Chapter 8 The Comparison of Two Populations COMPLETE BUSINESS STATISTICSby AMIR D. ACZEL & JAYAVEL...

Chapter 8Chapter 8The Comparison of Two PopulationsThe Comparison of Two Populations

COMPLETE BUSINESS STATISTICS

bybyAMIR D. ACZELAMIR D. ACZEL

&&JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN

7th edition.7th edition.

Prepared by Prepared by Lloyd Jaisingh, Morehead State Lloyd Jaisingh, Morehead State UniversityUniversity

McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved.

• Using Statistics• Paired-Observation Comparisons• A Test for the Difference between Two Population Means Using

Independent Random Samples• A Large-Sample Test for the Difference between Two Population

Proportions• The F Distribution and a Test for the Equality of Two Population

Variances

The Comparison of Two PopulationsThe Comparison of Two Populations888-2

• Explain the need to compare two population parameters• Conduct a paired-difference test for the difference in population means• Conduct an independent-samples test for the difference in population means• Describe why a paired-difference test is better than independent-samples test• Conduct a test for difference in population proportions• Test whether two population variances are equal• Use templates to carry out all tests

LEARNING OBJECTIVESLEARNING OBJECTIVES88

After studying this chapter you should be able to:After studying this chapter you should be able to:

8-3

• Inferences about differences between parameters of two populations Paired-Observations Observe the samesame group of persons or things

At two different times: “before” and “after” Under two different sets of circumstances or “treatments”

Independent Samples Observe differentdifferent groups of persons or things

At different times or under different sets of circumstances

8-1 Using Statistics8-4

• Population parameters may differ at two different times or under two different sets of circumstances or treatments because: The circumstances differ between times or treatments The people or things in the different groups are themselves different

• By looking at paired-observations, we are able to minimize the “between group” , extraneous variation.

8-2 Paired-Observation Comparisons8-5

Paired-Observation Comparisons of Means

hypothesis null under the sdifference of population theofmean size sample

sdifference for thedeviation standard samplesdifference for the average sample

1 ,0

testnsobservatio-paired for the statisticTest

0D

nDS

D

ndf

nDS

DD

t

hypothesis null under the sdifference of population theofmean size sample

sdifference for thedeviation standard samplesdifference for the average sample

1 ,0

testnsobservatio-paired for the statisticTest

0D

nDS

D

ndf

nDS

DD

t

8-6

A random sample of 16 viewers of Home Shopping Network was selected for an experiment. All viewers in the sample had recorded the amount of money they spent shopping during the holiday season of the previous year. The next year, these people were given access to the cable network and were asked to keep a record of their total purchases during the holiday season. Home Shopping Network managers want to test the null hypothesis that their service does not increase shopping volume, versus the alternative hypothesis that it does.

A random sample of 16 viewers of Home Shopping Network was selected for an experiment. All viewers in the sample had recorded the amount of money they spent shopping during the holiday season of the previous year. The next year, these people were given access to the cable network and were asked to keep a record of their total purchases during the holiday season. Home Shopping Network managers want to test the null hypothesis that their service does not increase shopping volume, versus the alternative hypothesis that it does.

Shopper Previous Current Diff 1 334 405 71 2 150 125 -25 3 520 540 20 4 95 100 5 5 212 200 -12 6 30 30 0 7 1055 1200 145 8 300 265 -35 9 85 90 510 129 206 7711 40 18 -2212 440 489 4913 610 590 -2014 208 310 10215 880 995 11516 25 75 50

Shopper Previous Current Diff 1 334 405 71 2 150 125 -25 3 520 540 20 4 95 100 5 5 212 200 -12 6 30 30 0 7 1055 1200 145 8 300 265 -35 9 85 90 510 129 206 7711 40 18 -2212 440 489 4913 610 590 -2014 208 310 10215 880 995 11516 25 75 50

H0: D 0H1: D > 0

df = (n-1) = (16-1) = 15

Test Statistic:

Critical Value: t0.05 = 1.753

Do not reject H0 if : t 1.753 Reject H0 if: t > 1.753

H0: D 0H1: D > 0

df = (n-1) = (16-1) = 15

Test Statistic:

Critical Value: t0.05 = 1.753

Do not reject H0 if : t 1.753 Reject H0 if: t > 1.753

t

D D

sD

n

0

Example 8-18-7

t

D D

sD

n

0 32 81 0

55 75

16

2 354.

