Statistics for business economics 7th by paul newbold chapter 17

Statistics for
Business and Economics
7th Edition

Chapter 17
Additional Topics in Sampling
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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

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Ch. 17-2


Types of Samples
(continued)

Samples

Probability Samples

Simple
Random

Cluster

(Chapter 6)

Non-Probability
Samples

Quota

Convenience

Stratified

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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
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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 + n 2 + . . . + n K = n

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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:
K

1

x st   N j x j
N j1

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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

1
 2
N

where

σˆ 2x j 


K

2 2

N
 j σˆ x j
j 1

s2j

(N j  n 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  z α/2σˆ x st  μ  x st  z α/2σˆ x st
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  N j x j
j1

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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
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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 pˆ 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   N jpˆ 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
where

1
 2
N

K

2 2
N
 j σˆ pˆ j
j 1

pˆ j (1 pˆ j ) (Nj  n j )
σˆ 

nj  1

Nj  1
2
pˆ j

is the estimate of the variance of the sample proportion in
the jth stratum
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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  z α/2σˆ pˆ st  P  pˆ st  z α/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

K

Nσ
i

i

i1

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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

n

Pi (1 Pi )

i1

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

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



2
σ
If the desired variance, x st , 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

2
N
σ
 j j
j1

n
Nσ


2
x st

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 st   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 , , x m
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

n x
i

m

i

x c  i1m

n

i

i 1

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

np

pˆ c  i1m

i i

n

i

i1

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

  ni (x i  x c )2 

M  m  i1



Mm n 2 
m 1





σˆ p2ˆ c

 m 2

  ni (Pi  pˆ c )2 
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

x c  z α/2σˆ x c  μ  x c  z α/2σˆ xc


for the population proportion

pˆ c  z α/2σˆ pˆ c  P  pˆ c  z α/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

(Chapter 6)

Non-Probability
Samples

Quota

Convenience

Stratified

Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall

Ch. 17-25