1 1 SlideSlide 2 2 SlideSlideChapter 10Chapter 10 Comparisons Involving Means Comparisons Involving MeansPart APart Ann Inferences About the Difference BetweenInferences About the Difference Between Two Population Means: Two Population Means: 1 1 and and 2 2 Known Knownnn Inferences About the Difference BetweenInferences About the Difference Between Two Population Means: Matched Samples Two Population Means: Matched Samplesnn Inferences About the Difference BetweenInferences About the Difference Between Two Population Means: Two Population Means: 1 1 and and 2 2 Unknown Unknown 3 3 SlideSlideInferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Two Population Means: 1 1 and and 2 2 Known KnownnnInterval Estimation of Interval Estimation of 1 1 2 2nnHypothesis Tests About Hypothesis Tests About 1 1 2 2 4 4 SlideSlideEstimating the Difference BetweenEstimating the Difference BetweenTwo Population MeansTwo Population Meansnn Let Let 1 1 equal the mean of population 1 and equal the mean of population 1 and 2 2 equal equal the mean of population 2. the mean of population 2.nn The difference between the two population means isThe difference between the two population means is 1 1 - - 2 2. .nn To estimate To estimate 1 1 - - 2 2, we will select a simple random, we will select a simple random sample of size sample of size n n1 1 from population 1 and a simple from population 1 and a simple random sample of size random sample of size n n2 2 from population 2. from population 2.nn Let equal the mean of sample 1 and equal theLet equal the mean of sample 1 and equal the mean of sample 2. mean of sample 2.n n The point estimator of the difference between theThe point estimator of the difference between the means of the populations 1 and 2 is . means of the populations 1 and 2 is . 5 5 SlideSlidenn Expected ValueExpected ValueSampling Distribution of Sampling Distribution of nnStandard Deviation (Standard Error)Standard Deviation (Standard Error)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 n n1 1 = sample size from population 1= sample size from population 1 n n2 2 = sample size from population 2 = sample size from population 2 6 6 SlideSlidennInterval EstimateInterval EstimateInterval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Known Knownwhere:where: 1 - 1 - is the confidence coefficient is the confidence coefficient 7 7 SlideSlidenn Example: Par, Inc.Example: Par, Inc. Interval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Known Known In a test of driving distance using a mechanicalIn a test of driving distance using a mechanicaldriving device, a sample of Par golf balls wasdriving device, a sample of Par golf balls wascompared with a sample of golf balls made by Rap,compared with a sample of golf balls made by Rap,Ltd., a competitor. The sample statistics appear on theLtd., a competitor. The sample statistics appear on 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.” 8 8 SlideSlidennExample: Par, Inc.Example: Par, Inc.Interval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Known KnownSample SizeSample SizeSample MeanSample MeanSample #1Sample #1Par, Inc.Par, Inc.Sample #2Sample #2Rap, Ltd.Rap, Ltd. 120 120 ballsballs 80 balls80 balls275 275 yards 258 yardsyards 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. 9 9 SlideSlideInterval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Known KnownnnExample: Par, Inc.Example: Par, Inc. Let us develop a 95% confidence interval estimateLet us develop a 95% confidence interval estimateof the difference between the mean driving distances ofof the difference between the mean driving distances ofthe two brands of golf ball.the two brands of golf ball. 1010 SlideSlideEstimating the Difference BetweenEstimating the Difference BetweenTwo Population MeansTwo Population Meansm m1 1 2 2 = difference between= difference between the mean distances the mean distancesx x1 1 - - x x2 2 = Point Estimate of = Point Estimate of m m1 1 2 2 Population 1Population 1Population 1Population 1Par, Inc. Golf BallsPar, Inc. Golf Balls 1 1 = mean driving = mean driving distance of Par distance of Pargolf ballsgolf ballsPopulation 2Population 2Population 2Population 2Rap, Ltd. Golf BallsRap, Ltd. Golf Balls 2 2 = mean driving = mean driving distance of Rap distance of Rapgolf ballsgolf balls Simple random sampleSimple random sample of of n n2 2 Rap golf balls Rap golf ballsx x2 2 = sample mean distance = sample mean distance for the Rap golf balls for the Rap golf balls Simple random sampleSimple random sample of of n n1 1 Par golf balls Par golf ballsx x1 1 = sample mean distance = sample mean distance for the Par golf balls for the Par golf balls 1111 SlideSlidePoint Estimate of Point Estimate of 1 1 - - 2 2 Point estimate of Point estimate of 1 1 - - 2 2 = =where:where: 1 1 = mean distance for the population = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls 2 2 = mean distance for the population = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls= 275 = 275 - - 258 258= 17 = 17 yardsyards 1212 SlideSlideInterval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 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 5.14 or 11.86 yards to 22.14 yardsor 11.86 yards to 22.14 yards 1313 SlideSlideHypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Known Knownn n HypothesesHypothesesLeft-tailedLeft-tailedRight-tailedRight-tailedTwo-tailedTwo-tailedn n Test StatisticTest Statistic 1414 SlideSlidennExample: 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? 