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Exercise 9.6.15 (Caffeine). This question continues with the caffeine data in Exercises 4.4.4 and 6.6.11. Start with the both-sides model Y = x9z' + R, where as before the Y is 2 x 28, the first column being the scores without caffeine, and the second being the scores with caffeine. The x is a 28 x 3 matrix for the three grades, with orthogonal polynomials. The linear vector is (-15 0,, 1;)' and the quadratic vector is (15 -1.810, 1;)'. The z looks at the sum and difference of scores: z = 1 1). (9.116) The goal of this problem is to use BIC to find a good model, choosing among the con- stant, linear and quadratic models for x, and the "overall mean" and "overall mean + difference models" for the scores. (a) For each of the 6 models, find the deviance, number of free parameters, BIC, and estimated probability. (b) Which model has highest probability? (c) What is the chance that the difference effect is in the model? (d) Find the MLE of A for the best model. 3. [7 pts] Exercise 9.6.15. Note: The "6 models" have the following pattern matrices P: overall mean overall mean + difference 0 constant 0 1 linear quadratic use the note and the dataset caffeine from the R package msos to answer exercise 9.6.15 from the book multivariate statistics old school