18Stratified Sampling for the Population Stratified Sampling for the Population Proportion:Example 16-1 (Continued) using Proportion:Example 16-1 (Continued) using the Templatethe Template

16-19

1. Preferably no more than 6 strata.2. Choose strata so that Cum f(x) is approximately constant for all strata (Cum f(x) is the cumulative square root of the frequency of X, the variable of interest).

Age Frequency (fi) 20-25 1 126-30 16 4 531-35 25 5 536-40 4 241-45 9 3 5

f(x) Cum f(x)

Rules for Constructing StrataRules for Constructing Strata

16-20

For optimum allocation of effort in stratified random sampling, minimize thecost for a given variance, or minimize the variance for a given cost.

Total Cost = Fixed Cost + Variable Cost C = C0 Cini

Optimum Allocation: nin

(Wi i ) / Ci(Wi i ) / Ci

If the cost per unit sampled is the same for all strata (Ci = c):

Neyman Allocation: nin

(Wi i )

(Wi i )

For optimum allocation of effort in stratified random sampling, minimize thecost for a given variance, or minimize the variance for a given cost.

Total Cost = Fixed Cost + Variable Cost C = C0 Cini

Optimum Allocation: nin

(Wi i ) / Ci(Wi i ) / Ci

If the cost per unit sampled is the same for all strata (Ci = c):

Neyman Allocation: nin

(Wi i )

(Wi i )

Optimum AllocationOptimum Allocation

16-21

1 0.4 1 4 0.4 0.200 0.329 0.235 2 0.5 2 9 1.0 0.333 0.548 0.588 3 0.1 3 16 0.3 0.075 0.123 0.176

i W W i isi Ci si W

isi

Ci

Optimum

Allocation

Neyman

Allocation

1.7 0.608

1 0.4 1 4 0.4 0.200 0.329 0.235 2 0.5 2 9 1.0 0.333 0.548 0.588 3 0.1 3 16 0.3 0.075 0.123 0.176

i W W i isi Ci si W

isi

Ci

Optimum

Allocation

Neyman

Allocation

1.7 0.608

Optimum Allocation: An ExampleOptimum Allocation: An Example

16-22

Optimum Allocation: An Example Optimum Allocation: An Example using the Templateusing the Template

16-23

7654321Group

Population DistributionPopulation Distribution

In stratified sampling a random sample (ni) is chosen from each segment of the population (Ni).

In stratified sampling a random sample (ni) is chosen from each segment of the population (Ni).

Sample Distribution Sample Distribution

In cluster sampling observations are drawn from m out of M areas or clusters of the population.

In cluster sampling observations are drawn from m out of M areas or clusters of the population.

16-4 Cluster Sampling16-4 Cluster Sampling

16-24

Cluster sampling estimator of :

Estimator of the variance of the sample mean:

s

where

=

2

Xn X

n

XM mMmn

n X X

m

nn

m

cl

i ii

m

ii

m

cl

i i cli

m

ii

m

1

1

2

2 2

1

1

1( )

( )


Estimator of the variance of the sample mean:

s

where

=

2

Xn X

n

XM mMmn

n X X

m

nn

m

cl

i ii

m

ii

m

cl

i i cli

m

ii

m

1

1

2

2 2

1

1

1( )

( )

Cluster Sampling: Estimating the Cluster Sampling: Estimating the Population MeanPopulation Mean

16-25


Estimator of the variance of the sample proportion:

s

2

p

Pn P

n

PM mMmn

n P P

m

cl

i ii

m

ii

m

cl

i i cli

m

( )( )

1

1

2

2 2

1

1


Estimator of the variance of the sample proportion:

s

2

p

Pn P

n

PM mMmn

n P P

m

cl

i ii

m

ii

m

cl

i i cli

m

( )( )

1

1

2

2 2

1

1

Cluster Sampling: Estimating the Cluster Sampling: Estimating the Population ProportionPopulation Proportion

