1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.

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1 IS 310 – Business Statistics IS 310 – Business Statistics IS 310 Business Statisti cs CSU Long Beach

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3 3 Slide IS 310 – Business Statistics Inferences About the Difference Between Two Population Means:  1 and  2 Known Interval Estimation of  1 –  2 Interval Estimation of  1 –  2 Hypothesis Tests About  1 –  2 Hypothesis Tests About  1 –  2

Transcript of 1 1 Slide IS 310 – Business Statistics IS 310 Business Statistics CSU Long Beach.

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IS 310 – Business StatisticsIS 310 – Business Statistics

IS 310Business Statistic

sCSU

Long Beach

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IS 310 – Business StatisticsIS 310 – Business Statistics

Chapter 10, Part AChapter 10, Part A Statistical Inferences About Means Statistical Inferences About Means

and Proportions with Two Populationsand Proportions with Two Populations

Inferences About the Difference BetweenInferences About the Difference Between Two Population Means: Two Population Means: 11 and and 22 Known Known

Inferences About the Difference BetweenInferences About the Difference Between Two Population Means: Matched SamplesTwo Population Means: Matched Samples

Inferences About the Difference BetweenInferences About the Difference Between Two Population Means: Two Population Means: 11 and and 22 Unknown Unknown

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Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Two Population Means: 1 1 and and 2 2 Known Known

Interval Estimation of Interval Estimation of 11 – – 22 Hypothesis Tests About Hypothesis Tests About 11 – – 22

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Estimating the Difference BetweenEstimating the Difference BetweenTwo Population MeansTwo Population Means

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.

x1 x2

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 2

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Expected ValueExpected Value

Sampling Distribution of Sampling Distribution of x x1 2

E x x( )1 2 1 2

Standard Deviation (Standard Error)Standard Deviation (Standard Error)

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

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Interval EstimateInterval Estimate

Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Known Known

2 21 2

1 2 / 21 2

x x zn n

where:where: 1 - 1 - is the confidence coefficient is the confidence coefficient

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Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Known Known

In a test of driving distance using a In a test of driving distance using a mechanicalmechanicaldriving device, a sample of Par golf balls wasdriving device, a sample of Par golf balls wascompared with a sample of golf balls made by compared with a sample of golf balls made by Rap,Rap,Ltd., a competitor. The sample statistics appear Ltd., a competitor. The sample statistics appear on theon thenext slide.next slide.

Par, Inc. is a manufacturerPar, Inc. is a manufacturerof golf equipment and hasof golf equipment and hasdeveloped a new golf balldeveloped a new golf ballthat has been designed tothat has been designed toprovide “extra distance.”provide “extra distance.”

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

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

Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Known Known

Sample SizeSample SizeSample MeanSample Mean

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

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

120 balls120 balls 80 balls80 balls275 yards 258 yards275 yards 258 yards

Based on data from previous driving distanceBased on data from previous driving distancetests, the two population standard deviations aretests, the two population standard deviations areknown with known with 1 1 = 15 yards and = 15 yards and 2 2 = 20 yards. = 20 yards.

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Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Known Known

Example: Par, Inc.Example: Par, Inc. Let us develop a 95% confidence interval Let us develop a 95% confidence interval estimateestimateof the difference between the mean driving of the difference between the mean driving distances ofdistances ofthe two brands of golf ball.the two brands of golf ball.

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Estimating the Difference BetweenEstimating the Difference BetweenTwo Population MeansTwo Population Means

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

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

Population 1Population 1Par, Inc. Golf BallsPar, Inc. Golf Balls11 = mean driving = mean driving

distance of Pardistance of Pargolf ballsgolf balls

Population 2Population 2Rap, Ltd. Golf BallsRap, Ltd. Golf Balls22 = mean driving = mean driving

distance of Rapdistance of Rapgolf ballsgolf balls

Simple random sampleSimple random sample of of nn22 Rap golf balls Rap golf ballsxx22 = sample mean distance = sample mean distance for the Rap golf ballsfor the Rap golf balls

Simple random sampleSimple random sample of of nn11 Par golf balls Par golf ballsxx11 = sample mean distance = sample mean distance for the Par golf ballsfor the Par golf balls

