Chapter 18Chapter 18

Additional Topics in Additional Topics in SamplingSampling

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Steps in Sampling StudySteps in Sampling Study

Step 1: Information Required?

Step 2: Relevant Population?

Step 3: Sample Selection?

Step 4: Obtaining Information?

Step 5: Inferences From

Step 6: Conclusions?

Examples of Nonsampling Examples of Nonsampling ErrorsErrors

The population actually sampled is not the relevant one.

Survey subjects may give inaccurate or dishonest answers.

Nonresponse to survey questions.

Simple Random SamplingSimple Random Sampling

Suppose that it is required to select a sample of n objects from a population of N objects. A simple random simple random samplesample procedure is one in which every possible sample of n objects is equally likely to be chosen.

Estimation of the Population Estimation of the Population Mean, Simple Random SampleMean, Simple Random Sample

Let X1, X2, . . . , Xn denote the values observed from a simple random sample of size n, taken from a population of N members with mean

(i) The sample mean is an unbiased estimator of the unbiased estimator of the population meanpopulation mean . The point estimate is:

(ii) An unbiased estimation procedure for the unbiased estimation procedure for the variance of the sample meanvariance of the sample mean yields the point estimate

(iii) Provided the sample size is large, 100(1 - )% confidence intervals for the population meanconfidence intervals for the population mean are given by

N

nN

n

sX

22̂

XX ZXZX ˆˆ 2/2/

n

iiXn

X1

1

Estimation of the Population Estimation of the Population Total, Simple Random SampleTotal, Simple Random Sample

Suppose a simple random sample of size n from a population of N is selected and that the quantity to be estimated is the population total N

(i) An unbiased estimation procedure for the population total N yields the point estimate NX.

(ii) An unbiased estimation procedure for the variance of our estimator of the population total yields the point estimate

(iii) Provided the sample size is large, 100(1 - )% confidence intervals for the population mean are obtained from XX NZXNNNZXN ˆˆ 2/2/

)(ˆ2

22 nNNn

sN X

Estimation of the Population Estimation of the Population Proportion, Simple Random Proportion, Simple Random

SampleSampleLet p be the proportion possessing a particular

characteristic in a random sample n observations from a population , a proportion, , of whose members possess that characteristic.

(i) The sample proportion, p, is an unbiased estimator of the population proportion, .

(ii) An unbiased estimation procedure for the variance of our estimator of the population total yields the point estimate

(iii) Provided the sample size is large, 100(1 - )% confidence intervals for the population proportion are given by pp ZpZp ˆˆ 2/2/

N

nN

n

ppp

)(

1

)1(ˆ 2

Stratified Random SamplingStratified Random Sampling

Suppose that a population of N individuals can be subdivided into K mutually exclusive and collectively exhaustive groups, or strata. Stratified random samplingStratified random sampling is the selection of independent simple random samples from each stratum of the population. If the K strata in the population contain N1, N2,. . ., NK members, then

There is no need to take the same number of sample members from every stratum. Denote the numbers in the sample by n1, n2, . . ., nK. Then the total number of sample members is

NNNN K 21

nnnn K 21

Estimation of the Population Estimation of the Population Mean, Stratified Random Mean, Stratified Random

SampleSampleSuppose that random samples of nj individuals are taken

from strata containing Nj individuals (j = 1, 2, . . ., K).

Let

Denote the sample means and variances in the strata by Xj and sj

2 and the overall population mean by .

(i) An unbiased estimation procedure for the overall population mean yields the point estimate:

K

jjjst XN

NX

1

1

K

j

K

jjj nnandNN

1 1

Estimation of the Population Estimation of the Population Mean, Stratified Random Mean, Stratified Random

SampleSample(continued)(continued)(ii) An unbiased estimation procedure for the variance of

our estimator of the overall population mean yields the point estimate

where

(iii) Provided the sample size is large, 100(1 - )% confidence intervals for the population mean confidence intervals for the population mean for stratified random samplesfor stratified random samples are obtained from

2

1

22

2 ˆ1

ˆjst X

K

jjX N

N

stst XstXst ZXZX ˆˆ 2/2/

j

jj

j

jX N

nN

n

sj

)(ˆ

22

Estimation of the Population Estimation of the Population Total, Stratified Random SampleTotal, Stratified Random Sample

Suppose that random samples of nj individuals from strata containing Nj individuals (j = 1, 2, . . ., K) are selected and that the quantity to be estimated is the population total, N.

