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Chap 20-1Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc.
Chapter 20
Sampling:Additional Topics in Sampling
Statistics for Business and Economics
6th Edition
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-2
Chapter Goals
After completing this chapter, you should be able to:
Explain the basic steps of a sampling study Describe sampling and nonsampling errors Explain simple random sampling and stratified sampling Analyze results from simple random or 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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-3
Steps of a 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?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-4
Sampling and Nonsampling Errors
A sample statistic is an estimate of an unknown population parameter
Sample evidence from a population is variable Sample-to-sample variation is expected
Sampling error results from the fact that we only see a subset of the population when a sample is selected
Statistical statements can be made about sampling error It can be measured and interpreted using confidence
intervals, probabilities, etc.
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-5
Sampling and Nonsampling Errors
Nonsampling error results from sources not related to the sampling procedure used
Examples: The population actually sampled is not the relevant
one Survey subjects may give inaccurate or dishonest
answers Nonresponse to survey questions
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-6
Probability Sample Items in the sample are chosen on the
basis of known probabilities
Nonprobability Sample Items included are chosen without
regard to their probability of occurrence
Types of Samples
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-7
Types of Samples
Quota
Samples
Non-Probability Samples
Judgement Convenience
(continued)
Probability Samples
Simple Random
Systematic
Stratified
Cluster
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-8
Simple Random Samples
Suppose that a sample of n objects is to be selected from a population of N objects
A simple random sample procedure is one in which every possible sample of n objects is equally likely to be chosen
Only sampling without replacement is considered here Random samples can be obtained from table of random
numbers or computer random number generators
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-9
Decide on sample size: n Divide frame of N individuals into groups of j
individuals: j=N/n Randomly select one individual from the 1st
group Select every jth individual thereafter
Systematic Sampling
N = 64
n = 8
j = 8
First Group
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-10
Finite Population Correction Factor
Suppose sampling is without replacement and the sample size is large relative to the population size
Assume the population size is large enough to apply the central limit theorem
Apply the finite population correction factor when estimating the population variance
N
nNfactor correction population finite
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-11
Estimating the Population Mean
Let a simple random sample of size n be taken from a population of N members with mean μ
The sample mean is an unbiased estimator of the population mean μ
The point estimate is:
n
1iix
n
1x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-12
Estimating the Population Mean
An unbiased estimation procedure for the variance of the sample mean yields the point estimate
Provided the sample size is large, 100(1 - )% confidence intervals for the population mean are given by
N
nN
n
sσ
22x
ˆ
xα/2xα/2 σzxμσzx ˆˆ
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-13
Estimating the Population Total
Consider a simple random sample of size n from a population of size N
The quantity to be estimated is the population total Nμ
An unbiased estimation procedure for the population total Nμ yields the point estimate NX
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-14
Estimating the Population Total
An unbiased estimator of the variance of the population total is
Provided the sample size is large, a 100(1 - )% confidence interval for the population total is
xα/2xα/2 σNzxNNμσNzxN ˆˆ
n)N(Nn
sσN
22x
2 ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-15
Confidence Interval for Population Total: Example
A firm has a population of 1000 accounts and wishes to estimate the total population value
A sample of 80 accounts is selected with average balance of $87.6 and standard deviation of $22.3
Find the 95% confidence interval estimate of the total balance
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-16
Example Solution
The 95% confidence interval for the population total balance is $82,912.52 to $92,287.16
2391.415718835σN
5718835)(1000)(92080
(22.3)n)N(N
n
sσN
x
222x
2
ˆ
ˆ
22.3s 87.6,x 80, n 1000,N
1.41)(1.96)(2396)(1000)(87.σNzxN xα/2 ˆ
92287.16Nμ82912.84
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-17
Estimating the Population Proportion
Let the true population proportion be P
Let be the sample proportion from n observations from a simple random sample
The sample proportion, , is an unbiased estimator of the population proportion, P
p̂
p̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-18
Estimating the Population Proportion
An unbiased estimator for the variance of the population proportion is
Provided the sample size is large, a 100(1 - )% confidence interval for the population proportion is
pα/2pα/2 σzpPσzp ˆˆˆˆˆˆ
N
n)(N
1n
)p(1pσ2
p
ˆˆ
ˆˆ
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-19
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-20
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-21
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:
K
1jjjst xN
N
1x
K
1j
K
1jjj nnandNN
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-22
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)
2x
K
1j
2j2
2x jst
σNN
1σ ˆˆ
stst xα/2stxα/2st σzxμσzx ˆˆ
j
jj
j
2j2
x N
)n(N
n
sσ
j
ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-23
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
1jjjst xNxN
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-24
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
(continued)
stα/2ststα/2st σNzxNNμσNzxN ˆˆ
2x
K
1j
2j
2x
2
ststσNσN ˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-25
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
K
1jjjst pN
N
1p ˆˆ
jp̂
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-26
• 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
(continued)
Estimation of the Population Proportion, Stratified Random Sample
2p
K
1j
2j2
2p jst
σNN
1σ ˆˆ
ˆˆ
j
jj
j
jj2p N
)n(N
1n
)p(1pσ
j
ˆˆ
ˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-27
Provided the sample size is large, 100(1 - )% confidence intervals for the population proportion for stratified random samples are obtained from
(continued)
Estimation of the Population Proportion, Stratified Random Sample
stst pα/2stpα/2st σzpPσzp ˆˆˆˆˆˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-28
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
N
N
n
n jj
nN
Nn j
j
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-29
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
nσN
σNn K
1iii
jjj
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-30
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(1PN
)P(1PNn K
1iiii
jjj
j
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-31
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-32
Sample Size, Mean, Simple Random Sampling
Consider estimating the mean of a population of N members, which has variance σ2
If the desired variance, of the sample mean is specified, the required sample size to estimate the population mean through simple random sampling is
22x
2
σ1)σ(N
Nσn
2xσ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-33
Sample Size, Mean, Simple Random Sampling
Often it is more convenient to specify directly the desired width of the confidence interval for the population mean rather than
Thus the researcher specifies the desired margin of error for the mean
Calculations are simple 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
2xσ
(continued)
xσ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-34
Required Sample Size Example
2000 items are in a population. If σ = 45, what sample size is needed to estimate the mean within ± 5 with 95% confidence?
