Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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© 2003 South-Western/Thomson Learning™© 2003 South-Western/Thomson Learning™

Slides Prepared bySlides Prepared byJOHN S. LOUCKSJOHN S. LOUCKS

St. Edward’s UniversitySt. Edward’s University

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Chapter 10Chapter 10 Statistical Inferences about Means Statistical Inferences about Means and Proportions for Two Populationsand Proportions for Two Populations

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 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 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

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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

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

where:where:

1 - 1 - is the confidence coefficient is the confidence coefficient Interval Estimate with Interval Estimate with 11 and and 22 Estimated by Estimated by ss11

and and ss22

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

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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.

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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 #2 Sample #2

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

Sample SizeSample Size nn11 = 120 balls = 120 balls nn22 = = 80 balls80 balls

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

Standard Dev.Standard Dev. ss11 = 15 yards = 15 yards ss22 = = 20 yards20 yards

x1x1 2x2x

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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


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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

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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 Estimated by Estimated by ss11 and and ss22

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 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.

x x zn n1 2 212

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )

x x zn n1 2 212

1

22

2

2 2

17 1 9615120

2080

/ .( ) ( )


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Using Excel to Develop an Interval Estimate Using Excel to Develop an Interval Estimate of of 11 – – 22: Large-Sample Case: Large-Sample Case

Formula WorksheetFormula WorksheetA B C D E

1 Par Rap Par, Inc. Rap, Ltd.2 195 226 Sample Size 120 803 230 198 Mean =AVERAGE(A2:A121) =AVERAGE(A2:A81)4 254 203 Stand. Dev. =STDEV(A2:A121) =STDEV(A2:A81)5 205 237 6 260 235 Confid. Coeff. 0.95 7 222 204 Lev. of Signif. =1-D6 8 241 199 z Value =NORMSINV(1-D7/2) 9 217 202 10 228 240 Std. Error =SQRT(D4 2̂*/D2+E4 2̂/E2)11 255 221 Marg. of Error =D8*D1012 209 206 13 251 201 Pt. Est. of Diff. =D3-E314 229 233 Lower Limit =D13-D1115 220 194 Upper Limit =D13+D11

Note: Rows 16-121 are not shown.Note: Rows 16-121 are not shown.

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Value WorksheetValue WorksheetA B C D E

1 Par Rap Par, Inc. Rap, Ltd.2 195 226 Sample Size 120 803 230 198 Mean 235 2184 254 203 Stand. Dev. 15 205 205 237 6 260 235 Confid. Coeff. 0.95 7 222 204 Lev. of Signif. 0.05 8 241 199 z Value 1.960 9 217 202 10 228 240 Std. Error 2.62211 255 221 Marg. of Error 5.13912 209 206 13 251 201 Pt. Est. of Diff. 1714 229 233 Lower Limit 11.8615 220 194 Upper Limit 22.14

Using Excel to Develop an Interval Estimate Using Excel to Develop an Interval Estimate of of 11 – – 22: Large-Sample Case: Large-Sample Case


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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 Assumed Known Assumed 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 ( )

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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 11 and and 22 Estimated by Estimated by ss11

and and ss22

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 ( )

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

(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 Sample #2#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 = 29.8 mpg = 27.3 mpgmpg

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

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


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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..


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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 0.3 mean mpg ratings of the two car types is from 0.3 to 4.7 mpg (with the M car having the higher mpg).to 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

. ( ) . . . ( )


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1 M Car J Car M Car J Car2 25.1 25.6 Sample Size 12 83 32.2 28.1 Mean =AVERAGE(A2:A13) =AVERAGE(B2:B9)4 31.7 27.9 Stand. Dev. =STDEV(A2:A13) =STDEV(B2:B9)5 27.6 25.3 6 28.5 30.1 Confid. Coeff. 0.95 7 33.6 27.5 Lev. of Signif. =1-D6 8 30.8 25.1 Deg. Freed. =D2+E2-2 9 26.2 28.8 z Value =TINV(D7,D8)10 29.0 11 31.0 Pool.Est.Var. =((D2-1)*D4 2̂+(E2-1)*E4^2)/D812 31.7 Std. Error =SQRT(D11*(1/D2+1/E2))13 30.0 Marg. of Error =D9*D1214 15 Pt. Est. of Diff. =D3-E316 Lower Limit =D15-D1317 Upper Limit =D15+D13

Using Excel to Develop an Interval Estimate Using Excel to Develop an Interval Estimate of of 11 – – 22: Small-Sample Case: Small-Sample Case