(i) An unbiased estimation procedure for the population total N yields the point estimate

(ii) An unbiased estimation procedure for the variance of our estimator of the population total yields the point estimate

(iii) Provided the sample size is large, 100(1 - )% confidence intervals for the population total for confidence intervals for the population total for stratified random samplesstratified random samples are obtained from

stststst NZXNNNZXN ˆˆ 2/2/

K

jjjst XNXN

1

2

1

222 ˆˆstst X

K

jjX NN

Estimation of the Population Estimation of the Population Proportion, Stratified Random Proportion, Stratified Random

SampleSampleSuppose that random samples of nj individuals from

strata containing Nj individuals (j = 1, 2, . . ., K) are obtained. Let j be the population proportion, and pj

the sample proportion, in the jth stratum, of those possessing a particular characteristic. If is the overall population proportion:

(i) An unbiased estimation procedure for yields

(ii) An unbiased estimation procedure for the variance of our estimator of the overall population proportion is

K

jjjst pN

Np

1

1

2

1

22

2 ˆ1

ˆjst p

K

jjp N

N

Estimation of the Population Estimation of the Population Proportion, Stratified Random Proportion, Stratified Random

SampleSample(continued)(continued)

where

is the estimate of the variance of the sample proportion in the jth stratum.

Provided the sample size is large, 100(1 - )% confidence intervals for the population proportion for stratified random samples are obtained from

stst pstpst ZpZp ˆˆ 2/2/

j

jj

j

jjp N

nN

n

ppj

)(

1

)1(ˆ 2

Proportional Allocation: Sample Proportional Allocation: Sample SizeSize

The proportion of sample members in any stratum is the same as the proportion of population members in the stratum. Thus, for the jth stratum,

So that the sample size for the jsample size for the jthth stratum using stratum using proportional allocationproportional allocation is

N

N

n

n jj

nN

Nn jj

Optimal Allocation: Sample Size Optimal Allocation: Sample Size for jfor jthth Stratum, Overall Stratum, Overall

Population Mean or TotalPopulation Mean or TotalIf it is required to estimate an overall population mean or total and if the population variances in the individual strata are denoted, j

2, it can be shown that the most precise estimators are obtained with optimal allocation. The sample size for the jsample size for the jthth stratum using optimal allocationstratum using optimal allocation is:

nN

Nn K

iii

jjj

1

Optimal Allocation: Sample Size Optimal Allocation: Sample Size for jfor jthth Stratum, Population Stratum, Population

ProportionProportionFor estimating the overall population proportion, estimators with the smallest possible variance are obtained by optimal allocation. The sample size sample size for the jfor the jthth stratum for population proportion stratum for population proportion using optimal allocationusing optimal allocation is:

nN

Nn K

iiii

jjj

j

1

)1(

)1(

Sample Size: Population Mean or Sample Size: Population Mean or Total, Simple Random SamplingTotal, Simple Random Sampling

Consider estimating the mean of a population of N members, which has variance 2. If the desired variance, of the sample mean is specified, the required sample size to estimate population mean through simple random sampling is

(i) Often it is more convenient to specify directly the width of confidence intervals for the population mean rather than . This is easily accomplished since, for example, a 95% confidence interval for the population mean will extend an approximate amount 1.96 on each side of the sample mean, X.

(ii) If the object of interest is the population total, the variance of the sample estimator os this quantity is N2 and that confidence intervals for it extend an approximate amount of 1.96N on each side of NX.

