Time to talk: The above t-theorem is useful for testing hyps about specific combinations of the mu_i. Note that the t is computed from sample means. But it would be even more useful if we could come up with a test (for hyps about specific combinations of mu_i that relied on SS values. After all, the original "omnibus" F-test was based on SS values: F = (SS_Tr / df ) / (SSE / df)

There is another (more practical) reason to come up with a test of means that involves SS. Suppose we have done the omnibus ANOVA F-test and have found a significant result. That implies that SS_Tr is larger than SSE. But we have no idea "why." Remember that the above t-test is for testing specific combinations of mu_i, but a large value of SS_Tr tells us that *some* specific combinations of the mu_i are nonzero; it doesn't tell us *which* specific combinations of mu_i (i.e., which contrasts) are nonzero.

So, let's come-up with a test of contrasts (not an omnibus test) that can allow us to write SS_Tr in terms of an SS from different contrasts. In other words, let's decompose SS_Tr into different components that show the contribution of specific contrasts (i.e., combinations of mu_i) to the whole SS_Tr.

hw_lect10_1:For the data shown in 3.22 (8th edition; or 3.20 in 7th edition), a) Do the means vary across the levels of y? Report a p-value. (By hand, ie not using lm()).b) Find SS_treatement, SS_E, F, and the corresponding p-value, but by R. Confirm that these are equal to those in part a.c) Make qqplots (of either the response or the residuals) for each of the 3 levels of X, and interpret the results. Use the "by hand" qq-plot code we developed so that you can superimpose multiple qqplots onto a single panel.d) Compute the p-value for testing whether mu1 and mu3 are different. State your conclusion. By hand. Note that even though you are testing only 2 means, the estimate of sigma^2 in the t statistic is the mse of the full model with the treatment factor having (a) levels.e) Suppose we are interested in whether mu2 = (mu1+mu3)/2. Construct an orthogonal contrast, and confirm that their contrast sum-of-squares add-up to SS_treatment found in part b. By hand.f) Suppose we are interested in whether mu2 = (mu1+mu3)/2. Construct a non-orthogonal contrast, and confirm that their contrast sum-of-squares does not add-up to SS_treatment found in part b. By hand.g) The SS_C in part e each have df = 1. Perform a t-test on each testing whether the contrast is non-zero. By hand.