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16-1
COMPLETE COMPLETE BUSINESS BUSINESS
STATISTICSSTATISTICSbyby
AMIR D. ACZELAMIR D. ACZEL
&&
JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN
66thth edition (SIE) edition (SIE)
16-2
Chapter 16 Chapter 16
Sampling MethodsSampling Methods
16-3
• Using Statistics• Nonprobability Sampling and Bias• Stratified Random Sampling• Cluster Sampling• Systematic Sampling• Nonresponse
Sampling MethodsSampling Methods1616
16-4
• Apply nonprobability sampling methods• Decide when to conduct a stratified
sampling method• Compute estimates from stratified sample
results• Decide when to conduct a cluster sampling
method
LEARNING OUTCOMESLEARNING OUTCOMES1616
After studying this chapter you should be able to:After studying this chapter you should be able to:
16-5
• Compute estimates from cluster sampling results
• Decide when to conduct a systematic sampling method
• Compute estimates from systematic sample results
• Avoid nonresponse biases in estimates
LEARNING OUTCOMES (2)LEARNING OUTCOMES (2)1616
After studying this chapter you should be able to:After studying this chapter you should be able to:
16-6
• Sampling methods that do not use samples with known probabilities of selection are know as nonprobability sampling methodsnonprobability sampling methods.
• In nonprobability sampling methods, there is no objective way of evaluating how far away from the population parameter the estimate may be.
• FrameFrame - a list of people or things of interest from which a random sample can be chosen.
16-2 Nonprobability Sampling and 16-2 Nonprobability Sampling and BiasBias
16-7
In stratified random samplingstratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2 +...+ Nm =N.
In stratified random samplingstratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2 +...+ Nm =N.
16-3 Stratified Random Sampling16-3 Stratified Random Sampling
2 Stratum
1 Stratum
mStratum
1N
2N
mN
The m strata are non-overlapping.The m strata are non-overlapping.
NNm
i i
1NN
m
i i
1
PopulationPopulation
16-8
In stratified random samplingstratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2 +...+ Nm =N.
In stratified random samplingstratified random sampling, we assume that the population of N units may be divided into m groups with Ni units in each group i=1,2,...,m. The m strata are nonoverlapping and together they make up the total population: N1 + N2 +...+ Nm =N.
7654321 Group
NiNi
Population DistributionPopulation Distribution7654321 Group
nini
Sample DistributionSample Distribution
In proportional allocationproportional allocation, the relative frequencies in the sample (ni/n) are the same as those in the population (Ni/N) .
In proportional allocationproportional allocation, the relative frequencies in the sample (ni/n) are the same as those in the population (Ni/N) .
16-3 Stratified Random Sampling 16-3 Stratified Random Sampling (Continued)(Continued)
16-9
m
i iX
iWX
is
iX
i
ni
n
if
Ni
N
iW
1st
:sampling random stratifiedin The
2 :i stratumin varianceSample
:i stratumin mean Sample
2 :i stratum of varianceTrue
2 :population theof varianceTrue
i :i stratumin mean True
:population ofmean True
:i stratumin fraction Sampling
:i stratum of weight True
mean population the ofestimator
m
i iX
iWX
is
iX
i
ni
n
if
Ni
N
iW
1st
:sampling random stratifiedin The
2 :i stratumin varianceSample
:i stratumin mean Sample
2 :i stratum of varianceTrue
2 :population theof varianceTrue
i :i stratumin mean True
:population ofmean True
:i stratumin fraction Sampling
:i stratum of weight True
mean population the ofestimator
Relationship Between the Population Relationship Between the Population and a Stratified Random Sampleand a Stratified Random Sample
16-10
1. If the estimator of the mean in each stratum, Xi , is then the stratified
estimator of the mean, Xst is an estimator of the population mean, .
2. If the samples in the different strata are drawn independently of each other, then the variance of the stratified estimator of the population mean, Xst , is given by:
( ) = Xii=1
m
If sampling in all strata is random, then the variance of Xst is further equal to:
( ) =i=1
m
When the sampling fractions, , are small and may be ignored, we have:
unbiased
unbiased,
( )
.
( )
V Xst Wi V
V Xst Wii
ni
fi
fi
2
3
22
1
V Xst Wii
ni
( ) =i=1
m 22
1. If the estimator of the mean in each stratum, Xi , is then the stratified
estimator of the mean, Xst is an estimator of the population mean, .
2. If the samples in the different strata are drawn independently of each other, then the variance of the stratified estimator of the population mean, Xst , is given by:
( ) = Xii=1
m
If sampling in all strata is random, then the variance of Xst is further equal to:
( ) =i=1
m
When the sampling fractions, , are small and may be ignored, we have:
unbiased
unbiased,
( )
.
