chapter 10 comparisons involving means part a estimation of the difference between the means of two...

Chapter 10 Comparisons Involving Means

Part A

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

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

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

• Point Estimator of the Difference between the Means of Two Populations

• Sampling Distribution of

• Interval Estimate of Large-Sample Case

• Interval Estimate of Small-Sample Case

x x1 2x x1 2

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 equalequal

the mean of population 2.the mean of population 2. The difference between the two population The difference between the two population means ismeans is 11 - - 22.. To estimate To estimate 11 - - 22, we will select a simple , we will select a simple randomrandom

sample of size sample of size nn11 from population 1 and a from population 1 and a simplesimple

random sample of size random sample of size nn22 from population 2. from population 2. Let equal the mean of sample 1 and Let equal the mean of sample 1 and

equal theequal the

mean of sample 2.mean of sample 2.

x1x1 x2x2

The point estimator of the difference between The point estimator of the difference between thethe

means of the populations 1 and 2 is .means of the populations 1 and 2 is .x x1 2x x1 2

Expected ValueExpected Value

Sampling Distribution ofSampling Distribution of x x1 2x x1 2

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

Standard DeviationStandard Deviation

x x n n1 2

12

1

22

2

x x n n1 2

12

1

22

2

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

• Interval Estimate with 1 and 2 Known

Interval Estimate of 1 - 2:Large-Sample Case (n1 > 30 and n2 > 30)

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

where:where:

1 - 1 - is the confidence coefficient is the confidence coefficient

Interval Estimate with Interval Estimate with 11 and and 22 Unknown Unknown

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 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

where:where:

• Example: Par, Inc. Par, Inc. is a manufacturer of golf

equipment and has developed

a new golf ball that has been

designed to provide “extra

distance.” In a test of driving

distance using a mechanical

driving device, a sample of

Par golf balls was compared with a sample of golf balls

made by Rap, Ltd., a competitor.

The sample statistics appear on the next slide.


• Example: Par, Inc.


Sample SizeSample Size

Sample MeanSample Mean

Sample Std. Dev.Sample Std. Dev.

Sample #1Sample #1Par, Inc.Par, Inc.

Sample #2Sample #2Rap, Ltd.Rap, Ltd.

120 balls120 balls 80 balls80 balls

235 yards 218 yards235 yards 218 yards


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

Point Estimate of the DifferencePoint Estimate of the DifferenceBetween Two Population MeansBetween Two Population Means

Point estimate of Point estimate of 11 2 2 ==x x1 2x x1 2

where:where:

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

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

= 235 = 235 218 218

= 17 yards= 17 yards

Substituting the sample standard deviations for the population standard deviation:

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

/ .( ) ( )

95% Confidence Interval Estimate of the Difference Between Two Population Means: Large-Sample Case, 1 and 2 Unknown

We are 95% confident that the difference betweenWe are 95% confident that the difference betweenthe mean driving distances of Par, Inc. balls and Rap,the mean driving distances of Par, Inc. balls and Rap,Ltd. balls is 11.86 to 22.14 yards.Ltd. balls is 11.86 to 22.14 yards.

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

Using Excel to Develop anUsing Excel to Develop anInterval Estimate of Interval Estimate of 11 – – 22: : Large-Sample Large-Sample

CaseCase Formula WorksheetFormula Worksheet

A B C D E1 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.

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 anUsing Excel to Develop anInterval Estimate of Interval Estimate of 11 – – 22: : Large-Sample Large-Sample

CaseCase


Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30)

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 ( )

where:where:

2 2 21 2 2 2 21 2 Interval Estimate withInterval Estimate with

Interval Estimate of 1 - 2:Small-Sample Case (n1 < 30 and/or n2 < 30)

• Interval Estimate with 2 Unknown

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 ( )

where:where:

Example: Specific MotorsExample: Specific Motors

Specific Motors of DetroitSpecific Motors of Detroit

has developed a new automobilehas developed a new automobile

known as the M car. 12 M carsknown as the M car. 12 M cars

and 8 J cars (from Japan) were roadand 8 J cars (from Japan) were road

tested to compare miles-per-gallon (mpg)tested to compare miles-per-gallon (mpg)

performance. The sample statistics are shown performance. The sample statistics are shown on theon the

next slide.next slide.

