Exam 10: Inferences for Two Population Means

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 simple 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 critical-value approach. $\text { (Note: } \left. \bar { x } _ { 1 } = 3.1125 , \bar { x } _ { 2 } = 3.4385 , s _ { 1 } = 0.4357 , s _ { 2 } = 0.5485 . \right)$

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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -The systolic blood pressure readings of ten subjects before and after following a particular diet for a month are shown in the following table. Subject Before After A 175 162 B 192 190 C 167 170 D 180 180 E 161 153 F 203 197 G 185 191 H 176 174 I 204 192 J 146 150 At the 1% level of significance, do the data provide sufficient evidence to conclude that the diet has an effect on systolic blood pressure?

Provide an appropriate response. -A statistician performs a paired Wilcoxon signed-rank test to compare the means of two populations using a paired sample. Suppose that the sum of the positive ranks is much larger than the sum of the negative ranks. Explain in your own words why this suggests that $\mu _ { 1 } > \mu _ { 2 } .$

Provide an appropriate response. -Suppose that $x$ is a normally distributed variable on each of two populations. Independent samples of sizes $n _ { 1 }$ and $n _ { 2 }$ , respectively, are selected from the two populations. Explain the difference between $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ and $\mu _ { 1 } - \mu _ { 2 }$ .

Provide an appropriate response. -Suppose a 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. Discuss the basic strategy for performing this hypothesis test, based on independent samples.

Provide an appropriate response. -Suppose that you want to perform a hypothesis test to compare the means of two populations using a paired sample. Preliminary data analyses of the sample of paired differences suggest that The distribution of the paired-difference variable is uniform . Given that the sample size is 16, Should you use a paired t-test, a paired Wilcoxon signed-rank test, or neither of these tests?

A statistician performs a Mann-Whitney test to compare the means of two populations using independent simple random samples. Suppose that the sum of the ranks for sample 1 is much larger than the sum of the ranks for sample 2 . Explain in your own words what this suggests about the relationship between $\mu _ { 1 }$ and $\mu _ { 2 }$ .

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

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 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 =11.9 =17.1 =4.1 =4.8 =14 =17 At the 5% significance level, do the data provide sufficient evidence to conclude that the mean time spent watching television by women is less than the mean time spent watching television by men? Use the critical-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 and a boxplot for the paired differences are given below.

Solve the problem. -A variable of two populations has a mean of $7.7$ and a standard deviation of $4.5$ for one of the populations and a mean of $8.6$ and a standard deviation of $3.3$ for the other population. For independent samples of sizes 4 and 5, respectively, find the standard deviation of $\bar { x } _ { 1 } - \bar { x } _ { 2 }$ . Round your answer to the nearest hundredth if necessary.

Summary statistics are given for independent simple random samples from two populations. Use the pooled t-intervalprocedure to obtain the specified confidence interval. - $\overline { \mathrm { x } } _ { 1 } = 72.9 , \mathrm {~s} _ { 1 } = 10.4 , \mathrm { n } _ { 1 } = 16 , \overline { \mathrm { x } } _ { 2 } = 68.0 , \mathrm {~s} _ { 2 } = 8.3 , \mathrm { n } _ { 2 } = 12$ Determine a $90 \%$ confidence interval.

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

Use the paired t-interval procedure to obtain the required confidence interval. You may assume that the conditions forusing the procedure are satisfied. -A test of abstract reasoning is given to a random sample of students before and after completing a formal logic course. The results are shown below. Before After 74 68 81 86 85 83 67 74 92 95 71 66 61 67 78 81 64 71 80 88 Determine a 95% confidence interval for the difference between the mean score before completing The course and the mean score after completing the course.

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 pairs for the proposed hypothesis test.

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. 997665 0 67789 420 1 0025 0 2 2 7 3 0 0 4 3

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 GPAs of students at two different colleges. Independent simple random samples of 8 students from college A and 13 students from college B yielded the following GPAs. College A College B 3.7 3.82.8 3.2 3.24.0 3.0 3.03.6 2.5 3.92.6 2.7 3.84.0 3.6 2.53.6 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 critical-value approach. $\left( \text { Note: } \bar { x } _ { 1 } = 3.1125 , \bar { x } _ { 2 } = 3.4385 , s _ { 1 } = 0.4357 , s _ { 2 } = 0.5485 . \right)$

The pooled t-test requires the assumption that the two population standard deviations be equal. What methods are available for checking this assumption? Which of these methods is not recommended and why is it not recommended?

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 one of the populations and that on the other population it is skewed to the right. Both 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.

Suppose x is a variable on each of two populations whose members can be paired and that a paired difference is the difference between the values of the variable x on the members of a pair. The mean of the paired differences equals the difference between the two population means.