..

2.131= t0.025

2.602= t0.01

1.753= t0.05

2.354=test statistic

50-5

0.4

0.3

0.2

0.1

0.0

t

f(t)

t Distribution: df=15

Nonrejection Region

Rejection Region

t = 2.354 > 1.753, so H0 is rejected and we conclude that there is evidence that shopping volume by network viewers has increased, with a p-value between 0.01 an 0.025. The Template output gives a more exact p-value

of 0.0163. See the next slide for the output.

t = 2.354 > 1.753, so H0 is rejected and we conclude that there is evidence that shopping volume by network viewers has increased, with a p-value between 0.01 an 0.025. The Template output gives a more exact p-value

of 0.0163. See the next slide for the output.

Example 8-1: Solution8-8

Example 8-1: Using the Template for Testing Paired Differences

Decision: Reject the null hypothesis

8-9

Example 8-1: Using Minitab for Testing Paired Differences

Decision: Reject the null hypothesis, P-value < 0.05

8-10

It has recently been asserted that returns on stocks may change once a story about a company appears in The Wall Street Journal column “Heard on the Street.” An investments analyst collects a random sample of 50 stocks that were recommended as winners by the editor of “Heard on the Street,” and proceeds to conduct a two-tailed test of whether or not the annualized return on stocks recommended in the column differs between the month before and the month after the recommendation. For each stock the analysts computes the return before and the return after the event, and computes the difference in the two return figures. He then computes the average and standard deviation of the differences.

It has recently been asserted that returns on stocks may change once a story about a company appears in The Wall Street Journal column “Heard on the Street.” An investments analyst collects a random sample of 50 stocks that were recommended as winners by the editor of “Heard on the Street,” and proceeds to conduct a two-tailed test of whether or not the annualized return on stocks recommended in the column differs between the month before and the month after the recommendation. For each stock the analysts computes the return before and the return after the event, and computes the difference in the two return figures. He then computes the average and standard deviation of the differences.

H0: D 0H1: D > 0

n = 50D = 0.1%sD = 0.05%

Test Statistic:

nDs

DDz 0

This test result is highly significant,and H0 may be rejected at any reasonable

level of significance.

p - value:

z

D D

sD

n

p z

0 0 1 0

0 05

50

14 14

14 14 0

.

..

( . )

This test result is highly significant,and H0 may be rejected at any reasonable


p - value:

z

D D

sD

n

p z

0 0 1 0

0 05

50

14 14

14 14 0

.

..

( . )

Example 8-28-11

.2z with 2 teapproximatmay we large, is size sample When theright, its to2 of areaan off cutsthat

freedom of degrees 1)-(non with distributi t theof value theis twhere

tD

:D differencemean for the interval confidence 100% )-(1A

2

2

n

sD

.2z with 2 teapproximatmay we large, is size sample When theright, its to2 of areaan off cutsthat

freedom of degrees 1)-(non with distributi t theof value theis twhere

tD

:D differencemean for the interval confidence 100% )-(1A

2

2

n

sD

Confidence Intervals for Paired Observations

8-12

0. value theincludenot does interval confidence is that thNote

]114.0 ,086.0[ 014.0 1.0

)0071)(.96.1( 1.0 50

0.051.96 0.1 z D

: 28 Examplein data for the interval confidence 95%

2

nsD

0. value theincludenot does interval confidence is that thNote

]114.0 ,086.0[ 014.0 1.0

)0071)(.96.1( 1.0 50

0.051.96 0.1 z D

: 28 Examplein data for the interval confidence 95%

2

nsD

Confidence Intervals for Paired Observations – Example 8-2

8-13

Hypothesis Test & Confidence Interval for Example 8-2 - Using the Template

Decision: Reject the null hypothesis. Confidence Interval

8-14

• When paired data cannot be obtained, use independent random samples drawn at different times or under different circumstances. Large sample test if:

