25 26 27 28 29 30 31 (Requires mathematical statistics.) Showing (4. 8 ). Suppose that n/N > O as n > 00, so that the fpc can be ignored. The central limit theorem tells us that under regularity conditions, 55:\" 5 52 123.5y _ _ >N 0, x , 323,\" RSxSy s; where [.3 denotes convergence in distribution. Show that the limiting distribution of Jar16, 5m) has mean 0 and variance S2 ZBRSISy + 3251. Show that if we consider approximations to the MSE 1n (4.8) and (4.17) to be accurate, then the variance of y, from ratio estimation is at least as large as the variance 015mg from regression estimation. HINT: Look at V(yr) V(y,eg) using the formulas In (4.8) and (4.17), and Show that the difference is non-negative. Prove (4.16). (Requires probability.) Prove (4.18). (Requires probability.) Let d,- = y; [jg + Bl(x,v fun. Show that for regression estimation, ln/N N signEU): \"$3 {:1 Nl 1'1;ng _yU] g _ As in Exercise 21, show that (E[reg jzu])2 is small compared to MSEIi-cg], when n is large. (Requires probability.) Consider the combined ratio estimator of the population total, in\