1 1 slide slides prepared by john s. loucks st. edward’s university © 2002 south-western /thomson...

1 1 Slide

Slide

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

© 2002 South-Western /Thomson Learning© 2002 South-Western /Thomson Learning

2 2 Slide

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Chapter 10Chapter 10 Comparisons Involving Means Comparisons Involving Means

11 = = 22 ? ?

ANOVAANOVA

Estimation of the Difference Between the Means of Estimation of the Difference Between the Means of

Two Populations: Independent Samples Two Populations: Independent Samples Hypothesis Tests about the Difference between theHypothesis Tests about the Difference between the

Means of Two Populations: Independent SamplesMeans of Two Populations: Independent Samples Inferences about the Difference between the Means Inferences about the Difference between the Means

of Two Populations: Matched Samplesof Two Populations: Matched Samples Inferences about the Difference between the Inferences about the Difference between the

Proportions of Two Populations:Proportions of Two Populations:

3 3 Slide

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Estimation of the Difference Between the Estimation of the Difference Between the Means Means

of Two Populations: Independent Samplesof Two Populations: Independent Samples Point Estimator of the Difference between the Point Estimator of the Difference between the

Means of Two PopulationsMeans of Two Populations Sampling DistributionSampling Distribution Interval Estimate of Interval Estimate of Large-Sample CaseLarge-Sample Case

Interval Estimate of Interval Estimate of Small-Sample CaseSmall-Sample Case

x x1 2x x1 2

4 4 Slide

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Point Estimator of the Difference BetweenPoint Estimator of the Difference Betweenthe Means of Two Populationsthe Means of Two Populations

Let Let 11 equal the mean of population 1 and equal the mean of population 1 and 22 equal the mean of population 2.equal the mean of population 2.

The difference between the two population The difference between the two population means is means is 11 - - 22..

To estimate To estimate 11 - - 22, we will select a simple , we will select a simple random sample of size random sample of size nn11 from population 1 from population 1 and a simple random sample of size and a simple random sample of size nn22 from from population 2.population 2.

Let equal the mean of sample 1 and Let equal the mean of sample 1 and equal the mean of sample 2.equal the mean of sample 2.

The point estimator of the difference between The point estimator of the difference between the means of the populations 1 and 2 is the means of the populations 1 and 2 is ..

x x1 2x x1 2

x1x1 x2x2

5 5 Slide

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Properties of the Sampling Distribution of Properties of the Sampling Distribution of

• Expected ValueExpected Value

• Standard DeviationStandard Deviation

where: where: 1 1 = standard deviation of population 1 = standard deviation of population 1

2 2 = standard deviation of population 2 = standard deviation of population 2

nn1 1 = sample size from population 1= sample size from population 1

nn22 = sample size from population 2 = sample size from population 2

Sampling Distribution of Sampling Distribution of x x1 2x x1 2

x x1 2x x1 2

E x x( )1 2 1 2 E x x( )1 2 1 2

x x n n1 2

12

1

22

2

x x n n1 2

12

1

22

2

6 6 Slide

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Interval Estimate with Interval Estimate with 11 and and 22 Known Known

where:where:

1 - 1 - is the confidence coefficient is the confidence coefficient Interval Estimate with Interval Estimate with 11 and and 22 Unknown Unknown

where:where:

Interval Estimate of Interval Estimate of 11 - - 22::Large-Sample Case (Large-Sample Case (nn11 >> 30 and 30 and nn22 >> 30) 30)

x x z x x1 2 2 1 2 /x x z x x1 2 2 1 2 /

x x z sx x1 2 2 1 2 /x x z sx x1 2 2 1 2 /

ssn

snx x1 2

12

1

22

2 s

sn

snx x1 2

12

1

22

2

7 7 Slide

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Example: Par, Inc.Example: Par, Inc.

