Download Statistics for Business and Economics, 7/e

Statistics for
Business and Economics
7th Edition
Chapter 17
Additional Topics in Sampling
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-1
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
Types of Samples
(continued)
Samples
Probability Samples
Simple
Random
Cluster
Non-Probability
Samples
Quota
Convenience
(Chapter 6)
Stratified
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-3
17.1
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
Population
Divided
into 4
strata
Sample
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-4
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
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
K
K
Nj  N and  n j  n
j1
j1

Denote the sample means and variances in the strata
by Xj and sj2 and the overall population mean by μ

An unbiased estimator of the overall population mean
μ is:
1 K
x st   Nj x j
N j1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-6
Estimation of the Population Mean,
Stratified Random Sample
(continued)

An unbiased estimator for the variance of the overall population
mean is
σˆ 2xst
where

1
 2
N
K
2 2
N
 j σˆ x j
j1
2
s
(N j  n j )
j
2
σˆ x j  
nj
Nj  1
Provided the sample size is large, a 100(1 - )% confidence
interval for the population mean for stratified random samples is
x st  zα/2σˆ xst  μ  x st  zα/2σˆ xst
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-7
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
K
Nx st   Nj x j
j1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-8
Estimation of the Population Total,
Stratified Random Sample
(continued)

An unbiased estimation procedure for the variance of
the estimator of the population total yields the point
estimate
K
N2σˆ 2xst   N2jσˆ 2xst
j1

Provided the sample size is large, 100(1 - )%
confidence intervals for the population total for
stratified random samples are obtained from
Nx st  z α/2Nσˆ st  Nμ  Nx st  z α/2Nσˆ st
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-9
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 p̂ j the
sample proportion, in the jth stratum
If P is the overall population proportion, an unbiased
estimation procedure for P yields
K
1
pˆ st   Njpˆ j
N j1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-10
Estimation of the Population
Proportion, Stratified Random Sample
(continued)
•
An unbiased estimation procedure for the
variance of the estimator of the overall population
proportion is
σˆ p2ˆ st
1 K 2ˆ2
 2  Nj σ pˆ j
N j1
where
pˆ j (1 pˆ j ) (N j  n j )
σˆ 

nj 1
Nj  1
2
pˆ j
is the estimate of the variance of the sample proportion in
the jth stratum
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-11
Estimation of the Population
Proportion, Stratified Random Sample
(continued)

Provided the sample size is large, 100(1 - )%
confidence intervals for the population proportion for
stratified random samples are obtained from
pˆ st  zα/2σˆ pˆ st  P  pˆ st  zα/2σˆ pˆ st
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-12
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,
nj
n


Nj
N
The sample size for the jth stratum using proportional
allocation is
nj 
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Nj
N
n
Ch. 17-13
Optimal Allocation
To estimate an overall population mean or total and if the
population variances in the individual strata are
denoted σj2 , the most precise estimators are obtained
with optimal allocation

The sample size for the jth stratum using optimal
allocation is
nj 
N jσ j
N σ
i1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
n
K
i
i
Ch. 17-14
Optimal Allocation
(continued)
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
nj 
N j Pj (1  Pj )
K
N
i1
i
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
n
Pi (1  Pi )
Ch. 17-15
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
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 σj2 denote the population variance in the jth stratum

An estimate of the overall population mean is desired

If the desired variance, σ 2xst , of the sample estimator is
specified, the required total sample size, n, can be
found
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-17
Sample Size for Stratified
Random Sampling: Mean
(continued)

For proportional allocation:
K
n
2
N
σ
 j j
j1
Nσ 2xs t

1 K
  N jσ 2j
N j1
For optimal allocation:

1 K
2
  N jσ j 

N  j1

n
1 K
2
Nσ x s t   N jσ 2j
N j1
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-18
17.2


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.
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Randomly selected
clusters for sample
Ch. 17-19
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 x1, x 2, , xm

Denote the proportions of cluster members possessing
an attribute of interest by P1, P2, . . . , Pm
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-20
Estimators for Cluster Sampling
(continued)

The objective is to estimate the overall population mean
µ and proportion P

Unbiased estimation procedures give
Mean
Proportion
m
xc 
n x
i1
m
i
n
i 1
m
i
i
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
p̂c 
n p
i1
m
i i
n
i1
i
Ch. 17-21
Estimators for Cluster Sampling
(continued)

Estimates of the variance of these estimators, following from
unbiased estimation procedures, are
Mean
σˆ 2xc
Proportion
 m 2
2 
n
(
x

x
)
 i i

c
M  m  i1



Mm n 2 
m 1




σˆ p2ˆ c
 m 2
2 
ˆ
n
(P

p
)
 i i

c
M  m  i1



Mm n 2 
m 1




m
Where n 
n
i1
m
i
is the average number of individuals in the sampled clusters
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-22
Estimators for Cluster Sampling
(continued)


Provided the sample size is large, 100(1 - )%
confidence intervals using cluster sampling are
for the population mean
xc  zα/2σˆ xc  μ  xc  zα/2σˆ xc

for the population proportion
pˆ c  zα/2σˆ pˆ c  P  pˆ c  zα/2σˆ pˆ c
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-23
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
Other Sampling Methods
(continued)
Samples
Probability Samples
Simple
Random
Cluster
Non-Probability
Samples
Quota
Convenience
(Chapter 6)
Stratified
Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall
Ch. 17-25
Nonprobabilistic Samples
(continued)

It may be simpler or less costly to use a nonprobability 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
Ch. 17-26
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