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Chapter 10Chapter 10 Statistical Inference About Means and Statistical Inference About Means and

Proportions With Two PopulationsProportions With Two Populations

Estimation of the Difference between the Means Estimation of the Difference between the Means of Two Populations: Independent Samples of Two Populations: Independent Samples

Hypothesis Tests about the Difference between Hypothesis Tests about the Difference between the Means of Two Populations: Independent the Means of Two Populations: Independent SamplesSamples

Inference about the Difference between the Inference about the Difference between the Means of Two Populations: Matched SamplesMeans of Two Populations: Matched Samples

Inference about the Difference between the Inference about the Difference between the Proportions of Two PopulationsProportions of Two Populations

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Estimation of the Difference Between the Estimation of the Difference Between the Means of Two Populations: Independent Means of Two Populations: Independent

SamplesSamples

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

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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 equal the mean of population 2.the mean of population 2.

The difference between the two population means isThe difference between the two population means is

11 - - 22..

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

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

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

21 xx

1x 2x

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Sampling Distribution of Sampling Distribution of

The sampling distribution of has the The sampling distribution of has the following properties. following properties.

Expected Value:Expected Value:

Standard Deviation:Standard Deviation:

wherewhere 11 = standard deviation of population 1 = standard deviation of population 1

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

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

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

21 xx

21 xx

2121 )( xxE

2

22

1

21

21 nnxx

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

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

wherewhere

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

wherewhere

212/21 xxzxx

212/21 xxszxx

2

22

1

21

21 n

s

n

ss xx

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

Par, Inc. is a manufacturer of golf equipment. ParPar, Inc. is a manufacturer of golf equipment. Par

has developed a new golf ball that has been designed tohas developed a new golf ball that has been designed to

provide “extra distance.” In a test of driving distanceprovide “extra distance.” In a test of driving distance

using a mechanical driving device, a sample of Par golfusing a mechanical driving device, a sample of Par golf

balls was compared with a sample of golf balls made byballs was compared with a sample of golf balls made by

Rap, Ltd., a competitor. The sample data is below.Rap, Ltd., a competitor. The sample data is below.

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

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

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

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

Standard DeviationStandard Deviation ss11 = 15 yards = 15 yards ss22 = 20 yards = 20 yardsx1x1 x2x2

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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.

21 xx

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

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

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

80

)20(

120

)15(96.117

22

2

22

1

21

2/21 nn

zxx

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

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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

wherewhere

Interval Estimate with Interval Estimate with 22 Unknown Unknown

wherewhere

212/21 xxzxx

212/21 xxstxx

2

)1()1(

21

222

2112

nn

snsns)

11(

21

2

21 nnss xx

)11

(21

2

21 nnxx

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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 data is below.gallon (mpg) performance. The sample data is below.

Sample #1 Sample #1 Sample #2Sample #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 mpg1.81 mpg

x2x2x1x1

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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

Example: Specific MotorsExample: Specific Motors

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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

For the small-sample case we will make the following For the small-sample case we will make the following assumption.assumption.

1. The miles per gallon rating must be normally 1. 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.

2. The variance in the miles per gallon rating must2. The 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..

Example: Specific MotorsExample: Specific Motors

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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

= 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 4.7mean mpg ratings of the two car types is from .3 to 4.7

mpg (with the M car having the higher mpg).mpg (with the M car having the higher mpg).

28.52812

)81.1(7)56.2(11

2

)1()1( 22

21

222

2112

nn

snsns

)8

1

12

1(28.5101.25.2)

11(

21

2025.21

nnstxx

Example: Specific MotorsExample: Specific Motors

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Hypothesis Tests About the DifferenceHypothesis Tests About the DifferenceBetween the Means of Two Populations: Between the Means of Two Populations:

Independent SamplesIndependent Samples

Hypothesis Forms:Hypothesis Forms:

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 Statistic: Test Statistic:

• Large-Sample CaseLarge-Sample Case

• Small-Sample CaseSmall-Sample Case

2221

21

2121 )()(

nn

xxz

)11(

)()(

212

2121

nns

xxt

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Par, Inc. is a manufacturer of golf equipment. ParPar, Inc. is a manufacturer of golf equipment. Par

has developed a new golf ball that has been designed tohas developed a new golf ball that has been designed to

provide “extra distance.” In a test of driving distanceprovide “extra distance.” In a test of driving distance

using a mechanical driving device, a sample of Par golfusing a mechanical driving device, a sample of Par golf

balls was compared with a sample of golf balls made byballs was compared with a sample of golf balls made by

Rap, Ltd., a competitor. The sample data is below.Rap, Ltd., a competitor. The sample data is below.

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

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

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

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

Standard DeviationStandard Deviation ss11 = 15 yards = 15 yards ss22 = 20 yards = 20 yardsx1x1 x2x2

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

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

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

golf ballsgolf balls

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

golf ballsgolf balls

Hypotheses:Hypotheses: HH00: : 1 1 - - 22 << 0 0

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

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

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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

Rejection Rule: Rejection Rule: Reject Reject HH00 if if zz > 2.33 > 2.33

Conclusion: Conclusion: Reject Reject HH00. We are at least 99% . We are at least 99% confidentconfident

that the mean driving distance of Par, Inc. golf balls isthat the mean driving distance of Par, Inc. golf balls is

greater than the mean driving distance of Rap, Ltd. greater than the mean driving distance of Rap, Ltd. golf balls.golf balls.

49.662.2

17

80)20(

120)15(

0)218235()()(22

2

22

1

21

2121

nn

xxz

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