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Point Estimate of Point Estimate of 11 - - 22

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

where: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

= 275 = 275 258 258= 17 yards= 17 yards

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x x zn n1 2 2

12

1

22

2

2 217 1 96 15

1202080

/ . ( ) ( )

Interval Estimation of Interval Estimation of 11 - - 22::11 and and 22 Known Known

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

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Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

HypothesesHypotheses

1 2 02 21 2

1 2

( )x x Dz

n n

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

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

Left-tailedLeft-tailed Right-tailedRight-tailed Two-tailedTwo-tailed Test StatisticTest Statistic

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

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

Can we conclude, usingCan we conclude, using = .01, that the mean driving= .01, that the mean drivingdistance of Par, Inc. golf ballsdistance of Par, Inc. golf ballsis greater than the mean drivingis greater than the mean drivingdistance of Rap, Ltd. golf balls?distance of Rap, Ltd. golf balls?

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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. Develop the hypotheses.1. Develop the hypotheses.

pp –Value and Critical Value Approaches –Value and Critical Value Approaches

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

2. Specify the level of significance.2. Specify the level of significance. = .01= .01

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3. Compute the value of the test statistic.3. Compute the value of the test statistic.

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

pp –Value and Critical Value Approaches –Value and Critical Value Approaches

1 2 02 21 2

1 2

( )x x Dz

n n

2 2(235 218) 0 17 6.492.62(15) (20)

120 80

z

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p p –Value Approach–Value Approach

4. Compute the 4. Compute the pp–value.–value.For For zz = 6.49, the = 6.49, the pp –value < .0001. –value < .0001.

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

5. Determine whether to reject 5. Determine whether to reject HH00..Because Because pp–value –value << = .01, we reject = .01, we 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.

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Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Known Known

5. Determine whether to reject 5. Determine whether to reject HH00..Because Because zz = 6.49 = 6.49 >> 2.33, we reject 2.33, we reject HH00..

Critical Value ApproachCritical Value Approach

For For = .01, = .01, zz.01.01 = 2.33 = 2.334. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.

Reject Reject HH00 if if zz >> 2.33 2.33

The sample evidence indicates the mean The sample evidence indicates the mean drivingdrivingdistance of Par, Inc. golf balls is greater than distance of Par, Inc. golf balls is greater than the meanthe meandriving distance of Rap, Ltd. golf balls.driving distance of Rap, Ltd. golf balls.

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Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Two Population Means: 1 1 and and 2 2

UnknownUnknown Interval Estimation of Interval Estimation of 11 – – 22 Hypothesis Tests About Hypothesis Tests About 11 – – 22

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Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Unknown Unknown

When When 1 1 and and 2 2 are unknown, we will: are unknown, we will:

• replace replace zz/2/2 with with tt/2/2. .

• use the sample standard deviations use the sample standard deviations ss11 and and ss22

as estimates of as estimates of 1 1 and and 2 2 , and , and

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2 21 2

1 2 / 21 2

s sx x tn n

Where the degrees of freedom for Where the degrees of freedom for tt/2/2 are: are:

Interval Estimation of Interval Estimation of 11 - - 22:: 1 1 and and 2 2 Unknown Unknown

Interval EstimateInterval Estimate

22 21 2

1 22 22 2

1 2

1 1 2 2

1 11 1

s sn n

dfs s

n n n n

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Example: Specific MotorsExample: Specific Motors

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

11 and and 2 2 Unknown Unknown

Specific Motors of DetroitSpecific Motors of Detroithas developed a new automobilehas developed a new automobileknown as the M car. 24 M carsknown as the M car. 24 M carsand 28 J cars (from Japan) were roadand 28 J cars (from Japan) were roadtested to compare miles-per-gallon (mpg) performance. tested to compare miles-per-gallon (mpg) performance. The sample statistics are shown on the next slide.The sample statistics are shown on the next slide.

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

11 and and 2 2 Unknown Unknown Example: Specific MotorsExample: Specific Motors

Sample SizeSample SizeSample MeanSample MeanSample Std. Dev.Sample Std. Dev.

