This chapter discusses additional sampling methods including stratified sampling, cluster sampling, and two-phase sampling. It provides formulas for estimating population means, totals, and proportions from stratified and cluster samples. Methods for determining optimal sample sizes to achieve desired levels of precision are also presented. Finally, the chapter addresses non-probability sampling techniques and their limitations compared to probability samples.

1. Chapter 17
Additional Topics in Sampling
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Statistics for
Business and Economics
7th Edition
Ch. 17-1

2. Chapter Goals
After completing this chapter, you should be
able to:
 Explain the difference between simple random sampling
and stratified sampling
 Analyze results from stratified samples
 Determine sample size when estimating population
mean, population total, or population proportion
 Describe other sampling methods
 Cluster Sampling, Two-Phase Sampling, Nonprobability Samples
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 17-2

3. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Types of Samples
Quota
Samples
Non-Probability
Samples
Convenience
(continued)
Probability Samples
Simple
Random
Stratified
Cluster
(Chapter 6)
Ch. 17-3

4. Stratified Sampling
Overview of stratified sampling:
 Divide population into two or more subgroups (called
strata) according to some common characteristic
 A simple random sample is selected from each subgroup
 Samples from subgroups are combined into one
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Population
Divided
into 4
strata
Sample
17.1
Ch. 17-4

5. Stratified Random Sampling
 Suppose that a population of N individuals can be
subdivided into K mutually exclusive and collectively
exhaustive groups, or strata
 Stratified random sampling is the selection of
independent simple random samples from each
stratum of the population.
 Let the K strata in the population contain N1, N2,. . .,
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 17-5

6. Estimation of the Population Mean,
Stratified Random Sample
 Let random samples of nj individuals be 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 μ
 An unbiased estimator of the overall population mean
μ is:
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall


K
1j
jjst xN
N
1
x
  

K
1j
K
1j
jj nnandNN
Ch. 17-6

7. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
2
x
K
1j
2
j2
2
x jst
σN
N
1
σ ˆˆ 

stst xα/2stxα/2st σzxμσzx ˆˆ 
1N
)n(N
n
s
σ
j
jj
j
2
j2
xj


ˆ
Ch. 17-7

8. Estimation of the Population Total,
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, Nμ
 An unbiased estimation procedure for the population
total Nμ yields the point estimate
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall


K
1j
jjst xNxN
Ch. 17-8

9. Estimation of the Population Total,
Stratified Random Sample
 An unbiased estimation procedure for the variance of
the estimator of the population total yields the point
estimate
 Provided the sample size is large, 100(1 - )%
confidence intervals for the population total for
stratified random samples are obtained from
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
stα/2ststα/2st σNzxNNμσNzxN ˆˆ 
2
x
K
1j
2
j
2
x
2
stst
σNσN ˆˆ 

Ch. 17-9

10. Estimation of the Population
Proportion, Stratified Random Sample
 Suppose that random samples of nj individuals from
strata containing Nj individuals (j = 1, 2, . . ., K) are
obtained
 Let Pj be the population proportion, and the
sample proportion, in the jth stratum
 If P is the overall population proportion, an unbiased
estimation procedure for P yields
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall


K
1j
jjst pN
N
1
p ˆˆ
jpˆ
Ch. 17-10

11. Estimation of the Population
Proportion, Stratified Random Sample
• An unbiased estimation procedure for the
variance of the estimator of the overall population
proportion is
where
is the estimate of the variance of the sample proportion in
the jth stratum
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
2
p
K
1j
2
j2
2
p jst
σN
N
1
σ ˆˆ
ˆˆ 

1N
)n(N
1n
)p(1p
σ
j
jj
j
jj2
pj






ˆˆ
ˆ ˆ
Ch. 17-11

12. Estimation of the Population
Proportion, Stratified Random Sample
 Provided the sample size is large, 100(1 - )%
confidence intervals for the population proportion for
stratified random samples are obtained from
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
stst pα/2stpα/2st σzpPσzp ˆˆ
ˆˆˆˆ 
Ch. 17-12