Interval Estimate of Interval Estimate of 11 - - 22: Large-Sample Case: Large-Sample Case

Par, Inc. is a manufacturer of golf Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball equipment and has developed a new golf ball that has been designed to provide “extra that has been designed to provide “extra distance.” In a test of driving distance using a distance.” In a test of driving distance using a mechanical driving device, a sample of Par golf mechanical driving device, a sample of Par golf balls was compared with a sample of golf balls balls was compared with a sample of golf balls made by Rap, Ltd., a competitor. made by Rap, Ltd., a competitor.

The sample statistics appear on the next The sample statistics appear on the next slide.slide.

8 8 Slide

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Interval Estimate of Interval Estimate of 11 - - 22: Large-Sample Case: Large-Sample Case

• Sample StatisticsSample Statistics

Sample #1 Sample #1 Sample #2Sample #2

Par, Inc. Par, Inc. Rap, LtdRap, Ltd..

Sample SizeSample Size nn11 = 120 balls = 120 balls nn22 = 80 = 80 ballsballs

MeanMean = 235 yards = 235 yards = 218 = 218 yardsyards

Standard Dev.Standard Dev. ss11 = 15 yards = 15 yards ss22 = 20 = 20 yardsyards

1x1x 2x2x

9 9 Slide

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Point Estimate of the Difference Between Two Point Estimate of the Difference Between Two Population MeansPopulation Means

11 = mean distance for the population of = mean distance for the population of

Par, Inc. golf ballsPar, Inc. golf balls

22 = mean distance for the population of = mean distance for the population of

Rap, Ltd. golf ballsRap, Ltd. golf balls

Point estimate of Point estimate of 11 - - 2 2 = = 235 - 218 = = = 235 - 218 = 17 yards.17 yards.

x x1 2x x1 2


10 10 Slide

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Point Estimator of the Difference Point Estimator of the Difference Between the Means of Two PopulationsBetween the Means of Two Populations

Population 1Population 1Par, Inc. Golf BallsPar, Inc. Golf Balls

11 = mean driving = mean driving distance of Pardistance of Par

golf ballsgolf balls

Population 1Population 1Par, Inc. Golf BallsPar, Inc. Golf Balls

11 = mean driving = mean driving distance of Pardistance of Par


Population 2Population 2Rap, Ltd. Golf BallsRap, Ltd. Golf Balls

22 = mean driving = mean driving distance of Rapdistance of Rap


Population 2Population 2Rap, Ltd. Golf BallsRap, Ltd. Golf Balls

22 = mean driving = mean driving distance of Rapdistance of Rap


11 – – 22 = difference between= difference between the mean distancesthe mean distances

Simple random sampleSimple random sample of of nn11 Par golf balls Par golf balls

xx11 = sample mean distance = sample mean distancefor sample of Par golf ballfor sample of Par golf ball

Simple random sampleSimple random sample of of nn11 Par golf balls Par golf balls

xx11 = sample mean distance = sample mean distancefor sample of Par golf ballfor sample of Par golf ball

Simple random sampleSimple random sample of of nn22 Rap golf balls Rap golf balls

xx22 = sample mean distance = sample mean distancefor sample of Rap golf ballfor sample of Rap golf ball

Simple random sampleSimple random sample of of nn22 Rap golf balls Rap golf balls

xx22 = sample mean distance = sample mean distancefor sample of Rap golf ballfor sample of Rap golf ball

xx11 - - xx22 = Point Estimate of = Point Estimate of 11 –– 22

11 11 Slide

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95% Confidence Interval Estimate of the Difference 95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Between Two Population Means: Large-Sample Case, Case, 11 and and 22 Unknown Unknown

Substituting the sample standard deviations Substituting the sample standard deviations for the population standard deviation:for the population standard deviation:

= 17 = 17 ++ 5.14 or 11.86 yards to 22.14 5.14 or 11.86 yards to 22.14 yards.yards.