( )
V Xst Wi V
V Xst Wii
ni
fi
fi
2
3
22
1
V Xst Wii
ni
( ) =i=1
m 22
Properties of the Stratified Estimator Properties of the Stratified Estimator of the Sample Meanof the Sample Mean
16-11
4. If the sample allocation is proportional for all i , then
( ) =1 - f
n i=1
m
which reduces to
( ) =1
n i=1
m
when the sampling fraction is small. In addition, if the population variances in all strata are equal, then
( ) =2
n
when the sampling fraction is small.
ni
nN
iN
V X st Wi
V X st Wi
V X st
i
i
2
2
4. If the sample allocation is proportional for all i , then
( ) =1 - f
n i=1
m
which reduces to
( ) =1
n i=1
m
when the sampling fraction is small. In addition, if the population variances in all strata are equal, then
( ) =2
n
when the sampling fraction is small.
ni
nN
iN
V X st Wi
V X st Wi
V X st
i
i
2
2
Properties of the Stratified Estimator Properties of the Stratified Estimator of the Sample Mean (continued)of the Sample Mean (continued)
16-12
)1(m
1=i
2i=)(2
:random is stratumeach in sampling If
iin data 1
2)(2i
:is , 2i
i, stratum of variancepopulation theofestimator unbiasedAn
2
i
i
fn
Si
W
stXS
in
iXX
S
)1(m
1=i
2i=)(2
:random is stratumeach in sampling If
iin data 1
2)(2i
:is , 2i
i, stratum of variancepopulation theofestimator unbiasedAn
2
i
i
fn
Si
W
stXS
in
iXX
S
When the Population Variance is When the Population Variance is UnknownUnknown
16-13
m
i in
is
in
in
iN
iN
m
i ini
s
in
iN
iN
stXsz
1 )1(
42
)/(
2
1=
2
)(
= df Effective
:freedom of degrees effective The
formula. above in the value- t theuse
unknown, are variancespopulation theand small, are sizes sample When the
)(st
x
:sampling
stratified using , mean, population for the interval confidence )100%-(1A
2
m
i in
is
in
in
iN
iN
m
i ini
s
in
iN
iN
stXsz
1 )1(
42
)/(
2
1=
2
)(
= df Effective
:freedom of degrees effective The
formula. above in the value- t theuse
unknown, are variancespopulation theand small, are sizes sample When the
)(st
x
:sampling
stratified using , mean, population for the interval confidence )100%-(1A
2
Confidence Interval for the Population Confidence Interval for the Population Mean in Stratified SamplingMean in Stratified Sampling
16-14
Population True SamplingNumber Weights Sample Fraction
Group of Firms (Wi) Sizes (fi) 1. Diversified service companies 100 0.20 20 0.202. Commercial banking companies 100 0.20 20 0.203. Financial service companies 150 0.30 30 0.304. Retailing companies 50 0.10 10 0.105. Transportation companies 50 0.10 10 0.106. Utilities 50 0.10 10 0.10
N = 500 n = 100
Population True SamplingNumber Weights Sample Fraction
Group of Firms (Wi) Sizes (fi) 1. Diversified service companies 100 0.20 20 0.202. Commercial banking companies 100 0.20 20 0.203. Financial service companies 150 0.30 30 0.304. Retailing companies 50 0.10 10 0.105. Transportation companies 50 0.10 10 0.106. Utilities 50 0.10 10 0.10
N = 500 n = 100
Stratum Mean Variance ni Wi Wixi
1 52.7 97650 20 0.2 10.54 156.240 2 112.6 64300 20 0.2 22.52 102.880 3 85.6 76990 30 0.3 25.68 184.776 4 12.6 18320 10 0.1 1.26 14.656 5 8.9 9037 10 0.1 0.89 7.230 6 52.3 83500 10 0.1 5.23 66.800
Estimated Mean: 66.12 532.582Estimated standard error of mean: 23.08
1 f
nWi si
2 95% Confdence Interval: x
st
66
z s Xst
2
12 1 96 23 0866 12 45 2420 88 111 36
( )
. ( . )( . )
. .[ . , . ]
95% Confdence Interval: x
st
66
z s Xst
2
12 1 96 23 0866 12 45 2420 88 111 36
( )
. ( . )( . )
. .[ . , . ]
Example 16-2Example 16-2
16-15
Example 16-2 Using the templateExample 16-2 Using the template
Observe that the computer gives a slightly more precise interval than the hand computation on the previous slide.
Observe that the computer gives a slightly more precise interval than the hand computation on the previous slide.