(Always round up)
So the required sample size is n = 270
269.39(45)51)(1999)(2.5
(2000)(45)
σ1)σ(N
Nσn
22
2
22x
2
N = 2000, 1.96 = 5 → = 2.551 xσ xσ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-35
Consider estimating the proportion P 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
Sample Size, Proportion,Simple Random Sampling
(continued)
P)P(11)σ(N
P)NP(1n
2p
ˆ
2pσ ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-36
The largest possible value for this expression occurs when the value of P is 0.25
A 95% confidence interval for the population proportion will extend an approximate amount 1.96 on each side of the sample proportion
Sample Size, Proportion,Simple Random Sampling
(continued)
0.251)σ(N
0.25Nn
2p
max
ˆ
pσ ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-37
Required Sample Size Example
How large a sample would be necessary to estimate the true proportion of voters who will vote for proposition A, within ±3%, with 95% confidence, from a population of 3400 voters?
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-38
Required Sample Size Example
Solution:
N = 34000
For 95% confidence, use z = 1.96
1.96 = .03 → = .015306
So use n = 1036
(continued)
spσ ˆ spσ ˆ
1035.47025153)(33999)(.0
00)(0.25)(340
0.251)σ(N
0.25Nn
22p
max
ˆ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-39
Sample Size, Mean, Stratified Sampling
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, , of the sample estimator is specified, the required total sample size, n, can be found
2xst
σ
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-40
Sample Size, Mean, Stratified Sampling
For proportional allocation:
For optimal allocation:
K
1j
2jj
2x
K
1j
2jj
σNN1
Nσ
σN
n
st
(continued)
K
1j
2jj
2x
K
1j
2jj
σNN1
Nσ
σNN1
n
st
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-41
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-42
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
m21 x,,x,x
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-43
Estimators for Cluster Sampling
The objective is to estimate the overall population mean µ and proportion P
Unbiased estimation procedures give
Mean Proportion
m
1ii
m
1iii
c
n
xnx
m
1ii
m
1iii
c
n
pnp̂
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-44
Estimates of the variance of these estimators, following from unbiased estimation procedures, are
Mean Proportion
1m
)xx(n
nMm
mMσ
m
1i
2ci
2i
22xc
ˆ
1m
)p(Pn
nMm
mMσ
m
1i
2ci
2i
22pc
ˆˆ
ˆ
m
nn
m
1ii
Where is the average number of individuals in the sampled clusters
Estimators for Cluster Sampling(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-45
Provided the sample size is large, 100(1 - )% confidence intervals using cluster sampling are
for the population mean
for the population proportion
cc xα/2cxα/2c σzxμσzx ˆˆ
cc pα/2cpα/2c σzpPσzp ˆˆˆˆˆˆ
Estimators for Cluster Sampling(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-46
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
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-47
Non-Probability Samples
Quota
Samples
Non-Probability Samples
Judgement Convenience
Probability Samples
Simple Random
Systematic
Stratified
Cluster
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-48
Non-Probability Samples
It may be simpler or less costly to use a non-probability based sampling method Judgement sample Quota sample Convience sample
These methods may still produce good estimates of population parameters
But … Are more subject to bias No valid way to determine reliability
(continued)
Statistics for Business and Economics, 6e © 2007 Pearson Education, Inc. Chap 20-49
Chapter Summary
Reviewed basic steps in a sampling study Defined sampling and nonsampling errors Examined probability sampling methods
Simple Random Sampling, Systematic Sampling, Stratified Random Sampling, 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