Exam 10: Inferences for Two Population Means

Suppose that you want to perform a hypothesis test based on independent simple random samples to compare the means of two populations. Assume that the variable under consideration is uniformly distributed on both of the populations and that the two population standard deviations are equal. Both sample sizes are small. 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 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 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 =12.6 =14.1 =3.9 =5.2 =14 =17 At the 5% significance level, do the data provide sufficient evidence to conclude that the mean amount of time spent watching television by women is less than the mean amount of time spent watching television by men? Use the critical-value approach.

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 GPAs of students at two different colleges. Independent random samples of 8 students from college A and 13 students from college B yielded the following GPAs: College A College B 3.7 3.8 2.8 3.2 3.2 4.0 3.0 3.0 3.6 2.5 3.9 2.6 2.7 3.8 4.0 3.6 2.5 3.6 2.8 3.9 3.4 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? Use the P-value approach. $\text { (Note: } \left. \bar { x } _ { 1 } = 3.1125 , \bar { x } _ { 2 } = 3.4385 , \mathrm {~s} _ { 1 } = 0.4357 , \mathrm {~s} _ { 2 } = 0.5485 . \right)$

Provide an appropriate response. -Suppose that you want to perform a hypothesis test based on independent simple random samples to compare the means of two populations. Several procedures are available, namely, the pooled t-test, the nonpooled t-test, and the Mann-Whitney test. Give circumstances for which the nonpooled t-test would be the most appropriate procedure. Be sure to comment on the sizes of the samples, the distributions of the variable under consideration on the populations, and the relative standard deviations of the distributions.

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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -Ten different families are tested for the number of gallons of water a day they use before and after viewing a conservation video. The results are shown below. Before 33 33 38 33 35 35 40 40 40 31 After 34 28 25 28 35 33 31 28 35 33 At the 5% significance level, do the data provide sufficient evidence to conclude that the mean amount of water use after the viewing differs from the mean amount of water use before the viewing? Use the critical-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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -Five students took a math test before and after tutoring. Their scores were as follows. Subject A B C D E Before 66 73 71 66 75 After 70 82 69 69 87 At the 1% significance level, do the data provide sufficient evidence to conclude that the mean score before tutoring differs from the mean score after tutoring? Use the critical-value approach.

A researcher wants to compare the mean systolic blood pressures for adults ages 20-29 and adults ages 30-39. Identify the variable under consideration and the two populations. Suppose the researcher wants to perform a hypothesis test to decide whether the mean systolic blood pressure for adults aged 20-29 is lower than the mean systolic blood pressure for adults aged 30-39. State the null and alternative hypotheses for the hypothesis test.

Suppose that you want to perform a hypothesis test based on independent simple random samples to compare the means of two populations. Further suppose that the variable under consideration is normally distributed on each of the two populations and that the population standard deviations are unknown. If the sample standard deviations are 3.8 and 6.1, respectively, and the sample sizes are 25 and 60, respectively, would you use the pooled or the nonpooled t-test? 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 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 critical-value approach. $\text { (Note: } \left. \bar { x } _ { 1 } = 64.744 , \bar { x } _ { 2 } = 62.556 , s _ { 1 } = 2.192 , s _ { 2 } = 2.697 . \right)$

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 109.7 112.3 109.5 118.3 111.1 120 117.1 114.6 After 110.3 111 107.1 119.1 109.3 120.1 113.5 110.7 At the 5% significance level, do the data provide sufficient evidence that the training helps to improve times for the 800 meters? Use the critical-value approach.

Suppose that x is a normally distributed variable on each of two populations. Independent samples of sizes n₁ and n₂, respectively, are selected from the two populations. The mean of all possible Differences between the two sample means is equal to which of the following?

Summary statistics are given for independent simple random samples from two populations. Use the pooled t-intervalprocedure to obtain the specified confidence interval. - $\overline { \mathrm { x } } _ { 1 } = 71.6 , \mathrm {~s} _ { 1 } = 3.9 , \mathrm { n } _ { 1 } = 11 , \overline { \mathrm { x } } _ { 2 } = 67.9 , \mathrm {~s} _ { 2 } = 3.4 , \mathrm { n } _ { 2 } = 9$ Determine a $99 \%$ confidence interval.

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 weight losses over a two-month period for people on two different diets. Independent simple random samples of 7 people on Diet A and 8 people on Diet B yielded the following weight losses (in pounds). Diet A Diet B 7 3 2 8 14 0 5 1 13 11 21 3 16 6 4 At the 10% significance level, do the data provide sufficient evidence to conclude that the mean weight loss for people on diet A differs from the mean weight loss for people on diet B?

Suppose that you want to perform a hypothesis test to compare the means of two populations using paired samples. How would you decide between the paired t-test and the paired Wilcoxon signed-rank test?

Suppose that you want to perform a hypothesis test based on independent simple random samples to compare the means of two populations. Several procedures are available, namely, the pooled t-test, the nonpooled t-test, and the Mann-Whitney test. Give circumstances for which the Mann-Whitney test would be the most appropriate procedure. Be sure to comment on the sizes of the samples, the distributions of the variable under consideration on the populations, and the relative shapes of the distributions.

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 wished to compare the drying times of two different types of paint. Independent simple 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 hours, were recorded. The summary statistics are as follows. \ Type A Type B =74.7 =64.4 =4.5 =5.1 =11 =9 At the 1% significance level, 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? 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 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 970 743 750 605 660 1640 At the $5 \%$ significance level, do the data provide sufficient evidence to conclude that at this particular company the mean salary of female employees is less than the mean salary of male employees? Use the critical-value approach. (Note: $\bar { x } _ { 1 } = 705.375 , \bar { x } _ { 2 } = 817.067$ , s1 = $183.855 , \mathrm {~s} _ { 2 } = 330.146$ .)

Which of the following statements regarding the equal-standard-deviations assumption of the pooled t-test is the most accurate?

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. -Test whether the mean grade for students from college A differs from the mean grade for students from College B. Independent simple random samples of 11 students from College A and 11 students from College B are selected. Grades of the students are shown in the table. Use a 0.05 level of significance. College A 3.2 4.0 2.4 2.6 2.0 1.8 1.3 0.0 0.5 1.4 2.9 College B 2.4 1.9 0.3 0.8 2.8 3.0 3.1 3.1 3.1 3.5 3.5

Suppose that you want to perform a hypothesis test based on independent simple random samples to compare the means of two populations. Assume that the variable under consideration is skewed to the right on both populations , that the two population standard deviations are equal, and that the sample sizes are large. 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.