Paired Comparisons of Means
A family counseling service offers a one-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 sam ples the cholesterol levels of 50 program participants both one day before and one month af ter the course is taken. It then summarizes the pairs of observations on the individuals by reporting the sample mean and sam ple standard deviation of the dif ferences between the before and af ter observations of the 50 program partici pants. Their finding were d = -11.73 and s = 3.89.
a. Assuming that the pairs of observations are iid out comes from some bivariate normal population dis tribution with before and after means /.,lb and µ0, define the appropriate likelihood function for the mean µ = /.,la - /.,lb and variance a2 of the population distribution of differences in pairs of choles terol measurements. Use this likelihood function to define a GLR size .OS test of the hypothesis that the meal planning program has no effect, i.e., a test for Ho : µ = 0 versus Ha: not H0• Test the hypothesis.
b. Can you define an LM test of the null hypothesis in (a)? Can you test the hypothesis with the informa tion 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 effec tiveness 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 per form if the observations were normally distributed.