Exam 10: Inferences for Two Population Means

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 investigating how much people read in their spare time and, in particular, whether people who own a personal computer read less in their spare time than those who do not own a computer. She selected independent simple random samples of 7 computer owners and 9 noncomputer owners, and she asked each person how many hours he or she had spent reading non-work-related books during the previous week. The results are as follows. Computer Owners Noncomputer Owners 1.7 1.8 0 6.6 2.6 0.3 0.9 0 6.2 0.2 0 2.3 0.5 8.8 2.2 1.4 At the 2.5% significance level, do the data provide sufficient evidence to conclude that the mean weekly reading time for computer owners is lower than the mean weekly reading time for noncomputer owners?

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.The standard deviation of all possible differences between the two sample means equals the square root of the difference of the population variances each divided by the corresponding sample size.

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 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: Sample 1 (women) Sample 2 (men) =12.3 =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 P-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 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 do not follow a vegetarian 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 a significance level of 0.01 to test the claim that the mean systolic blood pressure for vegetarians is lower than the mean systolic blood pressure for nonvegetarians. Use the P-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 resting pulse rates 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 exercise regularly were selected, and the resting pulse rate (in beats per minute)of each person was measured. The summary statistics are as follows. Do Not Exercise Do Exercise =73.5 =68.5 =10.9 =8.2 =16 =12 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.

Suppose that you want to perform a hypothesis test to compare the means of two populations using independent simple random samples. Suppose that preliminary data analyses of the samples Suggest that the distribution of the variable under consideration is skewed to the left for one Population and skewed to the right for the other. The sample sizes are 50 and 60. Should you use The pooled t-test, the nonpooled t-test, the Mann-Whitney test, or none of these tests?

Determine the null and alternative hypotheses for the proposed hypothesis test. -A pharmaceutical company wants to determine whether its new antianxiety medication has any effect on resting pulse rate. They will use a paired sample to determine whether the mean resting Pulse rate for adults not taking the antianxiety medication differs from the mean resting pulse rate For adults who are taking the antianxiety medication.

Provide an appropriate response. -Manufacturer X has designed athletic footwear which it hopes will improve the performance of athletes running the 100-meter sprint. It wishes to perform a hypothesis test to compare the mean times of athletes in the 100-meter sprint running with Manufacturer X's footwear and with other shoes. Do you think that a paired t-test or a pooled t-test would be more appropriate in this situation? Why? If you think that a paired test is preferable, explain what would constitute a pair.

Provide an appropriate response. -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 paired-difference variable for the proposed Hypothesis test.

A teacher is interested in performing a hypothesis test to compare the mean math score of the girls and the mean math score of the boys. She randomly selects 10 girls from the class and then randomly selects 10 boys. She arranges the girls' names alphabetically and uses this list to assign each girl a number between 1 and 10. She does the same thing for the boys. She pairs each girl with the boy having the same number. She then performs a paired t-test. Do you think that in this situation a paired t-test is appropriate? Explain your answer.

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 9.7 9.7 9.4 9.6 9.7 9.7 9.4 After 9.8 9.9 9.4 9.5 9.8 10 9.2 At the 1% significance level, do the data provide sufficient evidence to conclude that the training technique is effective in raising scores? Use the critical-value approach.

Suppose that you want to perform a hypothesis test based on a simple random paired sample to compare the means of two populations. Two procedures are available, namely, the paired t-test and the paired Wilcoxon signed-rank test. Give circumstances for which the Wilcoxon signed-rank test would be the more appropriate procedure. Be sure to comment on the size of the sample and the distribution of the paired-difference variable.

Suppose that you want to perform a hypothesis test based on independent simple random samples to compare the means of two populations. Normal probability plots for the two samples are linear and therefore suggest that the variable under consideration is normally distributed on each population. Is it then reasonable to proceed with a pooled t-test? If not, what additional information is required and what methods might be used to obtain this information?

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. -A person who commutes to work is choosing between two different routes. He tries the first route 11 times and the second route 12 times and records the time of each trip. The results (in minutes)are shown below. Use a significance level of 0.01 to test whether the mean time for route 1 differs from the mean time for route 2.

Use the Mann-Whitney Table to find the critical values. For a left-tailed or two-tailed test, you will also need the relation $\mathrm { M } _ { 1 - \mathrm { A } } = \mathrm { n } _ { \mathbf { 1 } } \left( \mathrm { n } _ { 1 } + \mathrm { n } _ { 2 } + 1 \right) - \mathrm { M } _ { \mathrm { A } } \text {. }$ - $\mathrm { n } _ { 1 } = 8 , \mathrm { n } _ { 2 } = 4$ , significance level $= 0.01$ , two-tailed test

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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -The table below shows the weights, in pounds, of seven subjects before and after following a particular diet for two months. Subject A B C D E F G Before 196 187 177 161 197 168 154 After 189 178 175 166 183 170 142 At the 1% significance level, do the data provide sufficient evidence to conclude that the diet is effective in reducing weight? 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 71 78 73 72 75 After 75 87 71 75 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 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 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 74 83 75 88 84 63 93 84 91 77 After 73 77 70 77 74 67 95 83 84 75 At the 5% significance level, do the data provide sufficient evidence to conclude that that the mean score after the course differs from the mean score before the course? 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 $\mathrm { H } _ { 0 } : \mu _ { 1 } = \mu _ { 2 }$ -The table below shows the weights, in pounds, of seven subjects before and after following a particular diet for two months. Subject A B C D E F G Before 164 199 196 191 193 184 168 After 157 190 194 196 179 186 156 At the 1% significance level, do the data provide sufficient evidence to conclude that the diet is effective in reducing weight? Use the P-value approach.

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. -A teacher uses two different CAI programs to remediate students. Independent simple random samples of 11 students from program I and 14 students from program II are selected. Results for each group on a standardized test are listed in the table below. At the 0.05 level of significance, do the data provide sufficient evidence to conclude that the mean score for students going through program I differs from the mean score for students going through program II? Program I Program II 60756163 66896877 86696470 84808187 728259 78739193 9495