7/26/2019 OPRE 6301-SYSM 6303 Chapter 13 Slides_students
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OPRE 6301/SYSM 6303Quantitative Introduction to Risk and
Uncertainty in Business
13-1
Chapter ThirteenInference about Comparing Two Populations
13-2
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F Distribution
13-3
F is defined as the value FA,n1,n2
AFFPA
)(21 ,, nn
F Distribution
Use the F-table to determine the value FA,n1,n2
13-4
12
21
,,
,,1
1
nn
nn
A
AF
F
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F Distribution
13-5
n1 = numerator degrees of freedom
n2 = denominator degrees of freedom
F Distribution
13-6
F0.05,5,7 = 3.97
166.004.6
11
4,8,05.0
8,4,95.0 F
F
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Difference in Two Means:Independent Samples
13-7
normal if populations are normal
approximately normal otherwise if samples sizes are large
21 xx
Difference in Two Means:Independent Samples
13-8
normal if populations are normal
approximately normal otherwise if samples sizes are large
21 xx
2121 )( xxE
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Difference in Two Means:Independent Samples
13-9
normal if populations are normal
approximately normal otherwise if samples sizes are large
21 xx
2121 )( xxE
2
2
2
1
2
1
21)(
nnxxV
Difference in Two Means:Independent Samples
13-10
normal if populations are normal
approximately normal otherwise if samples sizes are large
The standard error
21 xx
2121 )( xxE
2
22
1
21
21)(
nnxxV
2
2
2
1
2
1
nn
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Difference in Two Means:Independent Samples
13-11
Thus,
Is a standard normal(or approximately normal)
random variable
2
2
2
1
2
1
2121
nn
xxz
Difference in Two Means:Independent Samples
13-12
The interval estimator is
2
2
2
1
2
1221
nnzxx
7/26/2019 OPRE 6301-SYSM 6303 Chapter 13 Slides_students
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Difference in Two Means:Independent Samples
13-13
The interval estimator is
However, these equations are rarely used because valuesfor the population standard deviations are virtually always
unknown.
2
2
2
1
2
1221
nnzxx
Difference in Two Means:Independent Samples
13-14
The interval estimator is
However, these equations are rarely used because values
for the population standard deviations are virtually always
unknown.
Instead, we will use an estimate of standard error of thesampling distribution.
2
2
2
1
2
1221
nnzxx
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Difference in Two Means:Independent Samples
13-15
Test Statistic for 1-2 when variances are equal
and
2where
1121
21
2
2121
nn
nns
xxt
p
n
2
11
21
2
22
2
112
nn
snsnsp
Difference in Two Means:Independent Samples
13-16
Interval Estimator for 1-2 when variances are equal
2where11
21
21
2
,221
nnnn
stxxp
nn
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Difference in Two Means:Independent Samples
13-17
Test Statistic for 1-2 when variances are unequal
11
where
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
2
2
2
1
2
1
2121
n
ns
n
ns
nsns
n
s
n
s
xxt n
Difference in Two Means:Independent Samples
13-18
Test Statistic for 1-2 when variances are unequal
Interval Estimator for 1-2 when variances are unequal
11
where
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
2
2
2
1
2
1
2121
n
ns
n
ns
nsns
n
s
n
s
xxt n
11
where
2
2
2
2
2
1
2
1
2
1
2
2
2
21
2
1
2
2
2
1
2
1
,221
n
ns
n
ns
nsns
n
s
n
stxx n
n
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Difference in Two Means:Independent Samples
13-19
Testing the Population Variances
1:
1:
2
2
2
11
2
2
2
10
H
H
Difference in Two Means:Independent Samples
13-20
Testing the Population Variances
The test statistic is
which is F-distributed with
1:
1:
2
2
2
11
2
2
2
10
H
H
2
2
21
s
s
1
1
22
11
n
n
n
n
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Difference in Two Means:Independent Samples
13-21
Testing the Population Variances
This is a two-tailed test with rejection region
or
1:
1:
2
2
2
11
2
2
2
10
H
H
21 ,,2 nnFF
21 ,,21 nn FF
Difference in Two Means:Independent Samples
13-22
Lets investigate Examples 13.1 and 13.2
7/26/2019 OPRE 6301-SYSM 6303 Chapter 13 Slides_students
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Difference in Two Means:Independent Samples
13-23
Lets investigate Examples 13.1 and 13.2
Testing and Estimating aRatio of Two Variances
13-24
Lets investigate Example 13.7
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Difference in TwoPopulation Proportions
13-25
is normally distributed if
n1p1, n1(1-p1), n2p2, n2(1-p2) are all >5
21 pp
Difference in TwoPopulation Proportions
13-26
is normally distributed if
n1p1, n1(1-p1), n2p2, n2(1-p2) are all >5
21 pp
2121 )( ppppE
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Difference in TwoPopulation Proportions
13-27
is normally distributed if
n1p1, n1(1-p1), n2p2, n2(1-p2) are all >5
21 pp
2121 )( ppppE
2
22
1
11
21
11)(
n
pp
n
ppppV
Difference in TwoPopulation Proportions
13-28
is normally distributed if
n1p1, n1(1-p1), n2p2, n2(1-p2) are all >5
The standard error
21 pp
2121 )( ppppE
2
22
1
1121
11)(n
pp
n
ppppV
2
22
1
11
1121 n
pp
n
pppp
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Difference in TwoPopulation Proportions
13-29
Thus,
is approximately standard normally distributed.
2
22
1
11
2121
11
n
pp
n
pp
ppppz
Difference in TwoPopulation Proportions
13-30
Thus,
is approximately standard normally distributed.
Again, this equation is rarely used because values for the
population proportions are virtually always unknown.
2
22
1
11
2121
11
n
pp
n
pp
ppppz
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Difference in TwoPopulation Proportions
13-31
Test Statistic for p1 - p2
For H0: p1 - p2 = 0
21
21
111
nnpp
ppz
Difference in TwoPopulation Proportions
13-32
Test Statistic for p1 - p2
For H0: p1 - p2 = D, .
2
22
1
11
21
11
n
pp
n
pp
Dppz
0D
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Difference in TwoPopulation Proportions
13-33
Interval Estimator for p1 - p2
2
22
1
11
221
)1()1(
n
pp
n
ppzpp
Difference in TwoPopulation Proportions
13-34
Lets investigate Example 13.9
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