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Value WorksheetValue WorksheetA B C D E

1 M Car J Car M Car J Car2 25.1 25.6 Sample Size 12 83 32.2 28.1 Mean 29.8 27.34 31.7 27.9 Stand. Dev. 2.56 1.815 27.6 25.3 6 28.5 30.1 Confid. Coeff. 0.95 7 33.6 27.5 Lev. of Signif. 0.05 8 30.8 25.1 Deg. Freed. 18 9 26.2 28.8 z Value 2.10110 29.0 11 31.0 Pool.Est.Var. 5.276512 31.7 Std. Error 1.048513 30.0 Marg. of Error 2.202714 15 Pt. Est. of Diff. 2.483316 Lower Limit 0.280617 Upper Limit 4.6861

Using Excel to Develop an Interval Estimate Using Excel to Develop an Interval Estimate of of 11 – – 22: Small-Sample Case: Small-Sample Case

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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 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

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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.


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Hypothesis Tests about the Difference between Hypothesis Tests about the Difference between the Means of Two Populations: Large-Sample Casethe Means of Two Populations: Large-Sample Case

• Sample StatisticsSample Statistics

Sample #1 Sample #1 Sample #2 Sample #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

x1x1 x2x2

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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


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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 confident that the mean driving distance that the mean driving distance of Par, Inc. golf balls is greater than the mean of 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

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z

x x

n n

( ) ( ) ( )

( ) ( ) ..1 2 1 2

12

1

22

2

2 2

235 218 0

15120

2080

172 62

6 49


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Using Excel to Conduct a Hypothesis Test Using Excel to Conduct a Hypothesis Test about about 11 – – 22: Large Sample Case: Large Sample Case

Excel’s “Excel’s “zz-Test: Two Sample for Means” Tool-Test: Two Sample for Means” Tool

Step 1Step 1 Select the Select the ToolsTools pull-down menu pull-down menu

Step 2Step 2 Choose the Choose the Data AnalysisData Analysis option option

Step 3Step 3 Choose Choose zz-Test: Two Sample for -Test: Two Sample for MeansMeans

from the list of Analysis Toolsfrom the list of Analysis Tools

… … continuedcontinued

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Step 4Step 4 When the z-Test: Two Sample for MeansWhen the z-Test: Two Sample for Means

dialog box appears:dialog box appears:

Enter A1:A121 in the Enter A1:A121 in the Variable 1 RangeVariable 1 Range boxbox

Enter B1:B81 in the Enter B1:B81 in the Variable 2 RangeVariable 2 Range box box

Enter 0 in the Enter 0 in the Hypothesized Mean Hypothesized Mean DifferenceDifference boxbox

Enter 225 in the Enter 225 in the Variable 1 Variance Variable 1 Variance (known)(known) boxbox

Enter 400 in the Enter 400 in the Variable 2 Variance Variable 2 Variance (known)(known) boxbox



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Step 4Step 4 (continued) (continued)

Select Select LabelsLabels

Enter .01 in the Enter .01 in the AlphaAlpha box box

Select Select Output RangeOutput Range

Enter D4 in the Enter D4 in the Output RangeOutput Range box box

(Any upper left-hand corner cell (Any upper left-hand corner cell indicatingindicating

where the output is to begin may be where the output is to begin may be entered)entered)

Click Click OKOK


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Value WorksheetValue Worksheet


A B C D E F1 Par Rap Par, Inc. Rap, Ltd.2 195 226 Sample Variance 225 4003 230 198 4 254 203 z-Test: Two Sample for Means 5 205 237 6 260 235 Par, Inc. Rap, Ltd.7 222 204 Mean 235 2188 241 199 Known Variance 225 4009 217 202 Observations 120 8010 228 240 Hypothesized Mean Difference 011 255 221 z 6.48354560712 209 206 P(Z<=z) one-tail 4.50145E-1113 251 201 z Critical one-tail 2.32634192814 229 233 P(Z<=z) two-tail 9.00291E-1115 220 194 z Critical two-tail 2.575834515


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Hypothesis Tests about the Difference Hypothesis Tests about the Difference between the Means of Two Populations: Small-between the Means of Two Populations: Small-Sample CaseSample 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


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Hypothesis Tests about the Difference Hypothesis Tests about the Difference between the Means of Two Populations: Small-between the Means of Two Populations: Small-Sample CaseSample 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

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Using Excel to Conduct a Hypothesis Test Using Excel to Conduct a Hypothesis Test about about 11 – – 22: Small Sample Case: Small Sample Case

