Question 2 [3+3+3=9 points]: (Old Qualifying Exam Question) A randomized trial was con- ducted to investigate the relationship between a continuous response y and four treatments A, B, C, and D. The sample size was n = 200, with 50 observations in each of the four treatment groups. Let y be the 200 x 1 vector of response values, ordered so that the first 50 entries are for treatment group A, the next 50 for B, then C, and finally D. The regression model y = XB + e was fit, where X is the 200 x 4 design matrix given by 1 0 0 0 0 1 0 0 X = 0 0 1 0 0 0 0 1 and where each entry is a column vector of length 50 . The estimated regression coefficients were 3 = [37.5, -11.5, 1.0, -27.7], with standard errors 2.75, 3.89, 3.89, 3.89, and residual standard deviation o = 19.45. Also: 0.02 -0.02 -0.02 -0.02 (X'X) = -0.02 0.04 0.02 0.02 -0.02 0.02 0.04 0.02 0.02 0.02 0.02 0.04 (a) Interpret each of the four regression parameters. (b) What is an approximate 95% confidence interval for the mean difference in response between treatment groups B and A (so, the difference MB - MA) ? (c) What is an approximate 95% confidence interval for the mean response in treatment group B