Both n130 and n230 (Central Limit Theorem), or

Both populations are normal and 1 and 2 are both known

Small sample test if: Both populations are normal and 1 and 2 are unknown

8-3 A Test for the Difference between Two Population Means Using Independent Random Samples

8-15

• I: Difference between two population means is 0 1= 2

H0: 1 -2 = 0

H1: 1 -2 0

• II: Difference between two population means is less than 0 12

H0: 1 -2 0

H1: 1 -2 0

• III: Difference between two population means is less than D 1 2+D

H0: 1 -2 D

H1: 1 -2 D

Comparisons of Two Population Means: Testing Situations

8-16

• IV: Difference between two population means is greater than 0 12

H0: 1 -2 0

H1: 1 -2 0

• V: Difference between two population means is greater than D

1 2+ D

H0: 1 -2 D

H1: 1 -2 D

Comparisons of Two Population Means: Testing Situations

8-17

Large-sample test statistic for the difference between two population means:

The term (1- 2)0 is the difference between 1 an 2 under the null hypothesis. Is is equal to zero in situations I , II and IV, and it is equal to the prespecified value D in situations III and V. The term in the denominator is the standard deviation of the difference between the two sample means (it relies on the assumption that the two samples are independent).

Large-sample test statistic for the difference between two population means:

The term (1- 2)0 is the difference between 1 an 2 under the null hypothesis. Is is equal to zero in situations I , II and IV, and it is equal to the prespecified value D in situations III and V. The term in the denominator is the standard deviation of the difference between the two sample means (it relies on the assumption that the two samples are independent).

2

2

2

1

2

1

02121)()(

nn

xxz

Comparisons of Two Population Means: Test Statistic

8-18

212 =

452 =x

1200=n

Visa Preferred :1 Population

1

1

1

185 =

523 =x

800=n

Card Gold :2 Population

2

2

2

Is there evidence to conclude that the average monthly charge in the entire population of American Express Gold Card members is different from the average monthly charge in the entire population of Preferred Visa cardholders?

cesignifican of levelcommon any at rejected is 0

H

0 -7.926)<p(z :value-p

926.796.8

71

2346.80

71

800

2185

1200

2212

0)523452(

2

22

1

21

0)

21()

21(

021

:1

H

021

:0

H

nn

xxz

Two-Tailed Test for Equality of Two Population Means: Example 8-3

8-19

0.4

0.3

0.2

0.1

0.0z

f( z)

Standard Normal Distribution

NonrejectionRegion

RejectionRegion

-z0.01=-2.576 z0.01=2.576

Test Statistic=-7.926

RejectionRegion

0

Since the value of the test statistic is far below the lower critical point, the null hypothesis may be rejected, and we may conclude that there is a statistically significant difference between the average monthly charges of Gold Card and Preferred Visa cardholders.

Since the value of the test statistic is far below the lower critical point, the null hypothesis may be rejected, and we may conclude that there is a statistically significant difference between the average monthly charges of Gold Card and Preferred Visa cardholders.

Example 8-3: Carrying Out the Test8-20

Example 8-3: Using the Template

Decision: reject the null hypothesis.