1515 SlideSlideHH0 0: : 1 1 - - 2 2 0 0where: where: 1 1 = mean distance for the population = mean distance for the population of Par, Inc. golf balls of Par, Inc. golf balls 2 2 = mean distance for the population = mean distance for the population of Rap, Ltd. golf balls of Rap, Ltd. golf balls1. 1. Develop the hypotheses.Develop the hypotheses.n n p p Value and Critical Value Approaches Value and Critical Value ApproachesHypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Known Known2. 2. Specify the level of significance.Specify the level of significance. = .01 = .01 1616 SlideSlide3. 3. Compute the value of the test statistic.Compute the value of the test statistic.Hypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Known Knownn n p p Value and Critical Value Approaches Value and Critical Value Approaches 1717 SlideSliden n p p Value ApproachValue Approach4. 4. Compute the Compute the p pvalue.value.For For z z = 6.49, the = 6.49, the p p value .0001. value .0001. Hypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Known Known5. 5. Determine whether to reject Determine whether to reject HH0 0. .Because Because p pvalue value 2.33, we reject 2.33, we reject HH0 0. .n n Critical Value ApproachCritical Value ApproachFor For = .01, = .01, z z.01.014. 4. Determine the critical value and rejection rule.Determine the critical value and rejection rule.Reject Reject HH0 0 if if z z The sample evidence indicates the mean drivingThe sample evidence indicates the mean drivingdistance of Par, Inc. golf balls is greater than the meandistance of Par, Inc. golf balls is greater than the meandriving distance of Rap, Ltd. golf balls.driving distance of Rap, Ltd. golf balls. 1919 SlideSlideInferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Two Population Means: 1 1 and and 2 2 Unknown UnknownnnInterval Estimation of Interval Estimation of 1 1 2 2nnHypothesis Tests About Hypothesis Tests About 1 1 2 2 2020 SlideSlideInterval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Unknown UnknownWhen When 1 1 and and 2 2 are unknown, we will: are unknown, we will: replace replace z z /2/2 with with t t /2/2. . use the sample standard deviations use the sample standard deviations s s1 1 and and s s2 2as estimates of as estimates of 1 1 and and 2 2 , and , and 2121 SlideSlideWhere the degrees of freedom for Where the degrees of freedom for t t /2/2 are: are:Interval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Unknown UnknownnnInterval EstimateInterval Estimate 2222 SlideSlidennExample: Specific MotorsExample: Specific Motors Difference Between Two Population Means:Difference Between Two Population Means: 1 1 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. 2323 SlideSlideDifference Between Two Population Means:Difference Between Two Population Means: 1 1 and and 2 2 Unknown UnknownnnExample: Specific MotorsExample: Specific MotorsSample SizeSample SizeSample MeanSample MeanSample Std. Dev.Sample Std. Dev.Sample #1Sample #1M CarsM CarsSample #2Sample #2J CarsJ Cars 24 24 carscars 2 28 cars8 cars29.8 29.8 mpg 27.3 mpgmpg 27.3 mpg2.56 2.56 mpg 1.81 mpgmpg 1.81 mpg 2424 SlideSlideDifference Between Two Population Means:Difference Between Two Population Means: 1 1 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.nnExample: Specific MotorsExample: Specific Motors 2525 SlideSlidePoint estimate of Point estimate of 1 1 - - 2 2 = =Point Estimate of Point Estimate of 1 1 - - 2 2where:where: 1 1 = mean miles-per-gallon for the = mean miles-per-gallon for the population of M cars population of M cars 2 2 = mean miles-per-gallon for the = mean miles-per-gallon for the population of J cars population of J cars= 2.5 = 2.5 mpgmpg 2626 SlideSlideInterval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Unknown UnknownThe degrees of freedom for The degrees of freedom for t t /2/2 are: are:With With /2 = .05 and /2 = .05 and dfdf = 24, = 24, t t /2/2 2727 SlideSlideInterval Estimation of Interval Estimation of 1 1 - - 2 2: : 1 1 and and 2 2 Unknown Unknown 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 1.069 or 1.431 to 3.569 mpgor 1.431 to 3.569 mpg 2828 SlideSlideHypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Unknown Unknownnn HypothesesHypothesesLeft-tailedLeft-tailedRight-tailedRight-tailedTwo-tailedTwo-tailednn Test StatisticTest Statistic 2929 SlideSlidennExample: 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? 