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1. (10 points) A sleeping pill has been developed. To test its effectiveness, 20 subjects were divided in to two groups of 10. The first group (the control) is given a sugar pill. After one month of treatment, the first group was brought back for further evaluation, and the amount of sleep each of the 10 subjects had one given night was recorded. The sample mean for this group was 7.982 hours, and the sample standard deviation was 0.517 hours. The second group was also brought back for further evaluation. The sample mean for the second group was 8.945 hours, and the sample standard deviation was 0.519 hours. Let ux be the mean amount of sleep per night for those taking the sugar pill (or without treatment), and let py be the mean amount of sleep per night for those who take the pill (with treatment). a. Test Ho : px = py versus Ha : px < py. Use O = 0.01. b. Refer to (a). Report the p-value. c. Give a 99% upper confidence interval for "x - Py. d. Refer to (c). Based on your interval in (c), do you reject Ho : Px = Py in favor of Ha : 1x e. What assumptions are necessary in order for this test to be valid? 2. (10 points) A sleeping pill has been developed. To test its effectiveness, 20 subjects were divided in to two groups of 10. The first group (the control) is given a sugar pill. After one month of treatment, the first group was brought back for further evaluation, and the amount of sleep each of the 10 subjects had one given night was recorded. The sample mean for this group was 7.982 hours, and the sample standard deviation was 0.517 hours. The second group was also brought back for further evaluation. The sample mean for the second group was 8.945 hours, and the sample standard deviation was 0.519 hours. Let ux be the mean amount of sleep per night for those taking the sugar pill (or without treatment), and let Py be the mean amount of sleep per night for those who take the pill (with treatment). Let ox and o} be the population variances of those without treatment and those with treatment, respectively. a. Test the hypothesis Ho : ox = of versus Ha : ox # of. Use O = 0.05. b. Find a 95% confidence interval for the ratio 03/0} c. Use your interval in (a) to test Ho : o} = of versus Ha : ox # of. What is your conclusion? d. Explain why we would want to test that the variances are equal before performing a two sample T test as we did in problem 1.

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