Excel’s “Excel’s “tt-Test: Two Sample Assuming Equal -Test: Two Sample Assuming Equal Variances” ToolVariances” Tool



Step 3Step 3 Choose Choose tt-Test: Two Sample -Test: Two Sample Assuming Equal Assuming Equal Variances Variances from from the list of Analysis Toolsthe list of Analysis Tools

… … continued continued

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Step 4Step 4 When the When the tt-Test: Two Sample Assuming -Test: Two Sample Assuming Equal Variances dialog box Equal Variances dialog box

appears:appears:

Enter A1:A13 in the Enter A1:A13 in the Variable 1 RangeVariable 1 Range boxbox

Enter B1:B9 in the Enter B1:B9 in the Variable 2 RangeVariable 2 Range boxbox

Enter 0 in the Enter 0 in the Hypothesized Mean Hypothesized Mean DifferenceDifference box box



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Step 4Step 4 (continued) (continued)

Select Select LabelsLabels

Enter .01 in the Enter .01 in the AlphaAlpha box box

Select Select Output RangeOutput Range

Enter D1 in the Enter D1 in the Output RangeOutput Range box box

(Any upper left-hand corner cell (Any upper left-hand corner cell indicatingindicating

where the output is to begin may be where the output is to begin may be entered)entered)

Click Click OKOK


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A B C D E F1 M Car J Car t-Test: Two-Sample Assuming Equal Variances

2 25.1 25.63 32.2 28.1 M Car J Car

4 31.7 27.9 Mean 29.78333 27.3

5 27.6 25.3 Variance 6.556061 3.265714

6 28.5 30.1 Observations 12 8

7 33.6 27.5 Pooled Variance 5.276481

8 30.8 25.1 Hypothesized Mean Diff. 0

9 26.2 28.8 df 18

10 29.0 t Stat 2.368555

11 31.0 P(T<=t) one-tail 0.014626

12 31.7 t Critical one-tail 1.734063

13 30.0 P(T<=t) two-tail 0.029251

14 t Critical two-tail 2.100924

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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.

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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?

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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


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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


41 41 Slide

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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 19

2 9sd dndi

( ) ..

2

176 19

2 9

tds n

d

d

2 7 02 9 10

2 94..

.tds n

d

d

2 7 02 9 10

2 94..

.


42 42 Slide

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Using Excel to Conduct a Hypothesis Test Using Excel to Conduct a Hypothesis Test about about 11 – – 22: Matched Samples: Matched Samples

Excel’s “Excel’s “tt-Test: Paired Two Sample for Means” -Test: Paired Two Sample for Means” ToolTool



Step 3Step 3 Choose Choose tt-Test: Paired Two Sample -Test: Paired Two Sample for Meansfor Means

from the list of Analysis Toolsfrom the list of Analysis Tools


43 43 Slide

Slide


Excel’s “Excel’s “tt-Test: Paired Two Sample for Means” Tool-Test: Paired Two Sample for Means” ToolStep 4Step 4 When the When the tt-Test: Paired Two Sample for Means-Test: Paired Two Sample for Means

dialog box appears:dialog box appears: Enter B1:B11 in the Enter B1:B11 in the Variable 1 RangeVariable 1 Range box box Enter C1:C11 in the Enter C1:C11 in the Variable 2 RangeVariable 2 Range box box Enter 0 in the Enter 0 in the Hypothesized Mean Hypothesized Mean

DifferenceDifference boxbox Select Select LabelsLabels Enter .05 in the Enter .05 in the AlphaAlpha box box Select Select Output RangeOutput Range Enter E2 (your choice) in the Enter E2 (your choice) in the Output Output

RangeRange box box Click Click OKOK


44 44 Slide

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A B C D E F G1 Office UPX INTEX2 Seattle 32 25 t-Test: Paired Two Sample for Means3 L.A. 30 244 Boston 19 15 UPX INTEX5 Cleveland 16 15 Mean 17.7 156 N.Y.C. 15 13 Variance 62.011 31.77787 Houston 18 15 Observations 10 108 Atlanta 14 15 Pearson Correlation 0.96129 St. Louis 10 8 Hypothesized Mean Difference 010 Milwauk. 7 9 df 911 Denver 16 11 t Stat 2.936212 P(T<=t) one-tail 0.008313 t Critical one-tail 1.833114 P(T<=t) two-tail 0.016615 t Critical two-tail 2.2622

45 45 Slide

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Inferences about the Difference between Inferences about the Difference between the Proportions of Two Populationsthe Proportions of Two Populations

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

46 46 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

47 47 Slide

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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 ( ) ( )

48 48 Slide

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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 been the client’s product. The new campaign has been initiated with TV and newspaper advertisements initiated with TV and newspaper advertisements running for three weeks. A survey conducted running for three weeks. A survey conducted immediately after the new campaign showed 120 immediately after the new campaign showed 120 of 250 households “aware” of the client’s of 250 households “aware” of the client’s product.product.