8-21

84=

308=x

100=n

Duracell :1 Population

1

1

1

67=

254=x

100=n

Energizer :2 Population

2

2

2

Is there evidence to substantiate Duracell’s claim that their batteries last, on average, at least 45 minutes longer than Energizer batteries of the same size?

cesignifican of level

common any at rejected benot may 0

H

0.201=0.838)>p(z :value-p

838.075.10

9

45.115

9

100

267

100

284

45)254308(

2

22

1

21

0)

21()

21(

4521

:1

H

4521

:0

H

nn

xxz

Example 8-48-22

Is there evidence to substantiate Duracell’s claim that their batteries last, on average, at least 45 minutes longer than Energizer batteries of the same size?

Example 8-4 – Using the Template

P-value

8-23

A large-sample (1-)100% confidence interval for the difference between two population means, 1- 2 , using independent random samples:

A large-sample (1-)100% confidence interval for the difference between two population means, 1- 2 , using independent random samples:

2

22

1

21

2

)21

(nn

zxx

A 95% confidence interval using the data in example 8-3:


]56.88,44.53[800

2185

1200

221296.1)452523(

2

22

1

21

2

)21

( nn

zxx

Confidence Intervals for the Difference between Two Population Means

8-24

If we might assume that the population variances 12 and 2

2 are equal (even though unknown), then the two sample variances, s1

2 and s22, provide two separate estimators of

the common population variance. Combining the two separate estimates into a pooled estimate should give us a better estimate than either sample variance by itself.

If we might assume that the population variances 12 and 2

2 are equal (even though unknown), then the two sample variances, s1

2 and s22, provide two separate estimators of

the common population variance. Combining the two separate estimates into a pooled estimate should give us a better estimate than either sample variance by itself.

x1

** ** ** * ** * ** * *

}Deviation from the mean. One for each sample data point.

Sample 1

From sample 1 we get the estimate s12 with

(n1-1) degrees of freedom.

Deviation from the mean. One for each sample data point.

* * ** ** * * * * ** * *

x2

}

Sample 2

From sample 2 we get the estimate s22 with

(n2-1) degrees of freedom.

From both samples together we get a pooled estimate, sp2 , with (n1-1) + (n2-1) = (n1+ n2 -2)

total degrees of freedom.

A Test for the Difference between Two Population Means: Assuming Equal Population Variances

8-25

A pooled estimate of the common population variance, based on a sample variance s1

2 from a sample of size n1 and a sample variance s22 from a sample

of size n2 is given by:

The degrees of freedom associated with this estimator is:

df = (n1+ n2-2)

A pooled estimate of the common population variance, based on a sample variance s1

2 from a sample of size n1 and a sample variance s22 from a sample

of size n2 is given by:

The degrees of freedom associated with this estimator is:

df = (n1+ n2-2)

sn s n s

n np

2 1 1

2

2 2

2

1 2

1 12

( ) ( )

The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two samples. That is, larger weight is given to the variance from the larger sample.

The pooled estimate of the variance is a weighted average of the two individual sample variances, with weights proportional to the sizes of the two samples. That is, larger weight is given to the variance from the larger sample.

Pooled Estimate of the Population Variance

8-26

The estimate of the standard deviation of (x1 x2 is given by: sp2

)1

1

1

2n nThe estimate of the standard deviation of (x1 x2 is given by: sp2

)1

1

1

2n n

Test statistic for the difference between two population means, assuming equal population variances:

t =(x1 x2 1 2

sp2

where 1 2 is the difference between the two population means under the null

hypothesis (zero or some other number D).

The number of degrees of freedom of the test statistic is df = ( 1 (the

number of degrees of freedom associated with sp2

, the pooled estimate of the

population variance.

) ( )

( )

)

0

1

1

1

2

0

2 2

n n

n n

Test statistic for the difference between two population means, assuming equal population variances:

t =(x1 x2 1 2

sp2

where 1 2 is the difference between the two population means under the null

hypothesis (zero or some other number D).

The number of degrees of freedom of the test statistic is df = ( 1 (the

number of degrees of freedom associated with sp2

, the pooled estimate of the

population variance.