We are 95% confident that the difference between We are 95% confident that the difference between the mean driving distances of Par, Inc. balls and the mean driving distances of Par, Inc. balls and Rap, Ltd. balls lies in the interval of 11.86 to 22.14 Rap, Ltd. balls lies in the interval of 11.86 to 22.14 yards.yards.

x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

x x zn n1 2 2

12

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )


12 12 Slide

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Interval Estimate of Interval Estimate of 11 - - 22::Small-Sample Case (Small-Sample Case (nn11 < 30 and/or < 30 and/or nn22 < <

30)30) Interval Estimate with Interval Estimate with 22 Known Known

where:where:

x x z x x1 2 2 1 2 /x x z x x1 2 2 1 2 /

x x n n1 2

2

1 2

1 1 ( ) x x n n1 2

2

1 2

1 1 ( )

13 13 Slide

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Interval Estimate of Interval Estimate of 11 - - 22::Small-Sample Case (Small-Sample Case (nn11 < 30 and/or < 30 and/or nn22 < <

30)30) Interval Estimate with Interval Estimate with 22 Unknown Unknown

where:where:

x x t sx x1 2 2 1 2 /x x t sx x1 2 2 1 2 /

sn s n s

n n2 1 1

22 2

2

1 2

1 12

( ) ( )s

n s n sn n

2 1 12

2 22

1 2

1 12

( ) ( )s s

n nx x1 2

2

1 2

1 1 ( )s s

n nx x1 2

2

1 2

1 1 ( )

14 14 Slide

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Example: Specific MotorsExample: Specific Motors

Specific Motors of Detroit has developed a newSpecific Motors of Detroit has developed a new

automobile known as the M car. 12 M cars and 8 J carsautomobile known as the M car. 12 M cars and 8 J cars

(from Japan) were road tested to compare miles-per-(from Japan) were road tested to compare miles-per-

gallon (mpg) performance. The sample statistics are:gallon (mpg) performance. The sample statistics are:

Sample #1 Sample #1 Sample #2 Sample #2

M CarsM Cars J CarsJ Cars

Sample SizeSample Size nn11 = 12 cars = 12 cars nn22 = 8 cars = 8 cars

MeanMean = 29.8 mpg = 27.3 mpg = 29.8 mpg = 27.3 mpg

Standard DeviationStandard Deviation ss11 = 2.56 mpg = 2.56 mpg ss22 = 1.81 = 1.81 mpgmpg x2x2x1x1

15 15 Slide

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Point Estimate of the Difference Between Two Point Estimate of the Difference Between Two Population MeansPopulation Means

11 = mean miles-per-gallon for the population of = mean miles-per-gallon for the population of

M carsM cars

22 = mean miles-per-gallon for the population of = mean miles-per-gallon for the population of

J carsJ cars

Point estimate of Point estimate of 11 - - 2 2 = = 29.8 - 27.3 = = 29.8 - 27.3 = 2.5 mpg.= 2.5 mpg.

x x1 2x x1 2


16 16 Slide

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95% Confidence Interval Estimate of the Difference 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample CaseBetween Two Population Means: Small-Sample Case

We will make the following assumptions:We will make the following assumptions:

• The miles per gallon rating must be normally The miles per gallon rating must be normally

distributed for both the M car and the J car.distributed for both the M car and the J car.

• The variance in the miles per gallon rating mustThe variance in the miles per gallon rating must

be the same for both the M car and the J car.be the same for both the M car and the J car.

Using the Using the tt distribution with distribution with nn11 + + nn22 - 2 = 18 degrees - 2 = 18 degrees

of freedom, the appropriate of freedom, the appropriate tt value is value is tt.025.025 = 2.101. = 2.101.

We will use a weighted average of the two sampleWe will use a weighted average of the two sample

variances as the pooled estimator of variances as the pooled estimator of 22..


17 17 Slide

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95% Confidence Interval Estimate of the Difference 95% Confidence Interval Estimate of the Difference Between Two Population Means: Small-Sample Between Two Population Means: Small-Sample CaseCase

= 2.5 = 2.5 ++ 2.2 or .3 to 4.7 miles per gallon. 2.2 or .3 to 4.7 miles per gallon.

