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1、Chapter 10 Comparisons Involving Means Part 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 Abo
2、ut the Difference Between Two Population Means: s 1 and s 2 Known,Interval Estimation of m 1 m 2 Hypothesis Tests About m 1 m 2,Estimating the Difference Between Two Population Means,Let 1 equal the mean of population 1 and 2 equal the mean of population 2.,The difference between the two population
3、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 a
4、nd 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
5、 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 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 statisti
6、cs appear on the next slide.,Par, Inc. is a manufacturer of golf equipment and has developed a new golf ball that has been designed to provide “extra distance.”,Example: Par, Inc.,Interval Estimation of 1 - 2: s 1 and s 2 Known,Sample Size,Sample Mean,Sample #1 Par, Inc.,Sample #2 Rap, Ltd.,120 ball
7、s 80 balls,275 yards 258 yards,Based on data from previous driving distance tests, the two population standard deviations are known 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 estimate of the diff
8、erence between the mean driving distances of the two brands of golf ball.,Estimating the Difference Between Two 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 balls 2 = mea
9、n 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 between the 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
10、,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, using a = .01, that the mean driving distance of Par, Inc. golf balls is greater than the mean dr
11、iving distance 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.
12、 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 Kno
13、wn,5. Determine whether to reject H0.,Because pvalue a = .01, we reject H0.,At the .01 level of significance, the sample evidence indicates the mean driving distance of Par, Inc. golf balls is greater than the mean driving distance of Rap, Ltd. golf balls.,Hypothesis Tests About m 1 - m 2: s 1 and s
14、 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 driving distance of Par, Inc. golf balls is greater than th
15、e mean driving distance of Rap, Ltd. golf balls.,Inferences About the Difference Between Two Population Means: s 1 and s 2 Unknown,Interval Estimation of m 1 m 2 Hypothesis 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
16、ta/2.,use the sample standard deviations s1 and s2 as 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
17、 Detroit has developed a new automobile known as the M car. 24 M cars and 28 J cars (from Japan) were road tested 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,Sa
18、mple Size,Sample Mean,Sample Std. Dev.,Sample #1 M Cars,Sample #2 J 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% confidence interval estimate of the difference between the mpg performances of the two models
19、 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 cars 2 = 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 degree
20、s 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 between the miles-per-gallon performances of M cars and J cars is 1.431 to 3.569 mpg.,2.5 + 1.069 or 1.431 to 3.569 mpg,Hypothesis Tests
21、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 the miles-per-gallon (mpg) performance of M cars is greater than
22、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 significa
23、nce.,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.0
24、05 = 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
25、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% con
26、fident 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 vari
27、ation between sampled items is eliminated as a source of sampling error.,Inferences About the Difference Between Two Population Means: Matched Samples,Example: Express Deliveries,Inferences About the Difference Between Two Population Means: Matched Samples,A Chicago-based firm has documents that mus
28、t be quickly distributed to district offices throughout the U.S. The firm must decide between two delivery services, UPX (United Parcel Express) and INTEX (International Express), to transport its documents.,Example: Express Deliveries,Inferences About the Difference Between Two Population Means: Ma
29、tched Samples,In testing the delivery times of the two services, the firm sent two reports to a random sample of its district offices with one report carried by UPX and the other report carried by INTEX. Do the data on the next slide indicate a difference in mean delivery times for the two services?
30、 Use a .05 level of significance.,32 30 19 16 15 18 14 10 7 16,25 24 15 15 13 15 15 8 9 11,UPX,INTEX,Difference,District Office,Seattle Los Angeles Boston Cleveland New York Houston Atlanta St. Louis Milwaukee Denver,Delivery Time (Hours),7 6 4 1 2 3 -1 2 -2 5,Inferences About the Difference Between
31、 Two 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 Between Two Population Means: Matched Samples,p Value and Critical Value A
32、pproaches,2. Specify the level of significance.,a = .05,Inferences About the Difference Between Two 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 dif
33、ference 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 Between Two P
34、opulation Means: Matched Samples,p Value Approach,4. Determine the critical value and rejection rule.,Inferences About the Difference Between Two 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 10 Part A,