) ( )

( )

)

0

1

1

1

2

0

2 2

n n

n n

Using the Pooled Estimate of the Population Variance

8-27

0.12%= 1s

0.317%=1x

14=1n $66.00 = price Oil :1 Population

Do the data provide sufficient evidence to conclude that average percentage increase in the CPI differs when oil sells at these two different prices?


H

H

Critical point: t0.025

= 2.080

H0 may be rejected at the 5% level of significance

0 1 2 0

1 1 2 0

1 2 1 2 0

1 1 12

2 1 22

1 2 2

1

1

1

2

0 107

0 00247

0 107

0 04972 154

:

:

( ) ( )

( ) ( )

.

.

.

..

tx x

n s n s

n n n n

21)2914()221(n=df

0.11%= s0.21%=x9=n

$58.00 = price Oil :2 Population

2

2

2

n

Example 8-58-28



Decision: reject the null hypothesis.

8-29


Example 8-5: Using Minitab

Decision: reject the null hypothesis; p-value = 0.043.

8-30

Population 1: Before Reduction

n1 = 15

x1 = $6598

s1 = $844

The manufacturers of compact disk players want to test whether a small price reduction is enough to increase sales of their product. Is there evidence that the small price reduction is enough to increase sales of compact disk players?

cesignifican of level 10% at theeven rejected benot may 0

H

1.316=0.10

t:point Critical

91.096.298

272

25.89375

272

12

1

15

1

21215

2669)11(

2844)14(

0)65986870(

2

1

1

1

221

22

)12

(21

)11

(

0)

12()

12(

012

:1

H

012

:0

H

nnnn

snsn

xxt

Population 2: After Reduction

n = 12

x = $6870

s = $669

df = (n1

2

2

2

n2

2 15 12 2 25) ( )

Example 8-68-31

543210-1-2-3-4-5

0.4

0.3

0.2

0.1

0.0t

f(t)

t Distribution: df = 25

NonrejectionRegion

RejectionRegion

t0.10=1.316

Test Statistic=0.91

Since the test statistic is less than t0.10, the null hypothesis cannot be rejected at any reasonable level of significance. We conclude that the price reduction does not significantly affect sales.

Since the test statistic is less than t0.10, the null hypothesis cannot be rejected at any reasonable level of significance. We conclude that the price reduction does not significantly affect sales.

Example 8-6: Continued8-32


Decision: Do not reject the null hypothesis; p-value = 0.1858.

8-33



8-34

A (1-) 100% confidence interval for the difference between two population means, 1- 2 , using independent random samples and assuming equal population variances:

A (1-) 100% confidence interval for the difference between two population means, 1- 2 , using independent random samples and assuming equal population variances:

( )x x t sn np1 2

2

2 1

1

1

2

A 95% confidence interval using the data in Example 8-6:

A 95% confidence interval using the data in Example 8-6:

( ) ( ) . ( )( . ) [ . , . ]x x t s pn n

1 2

2

2 1

1

1

2

6870 6598 2 06 595835 0 15 343 85 887 85

Confidence Intervals Using the Pooled Variance

8-35

Confidence Intervals Using the Pooled Variance and the Template-Example 8-6

Confidence IntervalConfidence IntervalNOTE: The MINITAB outputs have the confidenceIntervals included in the output as well.

8-36

• Hypothesized difference is zero I: Difference between two population proportions is 0

• p1= p2

» H0: p1 -p2 = 0

» H1: p1 -p20

II: Difference between two population proportions is less than 0

• p1p2

» H0: p1 -p2 0

» H1: p1 -p2 > 0

• Hypothesized difference is other than zero: III: Difference between two population proportions is less than D

• p1 p2+D

» H0:p-p2 D

» H1: p1 -p2 > D

8-4 A Large-Sample Test for the Difference between Two Population Proportions

8-37

• Hypothesized difference is zero IV: Difference between two population proportions is greater than 0

