8-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)

8-1

COMPLETE COMPLETE BUSINESS BUSINESS

STATISTICSSTATISTICSbyby

AMIR D. ACZELAMIR D. ACZEL

&&

JAYAVEL SOUNDERPANDIANJAYAVEL SOUNDERPANDIAN

66thth edition (SIE) edition (SIE)

8-2

Chapter 8 Chapter 8

The Comparison of The Comparison of Two PopulationsTwo Populations

8-3

• 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 Populations88

8-4

• 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:

8-5

• Inferences about differences between parameters of two populationsPaired-ObservationsObserve 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 Statistics8-1 Using Statistics

8-6

• 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 Comparisons8-2 Paired-Observation Comparisons

8-7

freedom. of degrees 1)-(on with distributi t a has statistic the

, is differencemean population theand trueis hypothesis

null When the.hypothesis null under the differencemean

population theis symbol The ns.observatio of pairs of

number theis , size, sample theand s,difference theseof

deviation standard sample theis s ns,observatio ofpair

eachbetween difference average sample theis D where

: t testnsobservatio-paired for the statisticTest

0

0

0

D

n

n

ns

Dt

D

D

D

D

freedom. of degrees 1)-(on with distributi t a has statistic the

, is differencemean population theand trueis hypothesis

null When the.hypothesis null under the differencemean

population theis symbol The ns.observatio of pairs of

number theis , size, sample theand s,difference theseof

deviation standard sample theis s ns,observatio ofpair

eachbetween difference average sample theis D where

: t testnsobservatio-paired for the statisticTest

0

0

0

D

n

n

ns

Dt

D

D

D

D

Paired-Observation Comparisons Paired-Observation Comparisons of Meansof Means

8-8

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 8-1Example 8-1

8-9

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: SolutionExample 8-1: Solution

8-10

Example 8-1: Template for Testing Example 8-1: Template for Testing Paired DifferencesPaired Differences

8-11

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:

z

D D

sD

n

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

.

..

( . )


8-12

.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 Confidence Intervals for Paired ObservationsObservations

8-13

0. value theincludenot does interval confidence that thisNote

]114.0,086.0[014.01.0

)0071)(.96.1(1.050

0.051.960.1zD

: 28 Examplein data for the interval confidence 95%

2

nsD

0. value theincludenot does interval confidence that thisNote

]114.0,086.0[014.01.0

)0071)(.96.1(1.050

0.051.960.1zD

: 28 Examplein data for the interval confidence 95%

2

nsD

Confidence Intervals for Paired Confidence Intervals for Paired Observations – Example 8-2Observations – Example 8-2

8-14

Confidence Intervals for Paired Observations – Confidence Intervals for Paired Observations – Example 8-2 Using the TemplateExample 8-2 Using the Template

8-15

•When paired data cannot be obtained, use independentindependent 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 8-3 A Test for the Difference between Two Population Means Using Independent Random SamplesMeans Using Independent Random Samples

8-16

• 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 Comparisons of Two Population Means: Testing SituationsMeans: 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 and II, and it is equal to the prespecified value D in situation III. 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 and II, and it is equal to the prespecified value D in situation III. 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 Comparisons of Two Population Means: Test StatisticMeans: 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 Two-Tailed Test for Equality of Two Population Means: Example 8-3Population 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 TestExample 8-3: Carrying Out the Test

8-20

Example 8-3: Using the TemplateExample 8-3: Using the Template

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

Two-Tailed Test for Difference Between Two-Tailed Test for Difference Between Two Population Means: Example 8-4Two Population Means: Example 8-4

8-22

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?

Two-Tailed Test for Difference Between Two Two-Tailed Test for Difference Between Two Population Means: Example 8-4 – Using the Population Means: Example 8-4 – Using the TemplateTemplate

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 Confidence Intervals for the Difference between Two Population Meansbetween 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 A Test for the Difference between Two Population Means: Assuming Equal Population VariancesMeans: 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 Pooled Estimate of the Population VarianceVariance

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 Using the Pooled Estimate of the Population VariancePopulation Variance

8-27

Population 1: Oil price = $27.50

n1 = 14

x1 = 0.317%

s1 = 0.12%

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 nPopulation 2: Oil price = $20.00

n = 9

x = 0.21%

s = 0.11%

df = (n1

2

2

2

n2

2 14 9 2 21) ( )


8-28



P-value = P-value = 0.0430, so 0.0430, so reject Hreject H00 at at

the 5% the 5% significance significance level.level.

8-29

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) ( )


8-30


P-value = P-value = 0.1858, so do 0.1858, so do not reject Hnot reject H00

at the 5% at the 5% significance significance level.level.

8-31

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: ContinuedExample 8-6: Continued

8-32

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 Confidence Intervals Using the Pooled VarianceVariance

8-33

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

Confidence IntervalConfidence Interval

8-34

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

8-35

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

8-36

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

8-37

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.


8-38


P-value = P-value = 0.157, so do 0.157, so do not reject not reject HH00 at the 5% at the 5%

significance significance level.level.

8-39

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

8-40

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.


8-41


P-value = P-value = 0.0009, so 0.0009, so reject Hreject H00 at at

the 5% the 5% significance significance level.level.

8-42

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 Confidence Intervals for the Difference between Two Population Proportionsbetween Two Population Proportions

8-43

Confidence Intervals for the Difference between Two Confidence Intervals for the Difference between Two Population Proportions – Using the Template – Population Proportions – Using the Template – Using the Data from Example 8-8Using the Data from Example 8-8

8-44

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:

Fk

k

k k1 2

1

2

1

2

2

2

,

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

8-45

• 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 The FF Distribution Distribution

8-46

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

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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 Using the Table of the FF Distribution Distribution

8-47

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

8-48

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 Test Statistic for the Equality of Two Population VariancesPopulation Variances

8-49

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


8-50

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.


8-51

Example 8-9: Solution Using the Example 8-9: Solution Using the TemplateTemplate

Observe that the Observe that the p-value for the test p-value for the test is 0.0042 which is is 0.0042 which is less than 0.01. less than 0.01. Thus Thus the null hypothesis the null hypothesis must be rejected at must be rejected at this level of this level of significance of 0.01.significance of 0.01.

Observe that the Observe that the p-value for the test p-value for the test is 0.0042 which is is 0.0042 which is less than 0.01. less than 0.01. Thus Thus the null hypothesis the null hypothesis must be rejected at must be rejected at this level of this level of significance of 0.01.significance of 0.01.

8-52

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

8-53

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.


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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


8-54

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

Observe that the p-value for the test is 0.8304 which is larger than 0.05. Thus the Thus the null hypothesis null hypothesis cannot be rejected at cannot be rejected at this level of this level of significance of 0.05.significance of 0.05. That is, one can That is, one can assume equal assume equal variance.variance.

Observe that the p-value for the test is 0.8304 which is larger than 0.05. Thus the Thus the null hypothesis null hypothesis cannot be rejected at cannot be rejected at this level of this level of significance of 0.05.significance of 0.05. That is, one can That is, one can assume equal assume equal variance.variance.

8-55

The The FF Distribution Template Distribution Template

8-56

The Template for Testing Equality of The Template for Testing Equality of VariancesVariances

Documents

8-1 COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 6 th edition (SIE)