Exam 10: Inferences for Two Population Means

Preliminary data analyses indicates that use of a paired Wilcoxon signed-rank test is reasonable. Perform the hypothesistest by using a paired Wilcoxon signed-rank test. Assume that the null hypothesis is $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -In a study of the effectiveness of physical exercise in weight reduction, 12 subjects followed a program of physical exercise for two months. Their weights (in pounds)before and after this program are shown in the following table. Before 162 190 188 152 148 127 195 164 175 156 180 136 After 157 194 179 149 135 130 183 168 168 148 170 138 At the 5% significance level, do the data provide sufficient evidence to conclude that the exercise program is effective in reducing weight?

Solve the problem. -A company which designs sports shoes has made an improvement to their popular running shoe. They hope that athletes wearing the new running shoe will be able to run faster over short Distances. They will use a paired sample to determine whether the mean time to run 100 meters for Sprinters wearing the new running shoe is less than the mean time to run the 100 meters for Sprinters wearing the old running shoe. Identify the variable for the proposed hypothesis test.

Provide an appropriate response. -Suppose that you want to perform a hypothesis test based on a simple random paired sample to compare the means of two populations. Assume that the paired-difference variable has a symmetric bimodal distribution and that the sample size is 40. Identify the procedures that could be used to carry out the hypothesis test, that is, the procedures whose assumptions are satisfied. If more than one procedure could be used, which one would be the best? Explain your answer.

Preliminary data analyses indicate that you can reasonably consider the assumptions for using the Mann-Whitney testsatisfied. Assume that the null hypothesis is $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ . Use the Mann-Whitney test to carry out the requiredhypothesis test. -A researcher was interested in comparing the salaries of female and male employees at a particular company. Independent simple random samples of 8 female employees and 9 male employees yielded the following weekly salaries (in dollars). Female Male 495 480 660 605 914 1250 743 722 520 1150 760 518 1005 1540 556 750 530 At the 5% significance level, do the data provide sufficient evidence to conclude that at this company the mean salary of female employees is lower than the mean salary of male employees?

Independent simple random samples of sizes 120 and 150 are taken from two populations with the intent of performing a hypothesis test to compare the means. Prior data analyses have indicated that the two population standard deviations are equal; however, the variable under consideration is not normally distributed on either population. Is it reasonable to use the pooled t-test in this situation? Explain your answer.

Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t-proceduressatisfied. Perform the required hypothesis test by using either the critical-value approach or the P-value approach asindicated. -A researcher was interested in comparing the response times of two different cab companies. Companies A and B were each called at 50 randomly selected times. The calls to company A were made independently of the calls to company B. The response times for each call were recorded. The summary statistics were as follows: Company A Company B Mean response time 7.6 minutes 6.9 minutes Standard deviation 1.4 minutes 1.7 minutes At the 0.02 level of significance, do the data provide sufficient evidence to conclude that the mean response time for company A differs from the mean response time for company B? Use the P-value approach.

Suppose that you want to perform a hypothesis test to compare the means of two populations using independent simple random samples. In which of the situations below is it reasonable to use The Mann-Whitney test?

Preliminary data analyses indicate that you can reasonably consider the assumptions for using the Mann-Whitney test satisfied. Assume that the null hypothesis is $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ . Use the Mann-Whitney test to carry out the required hypothesis test. -A researcher was interested in comparing the heights of women in two different countries. Independent simple random samples of 9 women from country A and 9 women from country B yielded the following heights (in inches). Country A Country B 64.1 65.3 66.4 60.2 61.7 61.7 62.0 65.8 67.3 61.0 64.9 64.6 64.7 60.0 68.0 65.4 63.6 59.0 At the 2.5% significance level, do the data provide sufficient evidence to conclude that women in country A are, on average, taller than women in country B?

Suppose that you want to perform a hypothesis test to compare the means of two populations. Several procedures are available, namely, the pooled t-test, the nonpooled t-test, the Mann-Whitney test, the paired t-test, and the Wilcoxon signed-rank test. Before selecting the appropriate procedure, you need to look at the sample data to ascertain distribution type. What methods are available for doing this?

Solve the problem. -A nutritionist wants to investigate whether her new diet will be effective in helping women aged 30-40 to lose weight. She will use a paired sample to determine whether the mean weight of Women before going on this diet is greater than the mean weight of women after being on this diet For two months. Identify the two populations for the proposed hypothesis test.

Preliminary data analyses indicate that you can reasonably use nonpooled t-procedures on the given data. Apply anonpooled t-test to perform the required hypothesis test, using either the critical-value approach or the P-value approachas indicated. -A paint manufacturer wishes to compare the drying times of two different types of paint. Independent random samples of 11 cans of type A and 9 cans of type B were selected and applied to similar surfaces. The drying times (in minutes)were recorded. The summary statistics are as follows. Type A Type B =75.4 =64.7 =4.5 =5.1 =11 =9 Do the data provide sufficient evidence to conclude that the mean drying time for type A differs from the mean drying time for type B? Perform a t-test at the 1% significance level. Use the P-value approach.

