a. Show that the covariance matrix 19 30 2 127 30 57 5 23 E = 2 5 38 47 12 23 47 68. for the p = 4 random variables X1, X2, X3, and X4, can be generated by the m = 2 factor model X1 = 4F1 + F2 + $1 X2 = 7F1 + 2F2 + 82 X3 = -F1 + 6F2 + 83 XA = F1 + 8F2 + 84 where Cov(F;) = 1, Cov(E, F;) = 0, j = 1,2, and 0 0 01 = Cov(8) = 4 0 0 OOON 0 0 1 O O That is, write > in the form _ = LL' + , where L, is a matrix of the loadings. (8 pts) b. Calculate the communalities h?, i = 1, 2, 3 and interpret these quantities. (5 pts) c. Calculate Corr(X;, F; ) for i = 1, 2, 3, 4 and j = 1,2. What variable(s) might carry the greatest weight in "naming" the common factors? Why? (7 pts)