OPRE 6301-SYSM 6303 Chapter 13 Slides_students

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


13-8



21 xx

2121 )( xxE


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13-9



21 xx

2121 )( xxE

2

2

2

1

2

1

21)(

nnxxV


13-10



The standard error

21 xx

2121 )( xxE

2

22

1

21

21)(

nnxxV

2

2

2

1

2

1

nn


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13-11

Thus,

Is a standard normal(or approximately normal)

random variable

2

2

2

1

2

1

2121

nn

xxz


13-12

The interval estimator is

2

2

2

1

2

1221

nnzxx


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13-13


However, these equations are rarely used because valuesfor the population standard deviations are virtually always

unknown.

2

2

2

1

2

1221

nnzxx


13-14


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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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


13-16

Interval Estimator for 1-2 when variances are equal

2where11

21

21

2

,221

nnnn

stxxp

nn


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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


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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13-19

Testing the Population Variances

1:

1:

2

2

2

11

2

2

2

10

H

H


13-20


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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13-21


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


13-22

Lets investigate Examples 13.1 and 13.2


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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


13-26


n1p1, n1(1-p1), n2p2, n2(1-p2) are all >5

21 pp

2121 )( ppppE


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13-27


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


13-28


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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13-29

Thus,

is approximately standard normally distributed.

2

22

1

11

2121

11

n

pp

n

pp

ppppz


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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13-31

Test Statistic for p1 - p2

For H0: p1 - p2 = 0

21

21

111

nnpp

ppz


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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13-33

Interval Estimator for p1 - p2

2

22

1

11

221

)1()1(

n

pp

n

ppzpp


13-34

Lets investigate Example 13.9

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OPRE 6301-SYSM 6303 Chapter 13 Slides_students