• p1p2

» H0: p1 -p2 0

» H1: p1 -p2 < 0

• Hypothesized difference is other than zero: V: Difference between two population proportions is greater than D

• p1 p2+D

» H0:p-p2 D

» H1: p1 -p2 < D

8-4 A Large-Sample Test for the Difference between Two Population Proportions

8-38

A large-sample test statistic for the difference between two population proportions, when the hypothesized difference is zero:

where is the sample proportion in sample 1 and is the sample

proportion in sample 2. The symbol stands for the combined sample proportion in both samples, considered as a single sample. That is:

A large-sample test statistic for the difference between two population proportions, when the hypothesized difference is zero:

where is the sample proportion in sample 1 and is the sample

proportion in sample 2. The symbol stands for the combined sample proportion in both samples, considered as a single sample. That is:

zp p

p pn n

( )

( )

1 2

1 2

0

11 1

pxn1

1

1

When the population proportions are hypothesized to be equal, then a pooled estimator of the proportion ( ) may be used in calculating the test statistic.

When the population proportions are hypothesized to be equal, then a pooled estimator of the proportion ( ) may be used in calculating the test statistic.

pxn1

1

1

p

21

11ˆnn

xxp

p

Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Test Statistic

8-39

Carry out a two-tailed test of the equality of banks’ share of the car loan market in 1980 and 1995.

Population 1: 1980

n1 = 100

x1 = 53

p1 = 0.53

H

H

Critical point: z0.05

= 1.645

H0 may not be rejected even at a 10%


0 1 2 0

1 1 2 0

1 2 0

11

1

1

2

0 53 0 43

48 521

100

1

100

0 10

0 004992

0 10

0 070651 415

:

:

( )

( )

. .

(. )(. )

.

.

.

..

p p

p p

zp p

p pn n

Population 2: 1995

n = 100

x = 43

p = 0.43

x1 + x2

n1 n2

2

2

2

.p

53 43

100 1000 48

Comparisons of Two Population Proportions When the Hypothesized Difference Is Zero: Example 8-7

8-40

0.4

0.3

0.2

0.1

0.0z

f( z)


NonrejectionRegion

RejectionRegion

-z0.05=-1.645 z0.05=1.645

Test Statistic=1.415

RejectionRegion

0

Since the value of the test statistic is within the nonrejection region, even at a 10% level of significance, we may conclude that there is no statistically significant difference between banks’ shares of car loans in 1980 and 1995.

Since the value of the test statistic is within the nonrejection region, even at a 10% level of significance, we may conclude that there is no statistically significant difference between banks’ shares of car loans in 1980 and 1995.




8-42



8-43

Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such buyers when no sweepstakes are on.

Carry out a one-tailed test to determine whether the population proportion of traveler’s check buyers who buy at least $2500 in checks when sweepstakes prizes are offered as at least 10% higher than the proportion of such buyers when no sweepstakes are on.

Population 1: With Sweepstakes

n1 = 300

x1 = 120

p1 = 0.40

H

H

Critical point: z0.001

= 3.09

H0 may be rejected at any common level of significance.

0 1 2 0 10

1 1 2 0 10

1 2

11

1

1

21

2

2

0 40 0 20 0 10

0 40 0 60

300

0 20 80

700

0 10

0 032073 118

: .

: .

( )

( ) ( )

( . . ) .

( . )( . ) ( . )(. )

.

..

p p

p p

zp p D

p p

n

p p

n

Population 2: No Sweepstakes

n = 700

x = 140

p = 0.20

2

2

2

Comparisons of Two Population Proportions When the Hypothesized Difference Is Not Zero: Example 8-8

8-44

0.4

0.3

0.2

0.1

0.0z

f( z)


NonrejectionRegion

RejectionRegion

z0.001=3.09

Test Statistic=3.118

0

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on.