13. Proportional Allocation:
Sample Size
 One way to allocate sampling effort is to make the
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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
N
N
n
n jj

n
N
N
n
j
j 
Ch. 17-13

14. Optimal Allocation
To estimate an overall population mean or total and if the
population variances in the individual strata are
denoted σj
2 , the most precise estimators are obtained
with optimal allocation
 The sample size for the jth stratum using optimal
allocation is
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
n
σN
σN
n K
1i
ii
jj
j 

Ch. 17-14

15. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
n
)P(1PN
)P(1PN
n K
1i
iii
jjj
j 




Ch. 17-15

16. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 17-16

17. Sample Size for Stratified
Random Sampling: Mean
 Suppose that a population of N members is subdivided
in K strata containing N1, N2, . . .,NK members
 Let σj
2 denote the population variance in the jth stratum
 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, can be
found
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
2
xst
σ
Ch. 17-17

18. Sample Size for Stratified
Random Sampling: Mean
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 For proportional allocation:
 For optimal allocation:





 K
1j
2
jj
2
x
K
1j
2
jj
σN
N
1
Nσ
σN
n
st













 K
1j
2
jj
2
x
K
1j
2
jj
σN
N
1
Nσ
σN
N
1
n
st
(continued)
Ch. 17-18

19. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Population
divided into
16 clusters. Randomly selected
clusters for sample
17.2
Ch. 17-19

20. Estimators for Cluster Sampling
 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 members in
the m sampled clusters
 Denote the means of these clusters by
 Denote the proportions of cluster members possessing
an attribute of interest by P1, P2, . . . , Pm
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
m21 x,,x,x 
Ch. 17-20

21. Estimators for Cluster Sampling
 The objective is to estimate the overall population mean
µ and proportion P
 Unbiased estimation procedures give
Mean Proportion
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall




 m
1i
i
m
1i
ii
c
n
xn
x




 m
1i
i
m
1i
ii
c
n
pn
pˆ
(continued)
Ch. 17-21

22. Where is the average number of individuals in the sampled clusters
Estimators for Cluster Sampling
m
n
n
m
1i
i

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 Estimates of the variance of these estimators, following from
unbiased estimation procedures, are
Mean Proportion

















1m
)xx(n
nMm
mM
σ
m
1i
2
ci
2
i
2
2
xc
ˆ

















1m
)p(Pn
nMm
mM
σ
m
1i
2
ci
2
i
2
2
pc
ˆ
ˆ ˆ
(continued)
Ch. 17-22

23. Estimators for Cluster Sampling
cc xα/2cxα/2c σzxμσzx ˆˆ 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
 Provided the sample size is large, 100(1 - )%
confidence intervals using cluster sampling are
 for the population mean
 for the population proportion
cc pα/2cpα/2c σzpPσzp ˆˆ
ˆˆˆˆ 
(continued)
Ch. 17-23

24. Two-Phase Sampling
 Sometimes sampling is done in two steps
 An initial pilot sample can be done
 Disadvantage:
 takes more time
 Advantages:
 Can adjust survey questions if problems are noted
 Additional questions may be identified
 Initial estimates of response rate or population
parameters can be obtained
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 17-24

25. Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Other Sampling Methods
Quota
Samples
Non-Probability
Samples
Convenience
(continued)
Probability Samples
Simple
Random
Stratified
Cluster
(Chapter 6)
Ch. 17-25

26. Nonprobabilistic Samples
 It may be simpler or less costly to use a non-
probability based sampling method
 Quota sample
 Convenience sample
 These methods may still produce good
estimates of population parameters
 But …
 Are more subject to bias
 No valid way to determine reliability
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
(continued)
Ch. 17-26

27. 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
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall Ch. 17-27