A type of sampling (independent or paired), sample size(s), and a figure showing the results of preliminary data analyseson the sample(s)are provided. The intent is to employ the sample data to perform a hypothesis test to compare the meansof the two populations from which the data were obtained. Use the information provided to decide which procedureshould be applied. -Paired; n = 12. A normal probability plot for the paired differences is given below.

A researcher would like to conduct a hypothesis test to compare the mean SAT scores of students who have received extra coaching and the mean test score of students who have not received extra Coaching. He will randomly select 15 students and will use the difference between the student's Score after coaching and the student's score before coaching as the paired-difference variable. He Obtains some normal probability plots of the sample data to determine whether it would be Reasonable to use a paired t-test. In which of the following situations would it be reasonable to use A paired t-test ? A: The distribution of the before test scores appears normal, the distribution of the after test scores Appears normal, however the distribution of the paired differences does not appear normal. B: The distribution of the before test scores is nonnormal, the distribution of the after test scores is Nonnormal, however the distribution of the paired differences appears normal.

Summary statistics are given for independent simple random samples from two populations. Use the nonpooled t-test toconduct the required hypothesis test. - $\bar { x } _ { 1 } = 74.0 , \mathrm {~s} _ { 1 } = 4.5 , \mathrm { n } _ { 1 } = 11 , \overline { \mathrm { x } } _ { 2 } = 64.0 , \mathrm {~s} _ { 2 } = 5.1 , \mathrm { n } _ { 2 } = 9$ Perform a two-tailed hypothesis test using a significance level of $\alpha = 0.01$ .

Determine the null and alternative hypotheses for the proposed hypothesis test. -A company which designs sports shoes has made an improvement to their popular running shoe. They hope that athletes wearing the new running shoe will be able to run faster over short Distances. They will use a paired sample to determine whether the mean time to run 100 meters for Sprinters wearing the new running shoe is less than the mean time to run the 100 meters for Sprinters wearing the old running shoe.

Preliminary data analyses indicate that you can reasonably use nonpooled t-procedures on the given data. Apply anonpooled t-test to perform the required hypothesis test, using either the critical-value approach or the P-value approachas indicated. -A researcher was interested in comparing the salaries of female and male employees at a particular company. Independent simple random samples of 8 female employees and 15 male employees yielded the following weekly salaries (in dollars). Female Male 495 722 518 760 562 904 556 880 1150 904 520 805 520 500 480 1005 1250 970 743 750 605 At the $5 \%$ significance level, do the data provide sufficient evidence to conclude that at this company the mean salary of female employees is less than the mean salary of male employees? Use the critical-value approach. (Note: $\bar { x } _ { 1 } = 705.375 , \bar { x } _ { 2 } = 817.067 , \mathrm {~s} _ { 1 } = 183.855 , \mathrm {~s} _ { 2 } = 330.146 .$

Preliminary data analyses indicates that use of a paired Wilcoxon signed-rank test is reasonable. Perform the hypothesistest by using a paired Wilcoxon signed-rank test. Assume that the null hypothesis is $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -Eleven runners are timed in the 100-meter dash and are timed again one month later after following a new training program. The times (in seconds)are shown in the following table. Before 12.1 12.4 11.7 11.5 11.0 11.8 12.3 10.8 12.6 12.7 10.7 After 11.9 12.4 11.8 11.4 11.2 11.5 12.0 10.9 12.0 12.2 11.1 At the 10% significance level, do the data provide sufficient evidence to conclude that the training is effective in reducing times in the 100-meter dash?

A type of sampling (independent or paired), sample size(s), and a figure showing the results of preliminary data analyseson the sample(s)are provided. The intent is to employ the sample data to perform a hypothesis test to compare the meansof the two populations from which the data were obtained. Use the information provided to decide which procedureshould be applied. -Independent; $\mathrm { n } _ { 1 } = 12$ and $\mathrm { n } _ { 2 } = 12$ . A normal probability plot for each sample and a back-to-back stem-and-leaf diagram are given below. 2 4 75 3 632 4 2 5 5 25 2 6 057 752 7 12579 72 8

Preliminary data analyses indicate that you can reasonably consider the assumptions for using pooled t-proceduressatisfied. Perform the required hypothesis test by using either the critical-value approach or the P-value approach asindicated. -A researcher was interested in comparing the heights of women in two different countries. Independent simple random samples of 9 women from country A and 9 women from country B yielded the following heights (in inches). Country A Country B 64.1 65.3 66.4 60.2 61.7 61.7 62.0 65.8 67.3 61.0 64.9 64.6 64.7 60.0 68.0 65.4 63.6 59.0 At the 10% significance level, do the data provide sufficient evidence to conclude that the mean height of women in country A is greater than the mean height of women in country B? Use the P-value approach. $\text { (Note: } \bar { x } _ { 1 } = 64.744 , \bar { x } _ { 2 } = 62.556 , s _ { 1 } = 2.192 , s _ { 2 } = 2.697 \text {.) }$

Solve the problem. -A researcher wants to use a paired sample to determine whether the mean number of hours spent exercising per week for married men differs from the mean number of hours spent exercising per Week for married women. Identify the variable for the proposed hypothesis test.