Question #4 (both 454 & 854 - 10 points): We discussed in class the optimal/Neyman allocation for the n's over the H strata. This allocation minimizes VSTSRS (A) under the obvious constraint that n = n. However, it doesn't take into account another very important constraint that is cost. To this end, let ch be the cost of sampling one unit in stratum Uh; thus, the total cost is H &t= cann. h=1 (a) Show that, under STSRS, the n's that minimizes VSTSRS (A) under the constraint that n = n and for a fixed cost ct, with c = Chnh, are given by nh = Hint: use the Cauchy-Schwarz inequality. H Ct Wh Sun ch (Wh Sun ch) Ch h=1 (1) (b) Show that (1) simplifies to n Wh Sun nh.o H (Wh Sun) h=1 (i.e., the formula we saw in class), when ch = cVh's (i.e., when the cost is the same for all units of the population).