We are 95% confident that the difference between We are 95% confident that the difference between thethe

mean mpg ratings of the two car types is from .3 to mean mpg ratings of the two car types is from .3 to 4.7 mpg (with the M car having the higher mpg).4.7 mpg (with the M car having the higher mpg).

sn s n s

n n2 1 1

22 2

2

1 2

2 21 12

11 2 56 7 1 8112 8 2

5 28

( ) ( ) ( . ) ( . ).s

n s n sn n

2 1 12

2 22

1 2

2 21 12

11 2 56 7 1 8112 8 2

5 28

( ) ( ) ( . ) ( . ).

x x t sn n1 2 025

2

1 2

1 12 5 2 101 5 28

112

18

. ( ) . . . ( )x x t sn n1 2 025

2

1 2

1 12 5 2 101 5 28

112

18

. ( ) . . . ( )


18 18 Slide

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Hypothesis Tests About the Difference Hypothesis Tests About the Difference Between the Means of Two Populations: Between the Means of Two Populations:

Independent SamplesIndependent Samples HypothesesHypotheses

HH00: : 1 1 - - 22 << 0 0 HH00: : 1 1 - - 22 >> 0 0 HH00: : 1 1 - - 22 = 0 = 0

HHaa: : 1 1 - - 22 > 0 > 0 HHaa: : 1 1 - - 22 < 0 < 0 HHaa: : 1 1 - - 22 0 0

Test StatisticTest Statistic

Large-SampleLarge-Sample Small-SampleSmall-Sample

zx x

n n

( ) ( )1 2 1 2

12

1 22

2

zx x

n n

( ) ( )1 2 1 2

12

1 22

2

tx x

s n n

( ) ( )

( )1 2 1 2

21 21 1

t

x x

s n n

( ) ( )

( )1 2 1 2

21 21 1

19 19 Slide

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Hypothesis Tests About the Difference Hypothesis Tests About the Difference Between the Means of Two Populations: Between the Means of Two Populations: Large-Sample CaseLarge-Sample Case

Par, Inc. is a manufacturer of golf equipment Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has and has developed a new golf ball that has been designed to provide “extra distance.” In been designed to provide “extra distance.” In a test of driving distance using a mechanical a test of driving distance using a mechanical driving device, a sample of Par golf balls was driving device, a sample of Par golf balls was compared with a sample of golf balls made by compared with a sample of golf balls made by Rap, Ltd., a competitor. The sample statistics Rap, Ltd., a competitor. The sample statistics appear on the next slide.appear on the next slide.


20 20 Slide

Slide


Hypothesis Tests About the Difference Between the Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample CaseMeans of Two Populations: Large-Sample Case

• Sample StatisticsSample Statistics

Sample #1 Sample #1 Sample #2Sample #2

Par, Inc. Par, Inc. Rap, LtdRap, Ltd..

Sample SizeSample Size nn11 = 120 balls = 120 balls nn22 = 80 = 80 ballsballs

MeanMean = 235 yards = 235 yards = 218 yards = 218 yards

Standard Dev.Standard Dev. ss11 = 15 yards = 15 yards ss22 = 20 = 20 yardsyards

x1x1 x2x2

21 21 Slide

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Hypothesis Tests About the Difference Between the Hypothesis Tests About the Difference Between the Means of Two Populations: Large-Sample CaseMeans of Two Populations: Large-Sample Case

Can we conclude, using a .01 level of significance, Can we conclude, using a .01 level of significance, that the mean driving distance of Par, Inc. golf balls that the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, is greater than the mean driving distance of Rap, Ltd. golf balls?Ltd. golf balls?

11 = mean distance for the population of Par, Inc. = mean distance for the population of Par, Inc.


22 = mean distance for the population of Rap, Ltd. = mean distance for the population of Rap, Ltd.