3030 SlideSlideHH0 0: : 1 1 - - 2 2 0 0where: where: 1 1 = mean = mean mpgmpg for the population of M cars for the population of M cars 2 2 = mean = mean mpgmpg for the population of J cars for the population of J cars1. 1. Develop the hypotheses.Develop the hypotheses.n n p p Value and Critical Value Approaches Value and Critical Value ApproachesHypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Unknown Unknown 3131 SlideSlide2. 2. Specify the level of significance.Specify the level of significance.3. 3. Compute the value of the test statistic.Compute the value of the test statistic. = .05 = .05n n p p Value and Critical Value Approaches Value and Critical Value ApproachesHypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Unknown Unknown 3232 SlideSlideHypothesis Tests About Hypothesis Tests About 1 1 - - 2 2: : 1 1 and and 2 2 Unknown Unknownn n p p Value Approach Value Approach4. 4. Compute the Compute the p p value. value.The degrees of freedom for The degrees of freedom for t t are: are:Because Because t t = 4.003 = 4.003 t t.005.005 = 2.797, the = 2.797, the p pvalue .005.value .005. 3333 SlideSlide5. 5. Determine whether to reject Determine whether to reject HH0 0. . We are at least 95% confident that the miles-per-We are at least 95% confident that the miles-per-gallon (gallon (mpgmpg) performance of M cars is greater than ) performance of M cars is greater than the miles-per-gallon performance of J cars?.the miles-per-gallon performance of J cars?.n n p p Value Approach Value ApproachBecause Because p pvalue value 5. 5. Determine whether to reject Determine whether to reject HH0 0. .Because 4.003 Because 4.003 1.711, we reject 1.711, we reject HH0 0. . We are at least 95% confident that the miles-per-We are at least 95% confident that the miles-per-gallon (gallon (mpgmpg) performance of M cars is greater than ) performance of M cars is greater than the miles-per-gallon performance of J cars?.the miles-per-gallon performance of J cars?. 3535 SlideSliden n 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.n n This design often leads to a smaller sampling errorThis design often leads to a smaller sampling error than the independent-sample design because than the independent-sample design because variation between sampled items is eliminated as a variation between sampled items is eliminated 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 3636 SlideSlidennExample: Express DeliveriesExample: Express DeliveriesInferences 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. 3737 SlideSlidennExample: Express DeliveriesExample: Express DeliveriesInferences 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. 3838 SlideSlide32323030191916161515181814141010 7 716162525242415151515131315151515 8 8 9 91111UPXUPXINTEXINTEXDifferenceDifferenceDistrict OfficeDistrict OfficeSeattleSeattleLos AngelesLos AngelesBostonBostonClevelandClevelandNew YorkNew YorkHoustonHoustonAtlantaAtlantaSt. LouisSt. LouisMilwaukeeMilwaukeeDenverDenverDelivery Time (Hours)Delivery Time (Hours) 7 7 6 6 4 4 1 1 2 2 3 3 -1 -1 2 2 -2 -2 5 5Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samples 3939 SlideSlideHH0 0: : d d = 0= 0 HHa a: : d d Let Let d d = the mean of the = the mean of the differencedifference values for the values for the two delivery services for the population two delivery services for the population of district offices of district offices1. 1. Develop the hypotheses.Develop the hypotheses.Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samplesn n p p Value and Critical Value Approaches Value and Critical Value Approaches 4040 SlideSlide2. 2. Specify the level of significance.Specify the level of significance. = .05 = .05Inferences About the Difference BetweenInferences About the Difference BetweenTwo Population Means: Matched SamplesTwo Population Means: Matched Samplesn n p p Value and Critical Value Approaches Value and Critical Value Approaches3. 3. Compute the value of the test statistic.Compute the value of the test statistic. 4141 SlideSlide5. 5. Determine whether to reject Determine whether to reject HH0 0. . We are at least 95% confident that there is a We are at least 95% confident that there is a difference in mean delivery times for the two difference in mean delivery times for the two services?services?4. 4. Compute the Compute the p p value. value. For For t t = 2.94 and = 2.94 and dfdf = 9, the = 9, the p pvalue is betweenvalue is between.02 and .01. (This is a two-tailed test, so we double .02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.)the upper-tail areas of .01 and .005.)Because Because p pvalue value 5. 5. Determine whether to reject Determine whether to reject HH0 0. .Because Because t t = 2.94 = 2.94 2.262, we reject 2.262, we reject HH0 0. .We are at least 95% confident that there is a We are at least 95% confident that there is a difference in mean delivery times for the two difference in mean delivery times for the two services?services? 4343 SlideSlideEnd of Chapter 10End of Chapter 10Part APart A