STAT 235 Final, Spring 2004

ERICAN LiIBKARY OF VEIRUT The American University of Beirut STAT 235 FINAL EXAM Time = 1 hour and 30 minutes June 9, 2004 1. A regression analysis using an intercept and one independent variable gave the following: Y = 1.841246 + 0.10934.X, and 23) = [ 0.1240363 —0.002627 | (9) = | _9 992627 0.000099 MSE = 1.6360 (a) Construct a 95% confidence interval estimate of £y. (b) compute Y when X = 4 and its variance 2. (Simple linear regression through the origin) Let % = BX, + & with the usual assumptions of a linear regression model. (a) Show that f, = XVY/ 0X? (b) Show that s? = [0 Y¥?—8, 0 X,¥]/(n—1) is an unbiased estimator for a”. . For a simple linear regression model with intercept, Y; = Bo + BX; + ¢; with the usual assumptions. One may parametrize the model by considering the following model: Y, = a+ 3(X%;-X) +6: i=] So \ ; (a) Show that the least squares of B=8, anda=Y. (b) Show that Cov(a, 8) = 0. 4. Consider the following two linear models: where X and W are nx p matrices with linearly independent columns. Suppose further that VW = X.8, where B isa pxp non-singular matrix. (a) Show that G = BA. (b) Show that both models give the same predicted vector ¥ and the same residual vector €.