I

22

2

)1(

XN

Nn

2X

2X

X

2X

Sample Size: Population Sample Size: Population Proportion, Simple Random Proportion, Simple Random

SamplingSamplingConsider estimation of the proportion of individuals in a

population of size N who possess a certain attribute. If the desired variance, , of the sample proportion is specified, the required sample size to estimate the population proportion through simple random sampling is

The largest possible value for this expression, whatever the value of , is

A 95% confidence interval for the population proportion will extend an approximate amount 1.96 on each side of the sample proportion.

)1()1(

)1(2

XpN

Nn

2

Xp

25.0)1(

25.02max

Xp

N

Nn

Xp

Sample Size: Overall Mean Sample Size: Overall Mean (Stratum Population variances (Stratum Population variances Specified), Stratified SamplingSpecified), Stratified Sampling

Suppose that a population of N members is subdivided in K strata containing N1, N2, . . .,NK members. Let j

2

denote the population variance in the jth stratum, and suppose that an estimate of the overall population mean is desired. If the desired variance, , of the sample estimator is specified, the required total sample size, n, is as follows

(i) Proportional allocation:

K

jjjX

K

jjj

NN

N

N

n

st1

22

1

2

1

2

stX

Sample Size: Overall Mean Sample Size: Overall Mean (Stratum Population variances (Stratum Population variances Specified), Stratified SamplingSpecified), Stratified Sampling

(continued)(continued)(ii) Optimal allocation:

K

jjjX

K

jjj

NN

N

NN

n

st1

22

1

2

1

1

Estimators for Cluster Estimators for Cluster SamplingSampling

A population is subdivided into M clusters and a simple random sample of m of these clusters is selected and information is obtained from every member of the sampled clusters. Let n1, n2, . . ., nm denote the numbers of population members in the m sampled clusters. Denote the means of these clusters by and the proportions of cluster members possessing an attribute of interest by 1, 2, . . . ,m . The objective is to estimate the overall population mean and proportion .

(i) Unbiased estimation procedures give

and

m

ii

m

iii

c

n

XnX

1

1

mXXX ,,, 21

m

ii

m

iii

c

n

np

1

1

Estimators for Cluster Estimators for Cluster SamplingSampling

(continued)(continued)(ii) Estimates of the variance of these estimators,

following from unbiased estimation procedures, are,

and

1

)(ˆ 1

22

22

m

XXn

nMm

mM

m

icii

X c

1

)(ˆ 1

22

22

m

pn

nMm

mM

m

icii

pc

clusters sampled in the sindividual ofnumber average theis m

nn where

m

1ii

Estimation of the Population Estimation of the Population Mean, Cluster SamplingMean, Cluster Sampling

Provided the sample size is large, 100(1 - )% confidence intervals for the population mean confidence intervals for the population mean using cluster samplingusing cluster sampling are given by

cc XcXc ZXZX ˆˆ 2/2/

Estimation of the Population Estimation of the Population Proportion, Cluster SamplingProportion, Cluster Sampling

Provided the sample size is large, 100(1 - )% confidence intervals for the population confidence intervals for the population proportion using cluster samplingproportion using cluster sampling are given by

cc pcpc ZpZp ˆˆ 2/2/

Key WordsKey Words Cluster Sampling Estimation

Population Mean, Simple Random

Population Mean, Stratified Population Mean, Cluster Population Total, Simple

Random Population Total, Stratified Population Proportion,

Simple Random Population Proportion,

Stratified Population Proportion,

Cluster

Finite Population Correction factor

Nonprobabilistic Methods Nonsampling Error Optimal Allocation Proportional Allocation Quota Sampling Sampling Study Sampling Error Simple Random Sample Stratified Random

Sample

Key WordsKey Words(continued)(continued)

Sample Size Optimal Allocation, jth Stratum Optimal Allocation, Total Sample Population Mean, Simple Random &

Stratified Population Total, Simple Random Population Proportion, Simple Random &

Stratified Proportional Allocation, jth Stratum and

Total Sample Systematic Sampling Two-Phase Sampling