Chapter 10Comparisons Involving Means Part A商务统计 教学课件.ppt

Chapter 10 Comparisons Involving MeansPart A,Inferences About the Difference Between Two Population Means:s 1 and s 2 Known,Inferences About the Difference Between Two Population Means:Matched Samples,Inferences About the Difference Between Two Population Means:s 1 and s 2 Unknown,Inferences About the Difference BetweenTwo Population Means:s 1 and s 2 Known,Interval Estimation of m 1 m 2Hypothesis Tests About m 1 m 2,Estimating the Difference BetweenTwo Population Means,Let 1 equal the mean of population 1 and 2 equal the mean of population 2.,The difference between the two population means is 1-2.,To estimate 1-2,we will select a simple random sample of size n1 from population 1 and a simple random sample of size n2 from population 2.,Let equal the mean of sample 1 and equal the mean of sample 2.,The point estimator of the difference between the means of the populations 1 and 2 is.,Expected Value,Sampling Distribution of,Standard Deviation(Standard Error),where:1=standard deviation of population 1 2=standard deviation of population 2 n1=sample size from population 1 n2=sample size from population 2,Interval Estimate,Interval Estimation of 1-2:s 1 and s 2 Known,where:1-is the confidence coefficient,Example:Par,Inc.,Interval Estimation of 1-2:s 1 and s 2 Known,In a test of driving distance using a mechanicaldriving device,a sample of Par golf balls wascompared with a sample of golf balls made by Rap,Ltd.,a competitor.The sample statistics appear on thenext slide.,Par,Inc.is a manufacturerof golf equipment and hasdeveloped a new golf ballthat has been designed toprovide“extra distance.”,Example:Par,Inc.,Interval Estimation of 1-2:s 1 and s 2 Known,Sample Size,Sample Mean,Sample#1Par,Inc.,Sample#2Rap,Ltd.,120 balls 80 balls,275 yards 258 yards,Based on data from previous driving distancetests,the two population standard deviations areknown with s 1=15 yards and s 2=20 yards.,Interval Estimation of 1-2:s 1 and s 2 Known,Example:Par,Inc.,Let us develop a 95%confidence interval estimateof the difference between the mean driving distances ofthe two brands of golf ball.,Estimating the Difference BetweenTwo Population Means,m1 m2=difference between the mean distances,Point Estimate of 1-2,Point estimate of 1-2=,where:1=mean distance for the population of Par,Inc.golf balls2=mean distance for the population of Rap,Ltd.golf balls,=275-258,=17 yards,Interval Estimation of 1-2:1 and 2 Known,We are 95%confident that the difference betweenthe mean driving distances of Par,Inc.balls and Rap,Ltd.balls is 11.86 to 22.14 yards.,17+5.14 or 11.86 yards to 22.14 yards,Hypothesis Tests About m 1-m 2:s 1 and s 2 Known,Hypotheses,Left-tailed,Right-tailed,Two-tailed,Test Statistic,Example:Par,Inc.,Hypothesis Tests About m 1-m 2:s 1 and s 2 Known,Can we conclude,usinga=.01,that the mean drivingdistance of Par,Inc.golf ballsis greater than the mean drivingdistance of Rap,Ltd.golf balls?,H0:1-2 0,where:1=mean distance for the population of Par,Inc.golf balls 2=mean distance for the population of Rap,Ltd.golf balls,1.Develop the hypotheses.,p Value and Critical Value Approaches,Hypothesis Tests About m 1-m 2:s 1 and s 2 Known,2.Specify the level of significance.,a=.01,3.Compute the value of the test statistic.,Hypothesis Tests About m 1-m 2:s 1 and s 2 Known,p Value and Critical Value Approaches,p Value Approach,4.Compute the pvalue.,For z=6.49,the p value.0001.,Hypothesis Tests About m 1-m 2:s 1 and s 2 Known,5.Determine whether to reject H0.,Because pvalue a=.01,we reject H0.,At the.01 level of significance,the sample evidenceindicates the mean driving distance of Par,Inc.golfballs is greater than the mean driving distance of Rap,Ltd.golf balls.,Hypothesis Tests About m 1-m 2:s 1 and s 2 Known,5.Determine whether to reject H0.,Because z=6.49 2.33,we reject H0.,Critical Value Approach,For a=.01,z.01=2.33,4.Determine the critical value and rejection rule.,Reject H0 if z 2.33,The sample evidence indicates the mean drivingdistance of Par,Inc.golf balls is greater than the meandriving distance of Rap,Ltd.golf balls.,Inferences About the Difference BetweenTwo Population Means:s 1 and s 2 Unknown,Interval Estimation of m 1 m 2Hypothesis Tests About m 1 m 2,Interval Estimation of 1-2:s 1 and s 2 Unknown,When s 1 and s 2 are unknown,we will:,replace za/2 with ta/2.,use the sample standard deviations s1 and s2as estimates of s 1 and s 2,and,Where the degrees of freedom for ta/2 are:,Interval Estimation of 1-2:s 1 and s 2 Unknown,Interval Estimate,Example:Specific Motors,Difference Between Two Population Means:s 1 and s 2 Unknown,Specific Motors of Detroithas developed a new automobileknown as the M car.24 M carsand 28 J cars(from Japan)were roadtested to compare miles-per-gallon(mpg)performance.The sample statistics are shown on the next slide.,Difference Between Two Population Means:s 1 and s 2 Unknown,Example:Specific Motors,Sample