Statistics for business economics 7th by paul newbold chapter 17

Chapter GoalsAfter completing this chapter, you should be able to: and stratified sampling mean, population total, or population proportion  Cluster Sampling, Two-Phase Sampling, Nonpr

Chapter 17

Additional Topics in Sampling

Statistics for Business and Economics

7th Edition

Chapter Goals

After completing this chapter, you should be

able to:

and stratified sampling

mean, population total, or population proportion

 Cluster Sampling, Two-Phase Sampling, Nonprobability Samples

Stratified Sampling

Stratified Random Sampling

exhaustive groups, or strata

 Stratified random sampling is the selection of

stratum of the population

NK members, so that N1 + N2 + + NK = N

 Let the numbers in the samples be n1, n2, , nK

Then the total number of sample members is

n1 + n2 + + nK = n

Estimation of the Population Mean,

Stratified Random Sample

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

j j

N

1 x

K

1 j

j

j N and n nN

Estimation of the Population Mean,

Stratified Random Sample

 An unbiased estimator for the variance of the overall population

mean is

where

 Provided the sample size is large, a 100(1 - )% confidence

interval for the population mean for stratified random samples is

(continued)

2 x

K

1 j

2 j 2

x  ˆ    ˆ

1N

)n

(Nn

sσ

j

j j

j

2 j

Estimation of the Population Total,

Stratified Random Sample

strata containing Nj individuals (j = 1, 2, , K) are selected and that the quantity to be estimated is the

population total, Nμ

total Nμ yields the point estimate

j j

x N

Estimation of the Population Total,

Stratified Random Sample

the estimator of the population total yields the point estimate

confidence intervals for the population total for stratified random samples are obtained from

(continued)

st α/2

st st

α/2

st z N σ Nμ N x z N σ x

2 x

K

1 j

2 j

2 x

Estimation of the Population Proportion, Stratified Random Sample

strata containing Nj individuals (j = 1, 2, , K) are obtained

sample proportion, in the jth stratum

estimation procedure for P yields

j j

Estimation of the Population Proportion, Stratified Random Sample

variance of the estimator of the overall population proportion is

K

1 j

2 j 2

) n

(N 1

n

) p (1

p σ

j

j j

j

j j

Estimation of the Population Proportion, Stratified Random Sample

confidence intervals for the population proportion for stratified random samples are obtained from

(continued)

st

p α/2

st z σ P p z σ

p ˆ  ˆ ˆ   ˆ  ˆ ˆ

Proportional Allocation:

Sample Size

proportion of sample members in any stratum the same

as the proportion of population members in the stratum

 If so, for the jth stratum,

 The sample size for the jth stratum using proportional

allocation is

N

Nn

nj j



n N

N

nj  j 

Optimal Allocation

To estimate an overall population mean or total and if the population variances in the individual strata are

with optimal allocation

 The sample size for the jth stratum using optimal

allocation is

n σ

i i

Optimal Allocation

To estimate the overall population proportion, estimators with the smallest possible variance are obtained by optimal allocation

 The sample size for the jth stratum for population

proportion using optimal allocation is

(continued)

n )

P (1

P N

) P (1

P

N

1 i

i i

i

j j

Determining Sample Size

 The sample size is directly related to the size

of the variance of the population estimator

 If the researcher sets the allowable size of

the variance in advance, the necessary sample size can be determined

Sample Size for Stratified Random Sampling: Mean

in K strata containing N1, N2, ,NK members

found

2

xst

σ

Sample Size for Stratified Random Sampling: Mean

2 j j

2 x

K

1 j

2 j j

σ

N N

1 Nσ

σ

N n

2 j j

2 x

K

1 j

2 j j

σ

N N

1 Nσ

σ

N N

1 n

st

(continued)

Cluster Sampling

 Population is divided into several “clusters,”

each representative of the population

 A simple random sample of clusters is selected

 Generally, all items in the selected clusters are examined

 An alternative is to chose items from selected clusters using another probability sampling technique

Population

divided into

16 clusters Randomly selected

clusters for sample

17.2

Estimators for Cluster Sampling

random sample of m of these clusters is selected and information is obtained from every member of the

sampled clusters

the m sampled clusters

an attribute of interest by P1, P2, , Pm

m 2

1, x , , x

Estimators for Cluster Sampling

i

m

1 i

i i

c

n

x

n x

i

m

1 i

i i

c

n

p

n pˆ

(continued)

Where is the average number of individuals in the sampled clusters

Estimators for Cluster Sampling

m

n n

m 1 i i

)xx

(nn

Mm

m

Mσ

m

1 i

2 c i

2 i 2

)p(P

nn

Mm

m

Mσ

m

1 i

2 c i

2 i 2

Estimators for Cluster Sampling

c

x α/2

c z σ μ x z σ

confidence intervals using cluster sampling are

 for the population mean

 for the population proportion

c

c c α/2 p p

α/2

(continued)

Two-Phase Sampling

 Sometimes sampling is done in two steps

 An initial pilot sample can be done

 Disadvantage:

 Advantages:

parameters can be obtained

Other Sampling Methods

Nonprobabilistic Samples

 It may be simpler or less costly to use a

non-probability based sampling method

 These methods may still produce good

estimates of population parameters

 But …

(continued)

Chapter Summary

 Examined Stratified Random Sampling and

Cluster Sampling

 Identified Estimators for the population mean,

population total, and population proportion for different types of samples

 Determined the required sample size for

specified confidence interval width

 Examined nonprobabilistic sampling methods