Hocking (1985) describes an experiment for comparing five fluids that are supposed to prevent the buildup of lactic acid in long-distance runners. For reasons unrelated to the fluids, there were an unequal number of runners assigned to the treatments. The sample means, sample variances, and cell frequencies are shown in the following table: Summary statistics for Exercise 10.14 Fluid 123 4 5 33.3 32.6 30.9 29.0 26.1 13.1 14.2 12.2 13.9 14.2 7 10 8 6 10

a. Complete the analysis of variance table for testing the hypothesis of equal means.

b. Fluid #1 was plain water. Fluids #2 and #3 were a common sport drink, say, A, at low and high concentrations. Fluids #4 and #5 were another sport drink, say, B, at low and high concentrations. This description suggests the following contrasts: C: My + Hz + M 2 Cs: 1-3 Develop the sums of squares for testing these hypotheses and test for significance using the separate-t and Scheffe criteria. State the conclusions verbally. Note that the sums of squares for these contrasts do not add to the treatment sum of squares. Verify that this is not an orthogonal set of contrasts.c. Replace C, and C by C: N- C Test the associated hypotheses as in part

b. Compare the results with those for C, and C. Verify that C, C, C, and Cs are orthogonal contrasts.

d. Two other contrasts of interest are C7: Hz + H 2 Cs: 2 --+ the.. Note that Cs can be viewed as an have been used in place of, or in The use of more than p-1-4 Give a verbal description of these contrasts. interaction contrast. These contrasts might addition to, Ca and C in the initial study. contrasts suggests that the separate- criterion is not appropriate. Verify that Cs. C, C, and Cg form an orthogonal set.

e. Test all pairwise differences using the separate-t and Scheffe criteria. Note that the multiple comparison philosophy is not appropriate here because of the unequal cell frequencies but that we might apply the criterion to each of the pairs. Do so using the Neuman-Keuls criterion. Section 10.4