Exam 10: Inferences for Two Population Means

Apply the nonpooled t-interval procedure to obtain the required confidence interval. You may assume that theassumptions for using the procedure are satisfied. -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 Determine a $90 \%$ confidence interval for the difference, $\mu _ { 1 } - \mu _ { 2 }$ , between the mean height of women in country $A$ and the mean height of women in country $B$ . (Note: $\bar { x } _ { 1 } = 64.744 , \bar { x } _ { 2 } = 62.556 , s _ { 1 } = 2.192 , s _ { 2 } = 2.697 .$ )

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. -The forced vital capacity (FVC)is often used by physicians to assess a person's ability to move air in and out of their lungs. It is the maximum amount of air that can be exhaled after a deep breath. A Researcher wants to perform a hypothesis test to determine whether the mean forced vital capacity For adults who are smokers is less than the mean forced vital capacity for adults who are former Smokers. He will use a paired sample to determine whether forced vital capacity increases, on Average, when adults stop smoking.

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 paired-difference variable for the proposed hypothesis test.

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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -Students take a math test before and after tutoring. The data for a random sample of students is given below. Before 54 72 59 82 46 80 63 36 44 After 68 70 67 78 55 90 63 52 40 At the 2.5% level of significance, do the data provide sufficient evidence to conclude that the mean score before the tutoring is lower than the mean score after the tutoring?

Determine the null and alternative hypotheses for the proposed hypothesis test. -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.

Preliminary data analyses indicate that you can reasonably consider the assumptions for using the Mann-Whitney testsatisfied. Assume that the null hypothesis is $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ . Use the Mann-Whitney test to carry out the requiredhypothesis test. -A researcher was interested in comparing the GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 9 students from college B yielded the following GPAs. College A College B 2.8 3.6 3.2 3.9 3.7 3.3 2.5 4.0 2.6 3.8 3.0 2.9 4.0 3.6 2.7 4.0 3.5 At the 10% significance level, do the data provide sufficient evidence to conclude that the mean GPA of students at college A differs from the mean GPA of students at college B?

A researcher wishes to compare the incomes of men and women. She selects a random sample of 80 married women. She records the weekly income of each of these women. Using the husbands of these women as the sample of men, she also records the weekly income of each man. Incomes are known to be normally distributed for both populations (women and men). Furthermore, the two population standard deviations are known to be equal. Is it reasonable to use the Mann-Whitney test on these data? 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 resting pulse rate of people who exercise regularly and people who do not exercise regularly. Independent simple random samples of 16 people ages 30-40 who do not exercise regularly and 12 people ages 30-40 who do exercise regularly were selected and the resting pulse rate of each person was measured. The summary statistics are as follows. At the 2.5% significance level, do the data provide sufficient evidence to conclude that the mean resting pulse rate of people who do not exercise regularly is greater than the mean resting pulse rate of people who exercise regularly? Use the critical-value approach.

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 wishes to determine whether the systolic blood pressure of people who follow a vegetarian diet is, on average, lower than the systolic blood pressure of those who follow a nonvegetarian diet. Independent simple random samples of 85 vegetarians and 75 nonvegetarians yielded the following sample statistics: Vegetarians =85 =124.1 =38.7 Nonvegetarians =75 =138.7 =39.2 Use the sample data to test the claim that the mean systolic blood pressure for vegetarians is lower than the mean systolic blood pressure for nonvegetarians. Test the claim using a significance level of 0.01. 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. The two distributions of the variable under consideration have the same shape. How would you decide between the pooled t-test and the Mann-Whitney test?

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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -Students are given scores on both the mathematics section and the verbal section of a standardized test. The data for a random sample of students is given below. Mathematics 347 440 327 456 427 349 377 398 425 Verbal 285 378 243 371 340 271 294 322 385 At the 5% significance level, do the data provide sufficient evidence to conclude that there is a difference between the mean math score and the mean verbal score?

