Probability as statistic:

Let X1, . .., Xn be i.i.d. samples from Bernoulli(p). We want to find a uniformly most powerful test for the null hypothesis Ho : p = po against the one-sided alternative hypothesis H1 : p = p1 > po. (i) Compute the ratio L(po) /L(p1) of joint likelihood functions, where p1 > po. Show that L(po) n sk Po(1 - p1) 2i=1ti [1- po S k L(p1) pi(1 - po) 1- p1 xi2C := logk - nlog[(1 - po)/(1- p1)] i= 1 log[po(1 - p1)/ pi(1 - po)] (ii) Using the Neyman-Pearson lemma, conclude that a best critical region C for testing Ho : p = Po against H1 : p = p1 > Po is given by C= (x1, . .. , Xin) Exizcy. i=1 (iii) Let n = 30. Use CLT and (ii) and obtain a best critical region C for testing Ho : p = 0.2 against H1 : p = 0.5 at significance level a = 0.05