• HypothesesHypothesesHH00: : 1 1 - - 22 << 0 0

HHaa: : 1 1 - - 22 > 0 > 0


22 22 Slide

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Hypothesis Tests About the Difference Hypothesis Tests About the Difference Between the Means of Two Populations: Between the Means of Two Populations: Large-Sample CaseLarge-Sample Case

• Rejection RuleRejection Rule Reject Reject HH00 if if zz > 2.33 > 2.33

• ConclusionConclusion

Reject Reject HH00. We are at least 99% . We are at least 99% confident that the mean driving distance of confident that the mean driving distance of Par, Inc. golf balls is greater than the mean Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls.driving distance of Rap, Ltd. golf balls.

zx x

n n

( ) ( ) ( )

( ) ( ) ..1 2 1 2

12

1

22

2

2 2

235 218 0

15120

2080

172 62

6 49

z

x x

n n

( ) ( ) ( )

( ) ( ) ..1 2 1 2

12

1

22

2

2 2

235 218 0

15120

2080

172 62

6 49


23 23 Slide

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Hypothesis Tests About the Difference Hypothesis Tests About the Difference Between the Means of Two Populations: Between the Means of Two Populations: Small-Sample CaseSmall-Sample Case

Can we conclude, using a .05 level of Can we conclude, using a .05 level of significance, that the miles-per-gallon (significance, that the miles-per-gallon (mpgmpg) ) performance of M cars is greater than the performance of M cars is greater than the miles-per-gallon performance of J cars?miles-per-gallon performance of J cars?

11 = mean = mean mpgmpg for the population of M cars for the population of M cars

22 = mean = mean mpgmpg for the population of J cars for the population of J cars

• HypothesesHypothesesHH00: : 1 1 - - 22 << 0 0

HHaa: : 1 1 - - 22 > 0 > 0


24 24 Slide

Slide


Hypothesis Tests About the Difference Hypothesis Tests About the Difference Between the Means of Two Populations: Between the Means of Two Populations: Small-Sample CaseSmall-Sample Case

• Rejection RuleRejection Rule

Reject Reject HH00 if if tt > 1.734 > 1.734

(( = .05, d.f. = 18) = .05, d.f. = 18)

• Test StatisticTest Statistic

where:where:

tx x

s n n

( ) ( )

( )1 2 1 2

21 21 1

t

x x

s n n

( ) ( )

( )1 2 1 2

21 21 1

2 22 1 1 2 2

1 2

( 1) ( 1)

2

n s n ss

n n

2 22 1 1 2 2

1 2

( 1) ( 1)

2

n s n ss

n n

25 25 Slide

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Inference About the Difference Between Inference About the Difference Between the Means of Two Populations: Matched the Means of Two Populations: Matched

SamplesSamples With a With a matched-sample designmatched-sample design each sampled each sampled

item provides a pair of data values.item provides a pair of data values. The matched-sample design can be referred to The matched-sample design can be referred to

as as blockingblocking.. This design often leads to a smaller sampling This design often leads to a smaller sampling

error than the independent-sample design error than the independent-sample design because variation between sampled items is because variation between sampled items is eliminated as a source of sampling error.eliminated as a source of sampling error.

26 26 Slide

Slide

Example: Express DeliveriesExample: Express Deliveries

Inference About the Difference Between the Inference About the Difference Between the Means of Two Populations: Matched SamplesMeans of Two Populations: Matched Samples

A Chicago-based firm has documents that A Chicago-based firm has documents that must be quickly distributed to district offices must be quickly distributed to district offices throughout the U.S. The firm must decide throughout the U.S. The firm must decide between two delivery services, UPX (United between two delivery services, UPX (United Parcel Express) and INTEX (International Parcel Express) and INTEX (International Express), to transport its documents. In testing Express), to transport its documents. In testing the delivery times of the two services, the firm the delivery times of the two services, the firm sent two reports to a random sample of ten sent two reports to a random sample of ten district offices with one report carried by UPX district offices with one report carried by UPX and the other report carried by INTEX.and the other report carried by INTEX.

Do the data that follow indicate a Do the data that follow indicate a difference in mean delivery times for the two difference in mean delivery times for the two services?services?