Size,Sample Mean,Sample Std.Dev.,Sample#1M Cars,Sample#2J Cars,24 cars 28 cars,29.8 mpg 27.3 mpg,2.56 mpg 1.81 mpg,Difference Between Two Population Means:s 1 and s 2 Unknown,Let us develop a 90%confidenceinterval estimate of the differencebetween the mpg performances ofthe two models of automobile.,Example:Specific Motors,Point estimate of 1-2=,Point Estimate of m 1-m 2,where:1=mean miles-per-gallon for the population of M cars2=mean miles-per-gallon for the population of J cars,=29.8-27.3,=2.5 mpg,Interval Estimation of m 1-m 2:s 1 and s 2 Unknown,The degrees of freedom for ta/2 are:,With a/2=.05 and df=24,ta/2=1.711,Interval Estimation of m 1-m 2:s 1 and s 2 Unknown,We are 90%confident that the difference betweenthe miles-per-gallon performances of M cars and J carsis 1.431 to 3.569 mpg.,2.5+1.069 or 1.431 to 3.569 mpg,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,Hypotheses,Left-tailed,Right-tailed,Two-tailed,Test Statistic,Example:Specific Motors,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,Can we conclude,using a.05 level of significance,that themiles-per-gallon(mpg)performanceof M cars is greater than the miles-per-gallon performance of J cars?,H0:1-2 0,where:1=mean mpg for the population of M cars 2=mean mpg for the population of J cars,1.Develop the hypotheses.,p Value and Critical Value Approaches,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,2.Specify the level of significance.,3.Compute the value of the test statistic.,a=.05,p Value and Critical Value Approaches,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,p Value Approach,4.Compute the p value.,The degrees of freedom for ta are:,Because t=4.003 t.005=2.797,the pvalue.005.,5.Determine whether to reject H0.,We are at least 95%confident that the miles-per-gallon(mpg)performance of M cars is greater than the miles-per-gallon performance of J cars?.,p Value Approach,Because pvalue a=.05,we reject H0.,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,4.Determine the critical value and rejection rule.,Critical Value Approach,Hypothesis Tests About m 1-m 2:s 1 and s 2 Unknown,For a=.05 and df=24,t.05=1.711,Reject H0 if t 1.711,5.Determine whether to reject H0.,Because 4.003 1.711,we reject H0.,We are at least 95%confident that the miles-per-gallon(mpg)performance of M cars is greater than the miles-per-gallon performance of J cars?.,With a matched-sample design each sampled item provides a pair of data values.,This design often leads to a smaller sampling error than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.,Inferences About the Difference BetweenTwo Population Means:Matched Samples,Example:Express Deliveries,Inferences About the Difference BetweenTwo Population Means:Matched Samples,A Chicago-based firm hasdocuments that must be quicklydistributed to district officesthroughout the U.S.The firmmust decide between two deliveryservices,UPX(United Parcel Express)and INTEX(International Express),to transport its documents.,Example:Express Deliveries,Inferences About the Difference BetweenTwo Population Means:Matched Samples,In testing the delivery timesof the two services,the firm senttwo reports to a random sampleof its district offices with onereport carried by UPX and theother report carried by INTEX.Do the data on thenext slide indicate a difference in mean deliverytimes for the two services?Use a.05 level ofsignificance.,3230191615181410 716,25241515131515 8 911,UPX,INTEX,Difference,District Office,SeattleLos AngelesBostonClevelandNew YorkHoustonAtlantaSt.LouisMilwaukeeDenver,Delivery Time(Hours),7 6 4 1 2 3-1 2-2 5,Inferences About the Difference BetweenTwo Population Means:Matched Samples,H0:d=0 Ha:d,Let d=the mean of the difference values for the two delivery services for the population of district offices,1.Develop the hypotheses.,Inferences About the Difference BetweenTwo Population Means:Matched Samples,p Value and Critical Value Approaches,2.Specify the level of significance.,a=.05,Inferences About the Difference BetweenTwo Population Means:Matched Samples,p Value and Critical Value Approaches,3.Compute the value of the test statistic.,5.Determine whether to reject H0.,We are at least 95%confident that there is a difference in mean delivery times for the two services?,4.Compute the p value.,For t=2.94 and df=9,the pvalue is between.02 and.01.(This is a two-tailed test,so we double the upper-tail areas of.01 and.005.),Because pvalue a=.05,we reject H0.,Inferences About the Difference BetweenTwo Population Means:Matched Samples,p Value Approach,4.Determine the critical value and rejection rule.,Inferences About the Difference BetweenTwo Population Means:Matched Samples,Critical Value Approach,For a=.05 and df=9,t.025=2.262.,Reject H0 if t 2.262,5.Determine whether to reject H0.,Because t=2.94 2.262,we reject H0.,We are at least 95%confident that there is a difference in mean delivery times for the two services?,End of Chapter 10Part A,