Sample #1Sample #1M CarsM Cars

Sample #2Sample #2J CarsJ Cars

24 cars24 cars 2 28 cars8 cars29.8 mpg 27.3 mpg29.8 mpg 27.3 mpg2.56 mpg 1.81 mpg2.56 mpg 1.81 mpg

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

11 and and 2 2 Unknown Unknown

Let us develop a 90% confidenceLet us develop a 90% confidenceinterval estimate of the differenceinterval estimate of the differencebetween the mpg performances ofbetween the mpg performances ofthe two models of automobile.the two models of automobile.

Example: Specific MotorsExample: Specific Motors

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

Point Estimate of Point Estimate of 1 1 2 2

where:where:11 = mean miles-per-gallon for the = mean miles-per-gallon for the

population of M carspopulation of M cars22 = 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

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Interval Estimation of Interval Estimation of 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

The degrees of freedom for The degrees of freedom for tt/2/2 are: are:22 2

2 22 2

(2.56) (1.81)24 28

24.07 241 (2.56) 1 (1.81)

24 1 24 28 1 28

df

With With /2 = .05 and /2 = .05 and dfdf = 24, = 24, tt/2/2 = 1.711 = 1.711

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Interval Estimation of Interval Estimation of 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

2 2 2 21 2

1 2 / 21 2

(2.56) (1.81) 29.8 27.3 1.71124 28

s sx x tn n

We are 90% confident that the difference betweenWe are 90% confident that the difference betweenthe miles-per-gallon performances of M cars and J carsthe miles-per-gallon performances of M cars and J carsis 1.431 to 3.569 mpg.is 1.431 to 3.569 mpg.

2.5 2.5 ++ 1.069 or 1.431 to 3.569 mpg 1.069 or 1.431 to 3.569 mpg

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Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

HypothesesHypotheses

1 2 02 21 2

1 2

( )x x Dts sn n

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

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

Left-tailedLeft-tailed Right-tailedRight-tailed Two-tailedTwo-tailed Test StatisticTest Statistic

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Example: Specific MotorsExample: Specific Motors

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

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

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

HHaa: : 1 1 - - 22 > 0 > 0where: where: 11 = mean = mean mpgmpg for the population of M cars for the population of M cars22 = mean = mean mpgmpg for the population of J cars for the population of J cars

1. Develop the hypotheses.1. Develop the hypotheses. pp –Value and Critical Value Approaches –Value and Critical Value Approaches

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

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

3. Compute the value of the test statistic.3. Compute the value of the test statistic.

= .05= .05 pp –Value and Critical Value Approaches –Value and Critical Value Approaches

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

1 2 02 2 2 21 2

1 2

( ) (29.8 27.3) 0 4.003(2.56) (1.81)

24 28

x x Dts sn n

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Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

pp –Value Approach –Value Approach4. Compute the 4. Compute the pp –value. –value.The degrees of freedom for The degrees of freedom for tt are: are:

Because Because tt = 4.003 > = 4.003 > tt.005.005 = 1.683, the = 1.683, the pp–value < .005.–value < .005.

22 2

2 22 2

(2.56) (1.81)24 28

40.566 411 (2.56) 1 (1.81)

24 1 24 28 1 28

df

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5. Determine whether to reject 5. Determine whether to reject HH00..

We are at least 95% confident that the We are at least 95% confident that the miles-per-gallon (miles-per-gallon (mpgmpg) performance of M ) performance of M cars is greater than the miles-per-gallon cars is greater than the miles-per-gallon performance of J cars?.performance of J cars?.

pp –Value Approach –Value Approach

Because Because pp–value –value << = .05, we reject = .05, we reject HH00..

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

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4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule. Critical Value ApproachCritical Value Approach

Hypothesis Tests About Hypothesis Tests About 1 1 2 2:: 1 1 and and 2 2 Unknown Unknown

For For = .05 and = .05 and dfdf = 41, = 41, tt.05.05 = 1.683 = 1.683Reject Reject HH00 if if tt >> 1.683 1.683

5. Determine whether to reject 5. Determine whether to reject HH00..Because 4.003 Because 4.003 >> 1.683, we reject 1.683, we reject HH00..