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.001, the null hypothesis may be rejected, and we may conclude that the proportion of customers buying at least $2500 of travelers checks is at least 10% higher when sweepstakes are on.



Decision: Reject the null hypothesis; p-value = 0.0009.

8-46



8-47

A (1-) 100% large-sample confidence interval for the difference between two population proportions:

A (1-) 100% large-sample confidence interval for the difference between two population proportions:



( )

( ) ( )

p p z

p p

n

p p

n1 2

2

11

1

1

21

2

2

( )

( ) ( )

( . . ) .( . )( . ) ( . )( . )

. ( . )( . ) . . [ . , . ]

p p z

p p

n

p p

n1 2

2

11

1

1

21

2

20 4 0 2 1 96

0 4 0 6

300

0 2 0 8

700

0 2 1 96 0 0321 0 2 0 063 0 137 0 263

Confidence Intervals for the Difference between Two Population Proportions

8-48

Example 8-8 – Using the Template

Confidence IntervalConfidence Interval

8-49

Example 8-8 – Using Minitab

NOTE: In order to use Minitab to construct the confidence interval, you will have toMake sure that the “Not Equal” Alternative option is selected.

8-50

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each of which is divided by its own degrees of freedom.

The F distribution is the distribution of the ratio of two chi-square random variables that are independent of each other, each of which is divided by its own degrees of freedom.

An F random variable with k1 and k2 degrees of freedom:An F random variable with k1 and k2 degrees of freedom:

8-5 The F Distribution and a Test for Equality of Two Population Variances

2

22

121

2 ,1 k

kF kk

8-51

• The F random variable cannot be negative, so it is bound by zero on the left.

• The F distribution is skewed to the right.

• The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.

• The F random variable cannot be negative, so it is bound by zero on the left.

• The F distribution is skewed to the right.

• The F distribution is identified the number of degrees of freedom in the numerator, k1, and the number of degrees of freedom in the denominator, k2.

543210

1.0

0.5

0.0

F

F Distributions with different Degrees of Freedom

f(F

)

F(5,6)

F(10,15)

F(25,30)

The F Distribution8-52

Critical Points of the F Distribution Cutting Off a Right-Tail Area of 0.05

k1 1 2 3 4 5 6 7 8 9

k2

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.1810 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.0211 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.9012 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.8013 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.7114 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.6515 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59

3.01

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F0.05=3.01

f(F)

F Distribution with 7 and 11 Degrees of Freedom

F

The left-hand critical point to go along with F(k1,k2) is given by:

Where F(k1,k2) is the right-hand critical point for an F random variable with the reverse number of degrees of freedom.

1

2 1F k k,

Using the Table of the F Distribution8-53

The right-hand critical point read directly from the table of the F distribution is:

F(6,9) =3.37

The corresponding left-hand critical point is given by:

The right-hand critical point read directly from the table of the F distribution is:

F(6,9) =3.37

The corresponding left-hand critical point is given by:

1 1410

0 24399 6F , .

. 543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F

f(F)


F0.05=3.37F0.95=(1/4.10)=0.2439

0.05

0.05

0.90

Critical Points of the F Distribution: F(6, 9), = 0.10

8-54

Test statistic for the equality of the variances of two normallydistributed populations:

Fs

sn n1 21 1

1

2

2

2 ,

Test statistic for the equality of the variances of two normallydistributed populations:

Fs

sn n1 21 1

1

2

2

2 ,

I: Two-Tailed Test

• 1 = 2

• H0: 1 = 2

• H1: 2

II: One-Tailed Test

• 12

• H0: 1 2

• H1: 1 2

I: Two-Tailed Test

• 1 = 2

• H0: 1 = 2

• H1: 2

II: One-Tailed Test

• 12

• H0: 1 2

• H1: 1 2

Test Statistic for the Equality of Two Population Variances

8-55

The economist wants to test whether or not the event (interceptions and prosecution of insider traders) has decreased the variance of prices of stocks.