27 27 Slide

Slide

Delivery Time (Hours)Delivery Time (Hours)

District OfficeDistrict Office UPXUPX INTEXINTEX DifferenceDifference

SeattleSeattle 32 32 25 25 7 7

Los AngelesLos Angeles 30 30 24 24 6 6

BostonBoston 19 19 15 15 4 4

ClevelandCleveland 16 16 15 15 1 1

New YorkNew York 15 15 13 13 2 2

HoustonHouston 18 18 15 15 3 3

AtlantaAtlanta 14 14 15 15 -1 -1

St. LouisSt. Louis 10 10 8 8 2 2

MilwaukeeMilwaukee 7 7 9 9 -2 -2

Denver Denver 16 16 11 11 5 5


28 28 Slide

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Inference About the Difference Between the Means Inference About the Difference Between the Means of Two Populations: Matched Samplesof Two Populations: Matched Samples

Let Let d d = the mean of the = the mean of the differencedifference values for values for the the two delivery services for the two delivery services for the population of population of district offices district offices

• HypothesesHypotheses HH00: : d d = 0, = 0, HHaa: : dd

• Rejection RuleRejection Rule

Assuming the population of difference values Assuming the population of difference values is approximately normally distributed, the is approximately normally distributed, the tt distribution with distribution with nn - 1 degrees of freedom - 1 degrees of freedom applies. With applies. With = .05, = .05, tt.025.025 = 2.262 (9 degrees of = 2.262 (9 degrees of freedom).freedom).

Reject Reject HH00 if if tt < -2.262 or if < -2.262 or if tt > 2.262 > 2.262


29 29 Slide

Slide

Inference About the Difference Between the Inference About the Difference Between the Means of Two Populations: Matched SamplesMeans of Two Populations: Matched Samples

• ConclusionConclusion Reject Reject HH00. .

There is a significant difference between the There is a significant difference between the mean delivery times for the two services. mean delivery times for the two services.

ddni

( ... ).

7 6 510

2 7ddni ( ... )

.7 6 5

102 7

sd dndi

( ) ..

2

176 1

92 9s

d dndi

( ) ..

2

176 1

92 9

tds n

d

d

2 7 02 9 10

2 94.

..t

ds n

d

d

2 7 02 9 10

2 94.

..


30 30 Slide

Slide

Inferences About the Difference Inferences About the Difference Between the Proportions of Two Between the Proportions of Two

PopulationsPopulations Sampling Distribution of Sampling Distribution of Interval Estimation of Interval Estimation of pp11 - - pp22

Hypothesis Tests about Hypothesis Tests about pp11 - - pp22

p p1 2p p1 2

31 31 Slide

Slide

Expected ValueExpected Value

Standard DeviationStandard Deviation

Distribution FormDistribution Form

If the sample sizes are large (If the sample sizes are large (nn11pp11, , nn11(1 - (1 - pp11), ), nn22pp22,,

and and nn22(1 - (1 - pp22) are all greater than or equal to 5), the) are all greater than or equal to 5), thesampling distribution of can be approximatedsampling distribution of can be approximatedby a normal probability distribution. by a normal probability distribution.

Sampling Distribution of Sampling Distribution of p p1 2p p1 2

E p p p p( )1 2 1 2 E p p p p( )1 2 1 2

p pp pn

p pn1 2

1 1

1

2 2

2

1 1 ( ) ( ) p p

p pn

p pn1 2

1 1

1

2 2

2

1 1 ( ) ( )

p p1 2p p1 2

32 32 Slide

Slide

Interval Estimation of Interval Estimation of pp11 - - pp22

Interval EstimateInterval Estimate

Point Estimator of Point Estimator of

p p z p p1 2 2 1 2 /p p z p p1 2 2 1 2 /

p p1 2 p p1 2

sp pn

p pnp p1 2

1 1

1

2 2

2

1 1 ( ) ( )

sp pn

p pnp p1 2

1 1

1

2 2

2

1 1 ( ) ( )

33 33 Slide

Slide

Example: MRAExample: MRA

MRA (Market Research Associates) is MRA (Market Research Associates) is conducting research to evaluate the conducting research to evaluate the effectiveness of a client’s new advertising effectiveness of a client’s new advertising campaign. Before the new campaign began, a campaign. Before the new campaign began, a telephone survey of 150 households in the test telephone survey of 150 households in the test market area showed 60 households “aware” of market area showed 60 households “aware” of the client’s product. The new campaign has the client’s product. The new campaign has been initiated with TV and newspaper been initiated with TV and newspaper advertisements running for three weeks. A advertisements running for three weeks. A survey conducted immediately after the new survey conducted immediately after the new campaign showed 120 of 250 households campaign showed 120 of 250 households “aware” of the client’s product.“aware” of the client’s product.