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

49 49 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

50 50 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 52250

40 60150

. . .. (. ) . (. )

48 40 1 9648 52250

40 60150

51 51 Slide

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Using Excel to Develop Using Excel to Develop an Interval Estimate of an Interval Estimate of pp11 – – pp22


1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2)2 No Yes Sample Size 250 1503 Yes No No. of "Yes" =COUNTIF(A2:A251,"Yes") =COUNTIF(B2:B151,"Yes")4 Yes Yes Samp. Propor. =D3/D2 =E3/E25 No Yes 6 Yes No Confid. Coeff. 0.95 7 No No Lev. Of Signif. =1-D6 8 No Yes z Value =NORMSINV(1-D7/2) 9 Yes No

10 No No Std. Error =SQRT(D4*(1-D4)/D2+E4*(1-E4)/E2)11 Yes Yes Marg. of Error =D8*D1012 Yes No 13 Yes Yes Pt. Est. of Diff. =D4-E414 No Yes Lower Limit =D13-D1115 Yes Yes Upper Limit =D13+D11

Note: Rows 16-251 are not shownNote: Rows 16-251 are not shown..

52 52 Slide

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Using Excel to DevelopUsing Excel to Developan Interval Estimate of an Interval Estimate of pp11 – – pp22

A B C D E1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2)2 No Yes Sample Size 250 1503 Yes No No. of "Yes" 120 604 Yes Yes Samp. Propor. 0.48 0.405 No Yes 6 Yes No Confid. Coeff. 0.95 7 No No Lev. Of Signif. 0.05 8 No Yes z Value 1.960 9 Yes No

10 No No Std. Error 0.051011 Yes Yes Marg. of Error 0.099912 Yes No 13 Yes Yes Pt. Est. of Diff. 0.08014 No Yes Lower Limit -0.02015 Yes Yes Upper Limit 0.180


53 53 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

54 54 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

55 55 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

1 56

56 56 Slide

Slide


Using Excel to Conduct Using Excel to Conduct a Hypothesis Test about a Hypothesis Test about pp11 – – pp22

Formula WorksheetFormula Worksheet


A B C D E1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2)2 No Yes Sample Size 250 1503 Yes No No. of "Yes" =COUNTIF(A2:A251,"Yes") =COUNTIF(B2:B151,"Yes")4 Yes Yes Samp. Propor. =D3/D2 =E3/E25 No Yes 6 Yes No Lev of Signif. 0.05 7 No No Crit.Val. (upper) =NORMSINV(1-D7) 8 No Yes 9 Yes No Pt. Est. of Diff. =D4-E4

10 No No Hypoth. Value 011 Yes Yes12 Yes No Pool. Est. of p =(D2*D4+E2*E4)/(D2+E2)13 Yes Yes Standard Error =SQRT(D12*(1-D12)*(1/D2+1/E2))14 No Yes Test Statistic =(D9-D10)/D1315 Yes Yes p -Value =2*NORMSDIST(D14)16 Yes No Conclusion =IF(D15<D6,"Reject","Do Not Reject")

57 57 Slide

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Using Excel to ConductUsing Excel to Conducta Hypothesis Test about a Hypothesis Test about pp11 – – pp22

A B C D E1 Sur2 Sur1 Survey 2 (from Popul.1) Survey 1 (from Popul.2)2 No Yes Sample Size 250 1503 Yes No No. of "Yes" 120 604 Yes Yes Samp. Propor. 0.48 0.405 No Yes 6 Yes No Lev of Signif. 0.05 7 No No Crit.Val. (upper) 1.645 8 No Yes 9 Yes No Pt. Est. of Diff. 0.08

10 No No Hypoth. Value 011 Yes Yes12 Yes No Pool. Est. of p 0.45013 Yes Yes Standard Error 0.051414 No Yes Test Statistic 1.55715 Yes Yes p -Value 0.06016 Yes No Conclusion Do Not Reject


58 58 Slide

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End of Chapter 10End of Chapter 10

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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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