16-16
Stratified estimator of the population proportion, ,
The approximate variance of
V(
When the finite - population correction factors, must be considered:
V(
When proportional allocation is used:
V(
p
Pst Wi Pii
m
Pst
Pst WiPi Qinii
m
f
PstN
Ni
Ni
ni
Pi QiN
inii
m
Pstf
nWi Pi Qii
m
i
,
)
,
) ( )
( )
)
1
2
1
1
2
2
11
1
1
Stratified estimator of the population proportion, ,
The approximate variance of
V(
When the finite - population correction factors, must be considered:
V(
When proportional allocation is used:
V(
p
Pst Wi Pii
m
Pst
Pst WiPi Qinii
m
f
PstN
Ni
Ni
ni
Pi QiN
inii
m
Pstf
nWi Pi Qii
m
i
,
)
,
) ( )
( )
)
1
2
1
1
2
2
11
1
1
Stratified Sampling for the Population Stratified Sampling for the Population ProportionProportion
16-17
NumberGroup Wi ni fi InterestedMetropolitan 0.65 130 0.65 28 0.14 0.0005756Nonmetropolitan 0.35 70 0.35 18 0.09 0.0003099
Estimated proportion: 0.23 0.0008855Estimated standard error: 0.0297574
90% confidence interval:[0.181,0.279]
Wi pi
Wi pi qi
n
90% Confdence Interval: p
st ( )
. ( . )( . )
. .[ . , . ]
z s Pst
2
0 23 1 645 0 2970 23 0 0490 181 0 279
Stratified Sampling for the Population Stratified Sampling for the Population Proportion: Example 16-1 (Continued)Proportion: Example 16-1 (Continued)
16-18Stratified Sampling for the Population Stratified Sampling for the Population Proportion:Example 16-1 (Continued) using Proportion:Example 16-1 (Continued) using the Templatethe Template
16-19
1. Preferably no more than 6 strata.2. Choose strata so that Cum f(x) is approximately constant for all strata (Cum f(x) is the cumulative square root of the frequency of X, the variable of interest).
Age Frequency (fi) 20-25 1 126-30 16 4 531-35 25 5 536-40 4 241-45 9 3 5
f(x) Cum f(x)
Rules for Constructing StrataRules for Constructing Strata
16-20
For optimum allocation of effort in stratified random sampling, minimize thecost for a given variance, or minimize the variance for a given cost.
Total Cost = Fixed Cost + Variable Cost C = C0 Cini
Optimum Allocation: nin
(Wi i ) / Ci(Wi i ) / Ci
If the cost per unit sampled is the same for all strata (Ci = c):
Neyman Allocation: nin
(Wi i )
(Wi i )
For optimum allocation of effort in stratified random sampling, minimize thecost for a given variance, or minimize the variance for a given cost.
Total Cost = Fixed Cost + Variable Cost C = C0 Cini
Optimum Allocation: nin
(Wi i ) / Ci(Wi i ) / Ci
If the cost per unit sampled is the same for all strata (Ci = c):
Neyman Allocation: nin
(Wi i )
(Wi i )
Optimum AllocationOptimum Allocation
16-21
1 0.4 1 4 0.4 0.200 0.329 0.235 2 0.5 2 9 1.0 0.333 0.548 0.588 3 0.1 3 16 0.3 0.075 0.123 0.176
i W W i isi Ci si W
isi
Ci
Optimum
Allocation
Neyman
Allocation
1.7 0.608
1 0.4 1 4 0.4 0.200 0.329 0.235 2 0.5 2 9 1.0 0.333 0.548 0.588 3 0.1 3 16 0.3 0.075 0.123 0.176
i W W i isi Ci si W
isi
Ci
Optimum
Allocation
Neyman
Allocation
1.7 0.608
Optimum Allocation: An ExampleOptimum Allocation: An Example
16-22
Optimum Allocation: An Example Optimum Allocation: An Example using the Templateusing the Template
16-23
7654321Group
Population DistributionPopulation Distribution
In stratified sampling a random sample (ni) is chosen from each segment of the population (Ni).
In stratified sampling a random sample (ni) is chosen from each segment of the population (Ni).
Sample Distribution Sample Distribution
In cluster sampling observations are drawn from m out of M areas or clusters of the population.
In cluster sampling observations are drawn from m out of M areas or clusters of the population.