16-26


cl

z s Xcl

2

2183 1 96 15872183 2 4719 36 24 30

( )

. ( . )( . )

. .[ . , . ]


cl

z s Xcl

2

2183 1 96 15872183 2 4719 36 24 30

( )

. ( . )( . )

. .[ . , . ]

xi ni nixi xi-xcl (xi-xcl)2

21 8 168 -0.8333 0.694 0.0011822 8 176 0.1667 0.028 0.0000511 9 99 -10.8333 117.361 0.2526934 10 340 12.1667 148.028 0.3934828 7 196 6.1667 38.028 0.0495325 8 200 3.1667 10.028 0.0170618 10 180 -3.8333 14.694 0.0390624 12 288 2.1667 4.694 0.0179719 11 209 -2.8333 8.028 0.0258220 6 120 -1.8333 3.361 0.0032230 8 240 8.1667 66.694 0.1134626 9 234 4.1667 17.361 0.0373812 9 108 -9.8333 96.694 0.2081917 8 136 -4.8333 23.361 0.0397413 10 130 -8.8333 78.028 0.2074129 8 232 7.1667 51.361 0.0873824 8 192 2.1667 4.694 0.0079926 10 260 4.1667 17.361 0.0461518 10 180 -3.8333 14.694 0.0390622 11 242 0.1667 0.028 0.00009

3930 s2(Xcl)= 1.58691xcl = 21.83

M mMmn

n X Xm

i i cl

2

2 2

1( )

Cluster Sampling: Example 16-3Cluster Sampling: Example 16-3

16-27

Cluster Sampling: Example 16-3 Using Cluster Sampling: Example 16-3 Using the Templatethe Template

16-28

Cluster Sampling: Using the Template Cluster Sampling: Using the Template to Estimate Population Proportionto Estimate Population Proportion

16-29

Randomly select an element out of the first k elements in the population, and then select every kth unit afterwards until we have a sample of n elements.

Randomly select an element out of the first k elements in the population, and then select every kth unit afterwards until we have a sample of n elements.

Systematic sampling estimator of :

Estimator of the variance of the sample mean: s2

When the mean is constant within each stratum of k elements but different between strata:

s2

When the population is linearly increasing or decreasing with respect to the variable of interest:

s2

X sy

Xii

m

n

X syN n

NnS

X syN n

Nn

Xi Xi ki

n

n

X syN n

Nn

Xi Xi k Xi ki

n

n

1

2

2

1

2 1

2 22

1

6 2

( )

( )( )

( )

( )( )

( )

16-5 Systematic Sampling16-5 Systematic Sampling

16-30

s2

s2

A 95% confidence interval for the average price change for all stocks: s

X sy

Xii

m

n

X syN n

NnS

X sy X sy

1 0 5 0 36

2 2100 1002100 100

0 36 0 0034

1 96

0 5 1 96 0 00340 5 0114

0 386 0 614

. .

( )( )( )

. .

( . ) ( )

. ( . )( . )

. .[ . , . ]

s2

s2

A 95% confidence interval for the average price change for all stocks: s

X sy

Xii

m

n

X syN n

NnS

X sy X sy

1 0 5 0 36

2 2100 1002100 100

0 36 0 0034

1 96

0 5 1 96 0 00340 5 0114

0 386 0 614

. .

( )( )( )

. .

( . ) ( )

. ( . )( . )

. .[ . , . ]

Systematic Sampling: Example 16-4Systematic Sampling: Example 16-4

16-31

Systematic nonresponse can biasbias estimatesCallbacks of nonrespondentsOffers of monetary rewards for nonrespondentsRandom-response mechanism

Systematic nonresponse can biasbias estimatesCallbacks of nonrespondentsOffers of monetary rewards for nonrespondentsRandom-response mechanism

16-6 Nonresponse16-6 Nonresponse

16-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

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Transcript of 16-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)