70.223,24

01.0

01.223,24

05.0

0.322

s

24=2

n

After :2 Population

3.921

s

25=1

n

Before :1 Population

F

F

H

H

H0 may be rejected at a 1% level of significance.

0 1

2

2

2

1

2

1 1

2

2

2

1 1 2 1 24 23

12

22

9 3

3 031

:

:

, ,

.

..

F

n nF

s

s

Example 8-98-56

Distribution with 24 and 23 Degrees of Freedom

543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F0.01=2.7

f(F

)

F

Test Statistic=3.1

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.01, the null hypothesis may be rejected, and we may conclude that the variance of stock prices is reduced after the interception and prosecution of inside traders.

Since the value of the test statistic is above the critical point, even for a level of significance as small as 0.01, the null hypothesis may be rejected, and we may conclude that the variance of stock prices is reduced after the interception and prosecution of inside traders.


Example 8-9: Solution Using the Template


8-58

Population 1 Population 2

n1

= 14 n2

= 9

s12 s

22

0 122 0 112

0 05

13 83 28

0 10

13 82 50

. .

.

,.

.

,.

F

F

H

H

H0

may not be rejected at the 10% level of significance.

0 12

22

1 12

22

1 1 2 1 13 812

22

0122

0112119

:

:

, ,

.

..

F

n nF

s

s

Example 8-10: Testing the Equality of Variances for Example 8-5

8-59

Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject the null hypothesis. We conclude the two population variances are equal.

Since the value of the test statistic is between the critical points, even for a 20% level of significance, we can not reject the null hypothesis. We conclude the two population variances are equal.


543210

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

F

f(F)

F0.10=3.28F0.90=(1/2.20)=0.4545

0.10

0.10

0.80

Test Statistic=1.19


Template to test for the Difference between Two Population Variances: Example 8-10

Decision: Do not reject the null hypothesis; p-value = 0.8304;Assume equal variances..

8-61

Example 8-10: Using Minitab to Test for Equal Variances

Decision: Do not reject the null hypothesis; p-value = 0.830;Assume equal variances.

2

1

0.250.200.150.100.0595% Bonferroni Confidence Intervals for StDevs

Test Statistic 1.19P-Value 0.830

F-Test

Test for Equal Variances

2

1

0.250.200.150.100.0595% Bonferroni Confidence Intervals for StDevs


F-Test

Test for Equal Variances

Confidence intervals overlap with sample estimates in both – assume Equal variances.

8-62

The F Distribution Template 8-63

The Template for Testing Equality of Variances with data

Do not reject the Null hypothesis forEquality of variancesSince P-value = 0.6882

8-64

Using Minitab to test for the Equality of Variances with data

Do not reject the null hypothesis forequality of variancessince P-values are large for both the F-test and Levine’s test.

Do not reject the null hypothesis forequality of variancessince P-values are large for both the F-test and Levine’s test.

8-65

Using Minitab to test for the Equality of Variances with data

Do not reject the null hypothesis forequality of variancessince the confidenceintervals for the standard deviationsoverlap.

Do not reject the null hypothesis forequality of variancessince the confidenceintervals for the standard deviationsoverlap.

2

1

600500400300200

Sam

ple

95% Bonferroni Confidence Intervals for StDevs

2

1

120010008006004002000

Sam

ple

Data



F-Test

Levene's Test

Test for Equal Variances for Data

2

1

600500400300200

Sam

ple

95% Bonferroni Confidence Intervals for StDevs

2

1

120010008006004002000

Sam

ple

Data



F-Test

Levene's Test

Test for Equal Variances for Data

8-66

Chapter 8 The Comparison of Two Populations COMPLETE BUSINESS STATISTICSby AMIR D. ACZEL & JAYAVEL...

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Transcript of Chapter 8 The Comparison of Two Populations COMPLETE BUSINESS STATISTICSby AMIR D. ACZEL & JAYAVEL...