Paired Comparisons of Means. A family counseling service offers a 1-day program of training in meal planning which they claim is effective in reducing cholesterol levels of individuals completing the course. In order to assess the effectiveness of the program, a consumer advocacy group randomly samples the cholesterol levels of 50 program participants both one day before and one month after the course is taken. They then summarize the pairs of observations on the individuals by reporting the sample mean and sample standard deviation of the differences between the before and after observations of the 50 program participants. Their finding were \(\bar{d}=-11.73\) and \(s=3.89\).

(a) Assuming that the pairs of observations are iid outcomes from some bivariate normal population distribution with before and after means \(\mu_{\mathrm{b}}\) and \(\mu_{\mathrm{a}}\),

define the appropriate likelihood function for the mean \(\mu=\mu_{\mathrm{a}}-\mu_{\mathrm{b}}\) and variance \(\sigma^{2}\) of the population distribution of differences in pairs of cholesterol measurements. Use this likelihood function to define a GLR size .05 test of the hypothesis that the meal planning program has no effect, i.e. a test for \(H_{0}\) : \(\mu=0\) versus \(H_{a}\) : not \(H_{0}\). Test the hypothesis.

(b) Can you define an LM test of the null hypothesis in (a)? Can you test the hypothesis with the information available? If so, perform the test-if not, what other information would you need?

(c) Describe how you might test the hypothesis that the observations on paired differences are from a normal population distribution. What information would you need to test the hypothesis?

(d) Suppose that normality of the observations is not assumed. Can you define another test of the effectiveness of the meal planning program? If so, use the test to assess the effectiveness of the meal planning program. Discuss any differences that are required in the interpretation of the outcome of this test compared to the test in

(a) that you could perform if the observations were normally distributed.