We are at least 95% confident that the We are at least 95% confident that the miles-per-gallon (miles-per-gallon (mpgmpg) performance of M ) performance of M cars is greater than the miles-per-gallon cars is greater than the miles-per-gallon performance of J cars?.performance of J cars?.

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With a With a matched-sample designmatched-sample design each sampled item each sampled item provides a pair of data values.provides a pair of data values.

This design often leads to a smaller sampling This design often leads to a smaller sampling errorerror than the independent-sample design than the independent-sample design becausebecause variation between sampled items is variation between sampled items is eliminated as aeliminated as a source of sampling error.source of sampling error.

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

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Example: Express DeliveriesExample: Express Deliveries

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

A Chicago-based firm hasA Chicago-based firm hasdocuments that must be quicklydocuments that must be quicklydistributed to district officesdistributed to district officesthroughout the U.S. The firmthroughout the U.S. The firmmust decide between two deliverymust decide between two deliveryservices, UPX (United Parcel Express) and INTEXservices, UPX (United Parcel Express) and INTEX(International Express), to transport its documents.(International Express), to transport its documents.

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Example: Express DeliveriesExample: Express Deliveries

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

In testing the delivery timesIn testing the delivery timesof the two services, the firm sentof the two services, the firm senttwo reports to a random sampletwo reports to a random sampleof its district offices with oneof its district offices with onereport carried by UPX and thereport carried by UPX and theother report carried by INTEX. Do the data on theother report carried by INTEX. Do the data on thenext slide indicate a difference in mean deliverynext slide indicate a difference in mean deliverytimes for the two services? Use a .05 level oftimes for the two services? Use a .05 level ofsignificance.significance.

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32323030191916161515181814141010 771616

2525242415151515131315151515 88 991111

UPXUPX INTEXINTEX DifferenceDifferenceDistrict OfficeDistrict OfficeSeattleSeattleLos AngelesLos AngelesBostonBostonClevelandClevelandNew YorkNew YorkHoustonHoustonAtlantaAtlantaSt. LouisSt. LouisMilwaukeeMilwaukeeDenverDenver

Delivery Time (Hours)Delivery Time (Hours)

7 7 6 6 4 4 1 1 2 2 3 3 -1 -1 2 2 -2 -2 55

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

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HH00: : d d = 0= 0

HHaa: : dd Let Let d d = the mean of the = the mean of the differencedifference values for the values for the two delivery services for the populationtwo delivery services for the population of district officesof district offices

1. Develop the hypotheses.1. Develop the hypotheses.

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

pp –Value and Critical Value Approaches –Value and Critical Value Approaches

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2. Specify the level of significance.2. Specify the level of significance. = .05= .05

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

pp –Value and Critical Value Approaches –Value and Critical Value Approaches

3. Compute the value of the test statistic.3. Compute the value of the test statistic.

d dni ( ... ) .7 6 5

102 7

s d dndi

( ) . .2

176 1

92 9

2.7 0 2.942.9 10d

d

dts n

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5. Determine whether to reject 5. Determine whether to reject HH00..

We are at least 95% confident that We are at least 95% confident that there is a difference in mean delivery there is a difference in mean delivery times for the two services?times for the two services?

4. Compute the 4. Compute the pp –value. –value. For For tt = 2.94 and = 2.94 and dfdf = 9, the = 9, the pp–value is –value is betweenbetween.02 and .01. (This is a two-tailed test, so we .02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.)double the upper-tail areas of .01 and .005.)

Because Because pp–value –value << = .05, we reject = .05, we reject HH00..

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

pp –Value Approach –Value Approach

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4. Determine the critical value and rejection rule.4. Determine the critical value and rejection rule.

Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples

Critical Value ApproachCritical Value Approach

For For = .05 and = .05 and dfdf = 9, = 9, tt.025.025 = 2.262. = 2.262.Reject Reject HH00 if if tt >> 2.262 2.262

5. Determine whether to reject 5. Determine whether to reject HH00..Because Because tt = 2.94 = 2.94 >> 2.262, we reject 2.262, we reject HH00..

We are at least 95% confident that there We are at least 95% confident that there is a difference in mean delivery times for is a difference in mean delivery times for the two services?the two services?

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End of Chapter 10End of Chapter 10Part APart A