This question develops hypothesis tests for the difference between two population proportions. Let X ~ Binomial(n, p₁) and Y ~ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are: Ho P1 P2 HA: P1 P2. Please answer the following questions. (a) Find the generalized likelihood ratio statistic A for testing Ho vs. HA based on the data X and Y. == (b) Let X/n and p2 = Y/m. We now develop a test statistic based on p₁ - 2: P2(1-P2) m i. Show that E(p1-2) = P1 - P2 and Var(1-2) = P₁(1-p₁) + n ii. Under Ho, P₁ = P2 = p for some common value p. Show that : estimator of p under Ho. = X+Y is an unbiased m+n iii. Argue that, for n and m both large (say n ≥ 30 and m≥ 30) the statistic Ĥ1-Ĥ2 √√ô(1 − p) [½ + ½] has approximately a N(0,1) distribution under Ho. (c) Recent incidents of food contamination have caused great concern among consumers. The article "How Safe Is That Chicken?" (Consumer Reports, Jan 2010: 19-23) reported that 35 of 80 randomly selected Perdue brand broilers tested positively for either campylobacter or salmonella (or both), the leading bacterial causes of food-borne disease, whereas 66 of 80 Tyson brand broilers tested positive. Does it appear that the true proportion of non- contaminated Perdue broilers differs from that for the Tyson brand? Obtain a p-value from both the generalized likelihood ratio test (part (a)) and the approximate Z-test (developed in part (b)). Do the tests agree?