Apply the pooled t-interval procedure to obtain the required confidence interval. You may assume that the assumptionsfor using the procedure are satisfied. -A researcher was interested in comparing the amount of time spent watching television by women and by men. Independent simple random samples of 14 women and 17 men were selected and each Person was asked how many hours he or she had watched television during the previous week. The summary statistics are as follows. Women Men =12.0 =17.3 =4.8 =4.8 =14 =17 Determine a $95 \%$ confidence interval for the difference between the mean weekly television watching times of women and men.

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 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.

Preliminary data analyses indicates that use of a paired t-test is reasonable. Perform the hypothesis test by using eitherthe critical-value approach or the P-value approach as indicated. Assume that the null hypothesis is $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -A coach uses a new technique in training middle distance runners. The times, in seconds, for 8 different athletes to run 800 meters before and after this training are shown below. Athlete A B C D E F G H Before 117.6 110.6 109.5 112 117.6 116.8 111.3 111.8 After 118.2 109.3 107.1 112.8 115.8 116.9 107.7 107.9 At the 5% significance level, do the data provide sufficient evidence that the training helps to improve times for the 800 meters? Use the P-value approach.

Preliminary data analyses indicates that use of a paired t-test is reasonable. Perform the hypothesis test by using eitherthe critical-value approach or the P-value approach as indicated. Assume that the null hypothesis is $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -A coach uses a new technique to train gymnasts. 7 gymnasts were randomly selected and their competition scores were recorded before and after the training. The results are shown below. Subject A B C D E F G Before 9.6 9.4 9.7 9.5 9.4 9.4 9.6 After 9.7 9.6 9.7 9.4 9.5 9.7 9.4 At the 1% significance level, do the data provide sufficient evidence to conclude that the training technique is effective in raising scores? Use the P-value approach.

Summary statistics are given for independent simple random samples from two populations. Preliminary data analysesindicate that the variable under consideration is normally distributed on each population. Decide whether use of thepooled t-test and pooled t-interval procedure is reasonable. Explain your answer. - $\bar { x } _ { 1 } = 143.9 , \mathrm {~s} _ { 1 } = 17.2 , \mathrm { n } _ { 1 } = 7 , \bar { x } _ { 2 } = 212.9 , \mathrm {~s} _ { 2 } = 47.7 , \mathrm { n } _ { 2 } = 17$

Suppose a researcher wants to perform a hypothesis test to compare the mean systolic blood pressure for men with the mean systolic blood pressure for women, based on independent samples. She obtains a random sample of 30 married couples and determines the mean systolic blood pressure for the men, $\bar { x } _ { 1 }$ , and the mean systolic blood pressure for the women, $\bar { x } _ { 2 }$ . She formulates the hypotheses as follows $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ $\mathrm { H } _ { \mathrm { a } } : \mu _ { 1 } \neq \mu _ { 2 }$ and will reject the null hypothesis if $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ , the observed difference between the sample means is too large. She will use a two-means z-test. Identify the flaw in her methodology.

A researcher wishes to compare the incomes of men and women. She selects a random sample of 80 married women. She records the weekly income of each of these women. Using the husbands of these women as the sample of men, she also records the weekly income of each man. She obtains the following summary statistics. Women Men =520 =590 =220 =240 =80 =80 Incomes are known to be normally distributed for both populations (women and men). Furthermore, the two population standard deviations are known to be equal. Is it reasonable to use the pooled t-test on these data? Explain your answer.

In comparing the means of two populations, some methods are based on independent samples and some are based on paired samples. Explain the difference between independent and paired samples. Give an example of each type of sample.

Perform the required hypothesis test by using a Mann-Whitney test. Use a normal approximation for M. Find the meanand standard deviation of M, the value of M and of the test statistic, the critical value(s), and state your conclusion. -Independent simple random samples of 11 female employees and 11 male employees are selected from one company and the weekly salaries of the employees are recorded. The salaries (in dollars)are shown below. Use a significance level of 0.10 to test whether the mean salary for female employees differs from the mean salary for male employees.