Does the data support the position that Does the data support the position that the advertising campaign has provided an the advertising campaign has provided an increased awareness of the client’s product?increased awareness of the client’s product?

34 34 Slide

Slide


Point Estimator of the Difference Between the Point Estimator of the Difference Between the Proportions of Two PopulationsProportions of Two Populations

pp11 = proportion of the population of households = proportion of the population of households “ “aware” of the product aware” of the product afterafter the new the new

campaigncampaign

pp22 = proportion of the population of households = proportion of the population of households “ “aware” of the product aware” of the product beforebefore the new the new

campaign campaign = sample proportion of households “aware” of the= sample proportion of households “aware” of the

product product afterafter the new campaign the new campaign = sample proportion of households “aware” of the= sample proportion of households “aware” of the

product product beforebefore the new campaign the new campaign

p p p p1 2 1 2120250

60150

48 40 08 . . .p p p p1 2 1 2120250

60150

48 40 08 . . .

p1p1

p2p2

35 35 Slide

Slide


Interval Estimate of Interval Estimate of pp11 - - pp22: Large-Sample Case: Large-Sample Case

For For = .05, = .05, zz.025.025 = 1.96: = 1.96:

.08 .08 ++ 1.96(.0510) 1.96(.0510)

.08 .08 ++ .10 .10

or -.02 to +.18or -.02 to +.18

• ConclusionConclusion

At a 95% confidence level, the interval At a 95% confidence level, the interval estimate of the difference between the proportion estimate of the difference between the proportion of households aware of the client’s product before of households aware of the client’s product before and after the new advertising campaign is -.02 to and after the new advertising campaign is -.02 to +.18.+.18.

. . .. (. ) . (. )

48 40 1 9648 52

25040 60

150 . . .

. (. ) . (. )48 40 1 96

48 52250

40 60150

36 36 Slide

Slide


HypothesesHypotheses

HH00: : pp11 - - pp22 << 0 0

HHaa: : pp11 - - pp22 > 0 > 0 Test statisticTest statistic

Point Estimator of where Point Estimator of where pp11 = = pp22

where:where:

zp p p p

p p

( ) ( )1 2 1 2

1 2

zp p p p

p p

( ) ( )1 2 1 2

1 2

s p p n np p1 21 1 11 2 ( )( )s p p n np p1 21 1 11 2 ( )( )

pn p n pn n

1 1 2 2

1 2

pn p n pn n

1 1 2 2

1 2

p p1 2 p p1 2

37 37 Slide

Slide



Can we conclude, using a .05 level of Can we conclude, using a .05 level of significance, that the proportion of households significance, that the proportion of households aware of the client’s product increased after the aware of the client’s product increased after the new advertising campaign?new advertising campaign?

pp11 = proportion of the population of households = proportion of the population of households

“ “aware” of the product after the new aware” of the product after the new campaigncampaign

pp22 = proportion of the population of households = proportion of the population of households

“ “aware” of the product before the new aware” of the product before the new campaign campaign

• HypothesesHypotheses HH00: : pp1 1 - - pp22 << 0 0

HHaa: : pp1 1 - - pp22 > 0 > 0

38 38 Slide

Slide



• Rejection RuleRejection Rule Reject Reject HH00 if if zz > 1.645 > 1.645

• Test StatisticTest Statistic

• ConclusionConclusion Do not reject Do not reject HH00. .

p

250 48 150 40250 150

180400

45(. ) (. )

.p

250 48 150 40250 150

180400

45(. ) (. )

.

sp p1 245 55 1

2501150 0514 . (. )( ) .sp p1 2

45 55 1250

1150 0514 . (. )( ) .

z

(. . )

..

..

48 40 00514

080514

1 56z

(. . )

..

..

48 40 00514

080514

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1 1 slide slides prepared by john s. loucks st. edward’s university © 2002 south-western /thomson...

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