16-4 Cluster Sampling16-4 Cluster Sampling
16-24
Cluster sampling estimator of :
Estimator of the variance of the sample mean:
s
where
=
2
Xn X
n
XM mMmn
n X X
m
nn
m
cl
i ii
m
ii
m
cl
i i cli
m
ii
m
1
1
2
2 2
1
1
1( )
( )
Cluster sampling estimator of :
Estimator of the variance of the sample mean:
s
where
=
2
Xn X
n
XM mMmn
n X X
m
nn
m
cl
i ii
m
ii
m
cl
i i cli
m
ii
m
1
1
2
2 2
1
1
1( )
( )
Cluster Sampling: Estimating the Cluster Sampling: Estimating the Population MeanPopulation Mean
16-25
Cluster sampling estimator of :
Estimator of the variance of the sample proportion:
s
2
p
Pn P
n
PM mMmn
n P P
m
cl
i ii
m
ii
m
cl
i i cli
m
( )( )
1
1
2
2 2
1
1
Cluster sampling estimator of :
Estimator of the variance of the sample proportion:
s
2
p
Pn P
n
PM mMmn
n P P
m
cl
i ii
m
ii
m
cl
i i cli
m
( )( )
1
1
2
2 2
1
1
Cluster Sampling: Estimating the Cluster Sampling: Estimating the Population ProportionPopulation Proportion
16-26
95% Confdence Interval: x
cl
z s Xcl
2
2183 1 96 15872183 2 4719 36 24 30
( )
. ( . )( . )
. .[ . , . ]
95% Confdence Interval: x
cl
z s Xcl
2
2183 1 96 15872183 2 4719 36 24 30
( )
. ( . )( . )
. .[ . , . ]
xi ni nixi xi-xcl (xi-xcl)2
21 8 168 -0.8333 0.694 0.0011822 8 176 0.1667 0.028 0.0000511 9 99 -10.8333 117.361 0.2526934 10 340 12.1667 148.028 0.3934828 7 196 6.1667 38.028 0.0495325 8 200 3.1667 10.028 0.0170618 10 180 -3.8333 14.694 0.0390624 12 288 2.1667 4.694 0.0179719 11 209 -2.8333 8.028 0.0258220 6 120 -1.8333 3.361 0.0032230 8 240 8.1667 66.694 0.1134626 9 234 4.1667 17.361 0.0373812 9 108 -9.8333 96.694 0.2081917 8 136 -4.8333 23.361 0.0397413 10 130 -8.8333 78.028 0.2074129 8 232 7.1667 51.361 0.0873824 8 192 2.1667 4.694 0.0079926 10 260 4.1667 17.361 0.0461518 10 180 -3.8333 14.694 0.0390622 11 242 0.1667 0.028 0.00009
3930 s2(Xcl)= 1.58691xcl = 21.83
M mMmn
n X Xm
i i cl
2
2 2
1( )
Cluster Sampling: Example 16-3Cluster Sampling: Example 16-3
16-27
Cluster Sampling: Example 16-3 Using Cluster Sampling: Example 16-3 Using the Templatethe Template
16-28
Cluster Sampling: Using the Template Cluster Sampling: Using the Template to Estimate Population Proportionto Estimate Population Proportion
16-29
Randomly select an element out of the first k elements in the population, and then select every kth unit afterwards until we have a sample of n elements.
Randomly select an element out of the first k elements in the population, and then select every kth unit afterwards until we have a sample of n elements.
Systematic sampling estimator of :
Estimator of the variance of the sample mean: s2
When the mean is constant within each stratum of k elements but different between strata:
s2
When the population is linearly increasing or decreasing with respect to the variable of interest:
s2
X sy
Xii
m
n
X syN n
NnS
X syN n
Nn
Xi Xi ki
n
n
X syN n
Nn
Xi Xi k Xi ki
n
n
1
2
2
1
2 1
2 22
1
6 2
( )
( )( )
( )
( )( )
( )
16-5 Systematic Sampling16-5 Systematic Sampling
16-30
s2
s2
A 95% confidence interval for the average price change for all stocks: s
X sy
Xii
m
n
X syN n
NnS
X sy X sy
1 0 5 0 36
2 2100 1002100 100
0 36 0 0034
1 96
0 5 1 96 0 00340 5 0114
0 386 0 614
. .
( )( )( )
. .
( . ) ( )
. ( . )( . )
. .[ . , . ]
s2
s2
A 95% confidence interval for the average price change for all stocks: s
X sy
Xii
m
n
X syN n
NnS
X sy X sy
1 0 5 0 36
2 2100 1002100 100
0 36 0 0034
1 96
0 5 1 96 0 00340 5 0114
0 386 0 614
. .
( )( )( )
. .
( . ) ( )
. ( . )( . )
. .[ . , . ]
Systematic Sampling: Example 16-4Systematic Sampling: Example 16-4
16-31
Systematic nonresponse can biasbias estimatesCallbacks of nonrespondentsOffers of monetary rewards for nonrespondentsRandom-response mechanism
Systematic nonresponse can biasbias estimatesCallbacks of nonrespondentsOffers of monetary rewards for nonrespondentsRandom-response mechanism
16-6 Nonresponse16-6 Nonresponse
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