Problem 2 (10 pts) Matrix results for multiple linear regression This problem can be hand written or you can format your work in LaTeX. If handwritten, please scan to pdf les and include with this document. Clearly label the solutions (e.g. Prob- lem 2(a) etc), and indicate in this document where those solutions are located in the attached les. Alternatively, you can save each part's solution as an image le and insert those in this document using the method described under the assignment link. Suppose we have a designed experiment with 4 observations. We t a linear model of the form y = X + e with yl 1 1 61 = 92 X = 1 1 = 30 e 62 y 293 ' 1 1 3 (51 ' 63 294 1 1 e4 We make the working assumption that the errors 6,- are uncorrelated random variables with E(e.,) = O and Var(e,) = 02. (a) (2 pts) Calculate the matrix XTX. Answer: (b) (2 pts) Show how the components of the ordinary least squares estimate B depend on y1, y2, y3, 94. You should get simple, explicit formulas for 60 and 81 Answer: (c) (2 pts) Show that the ordinary least squares estimators 30 and {31 are uncorrelated with each other for this design, in other words, show Cov(;30, ,81) = 0. Answer: ((1) (2 pts) Compute the hat matrix, H = X(XtX)1Xt. Answer: (e) (2 pts) Compute explicit simplied formulas for the tted values Q1, 1122, 333, a