Difference Between Two Population Difference Between Two Population Means:Means:

Small Sample CaseSmall Sample Case

Difference Between Two Population Difference Between Two Population Means:Means:

Small Sample CaseSmall Sample Case Example: Specific MotorsExample: Specific Motors




Sample #1Sample #1M CarsM Cars

Sample #2Sample #2J CarsJ Cars

12 cars12 cars 8 cars8 cars

29.8 mpg 27.3 mpg29.8 mpg 27.3 mpg

2.56 mpg 1.81 mpg2.56 mpg 1.81 mpg

Point estimate of Point estimate of 11 2 2 ==x x1 2x x1 2

Point Estimate of the DifferenceBetween Two Population Means

where:where:

11 = mean miles-per-gallon for the = mean miles-per-gallon for the population of M carspopulation of M cars

22 = mean miles-per-gallon for the = mean miles-per-gallon for the population of J carspopulation of J cars

= 29.8 - 27.3= 29.8 - 27.3

= 2.5 mpg= 2.5 mpg

We will make the following assumptions:

95% Confidence Interval Estimate of the Difference Between Two Population Means:

Small-Sample Case

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

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

• The miles per gallon rating is normally The miles per gallon rating is normally

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

95% Confidence Interval Estimate of the 95% Confidence Interval Estimate of the Difference Between Two Population Difference Between Two Population

Means: Means: Small-Sample CaseSmall-Sample Case

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..

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

( ) ( ) ( . ) ( . ).

2.5 + 2.2 or .3 to 4.7 miles per gallon

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

. ( ) . . . ( )

95% Confidence Interval Estimate of the Difference Between Two Population Means:

Small-Sample Case

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

of freedom, the appropriate of freedom, the appropriate tt value is value is tt.025.025 = = 2.101.2.101.

We are 95% confident that the difference betweenWe are 95% confident that the difference between the mean mpg ratings of the two car types is .3 to 4.7the mean mpg ratings of the two car types is .3 to 4.7 mpg (with the M car having the higher mpg).mpg (with the M car having the higher mpg).

Formula WorksheetFormula 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 =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 anUsing Excel to Develop anInterval Estimate of Interval Estimate of 11 – – 22: : Small-SampleSmall-Sample

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 anUsing Excel to Develop anInterval Estimate of Interval Estimate of 11 – – 22: : Small-SampleSmall-Sample

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

Independent Samples

1 2: 0aH 1 2: 0aH

• Hypotheses

• Test Statistic

Large-SampleLarge-SampleSmall-SampleSmall-Sample

0 1 2: 0H 0 1 2: 0H 0 1 2: 0H 0 1 2: 0H

1 2: 0aH 1 2: 0aH 0 1 2: 0H 0 1 2: 0H

1 2: 0aH 1 2: 0aH

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

zx x

n n

( ) ( )1 2 1 2

12

1 22

2

zx x

n n

( ) ( )1 2 1 2

12

1 22

2

Example: Par, Inc.Example: Par, Inc. Recall that Par, Inc. hasRecall that Par, Inc. has

developed a new golf ball that developed a new golf ball that

was designed to provide “extrawas designed to provide “extra

distance.” A sample of Par golfdistance.” A sample of Par golf

balls was compared with a sample of golf balls balls was compared with a sample of golf balls mademade

by Rap, Ltd., a competitor.by Rap, Ltd., a competitor.

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

Hypothesis Tests About the DifferenceHypothesis Tests About the DifferenceBetween the Means of Two Populations:Between the Means of Two Populations:Independent Samples, Large-Sample CaseIndependent Samples, Large-Sample Case

Example: Par, Inc.Example: Par, Inc. Can we conclude, using Can we conclude, using = .01, = .01,

that the mean driving distance ofthat the mean driving distance of

Par, Inc. golf balls is greater thanPar, Inc. golf balls is greater than

the mean driving distance ofthe mean driving distance of

Rap, Ltd. golf balls?Rap, Ltd. golf balls?





Sample #1Sample #1Par, Inc.Par, Inc.

Sample #2Sample #2Rap, Ltd.Rap, Ltd.

120 balls120 balls 80 balls80 balls



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

HHaa: : 1 1 - - 22 > 0 > 0where: where: 11 = mean distance for the population = mean distance for the population of Par, Inc. golf ballsof Par, Inc. golf balls22 = mean distance for the population = mean distance for the population of Rap, Ltd. golf ballsof Rap, Ltd. golf balls

1. Determine the hypotheses.1. Determine the hypotheses.

Using the Test StatisticUsing the Test Statistic



2. Specify the level of significance.2. Specify the level of significance.

3. Select the test statistic.3. Select the test statistic.

= .01= .01

4. State the rejection rule.4. State the rejection rule.Reject Reject HH00 if if zz > 2.33 > 2.33


1 2 1 2

2 21 2

1 2

( ) ( )x xz

n n

1 2 1 2

2 21 2

1 2

( ) ( )x xz

n n


5. Compute the value of the test statistic.5. Compute the value of the test statistic.


1 2 1 2

2 21 2

1 2

( ) ( )x xz

n n

1 2 1 2

2 21 2

1 2

( ) ( )x xz

n n

2 2

(235 218) 0 17 6.49

2.62(15) (20)120 80

z

2 2

(235 218) 0 17 6.49

2.62(15) (20)120 80

z


6. Determine whether to reject 6. Determine whether to reject HH00..

At the .01 level of significance, the sample At the .01 level of significance, the sample evidenceevidenceindicates the mean driving distance of Par, Inc. indicates the mean driving distance of Par, Inc. golfgolfballs is greater than the mean driving distance balls is greater than the mean driving distance of Rap,of Rap,Ltd. golf balls.Ltd. golf balls.

zz = 6.49 > = 6.49 > zz.01.01 = 2.33, so we reject = 2.33, so we reject HH00..


Using Excel to Conduct aUsing Excel to Conduct aHypothesis Test about Hypothesis Test about 11 – – 22: : Large Sample Large Sample

CaseCase 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 menu menu

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

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

from the list of Analysis Toolsfrom the list of Analysis Tools

… … continuedcontinued

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


CaseCase

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

dialog box appears:dialog box appears:

… … continuedcontinued

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

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

Type 0 in the Type 0 in the Hypothesized MeanHypothesized Mean

DifferenceDifference box boxType 225 in the Type 225 in the Variable 1 VarianceVariable 1 Variance

(known)(known) box boxType 400 in the Type 400 in the Variable 2 VarianceVariable 2 Variance

(known)(known) box box

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


CaseCase

Click Click OKOK

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

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

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

Type .01 in the Type .01 in the AlphaAlpha box boxSelect Select LabelsLabels

Step 4Step 4 (continued) (continued)


CaseCase

Value WorksheetValue Worksheet


CaseCase

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


Using the Using the p p ValueValue

4. Compute the value of the test statistic.4. Compute the value of the test statistic.

5. Compute the 5. Compute the pp–value.–value.

The Excel worksheet states The Excel worksheet states pp-value = 4.501E-11-value = 4.501E-11

6. Determine whether to reject 6. Determine whether to reject HH00..

Because Because pp–value < –value < = .01, we reject = .01, we reject HH00..

The Excel worksheet states The Excel worksheet states zz = 6.48 = 6.48


CaseCase

chapter 10 comparisons involving means part a estimation of the difference between the means of two...

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