Let X and Y be the number of successes in two sets of n binomial trials with probabilities p1 and p2 of success.

(i) The most powerful test of the hypothesis H : p2 ≤ p1 against an alternative

(p 1, p 2) with p 1 < p 2 and p 1 + p 2 = 1 at level α < 1 2 rejects when Y − X > C and with probability γ when Y − X = C.

(ii) This test is not UMP against the alternatives p1 < p2.

[(i): Take the distribution  assigning probability 1 to the point p1 = p2 = 1 2 as an a priori distribution over H. The most powerful test against (p 1, p 2) is then the one proposed above. To see that  is least favorable, consider the probability of rejection β(p1, p2) for p1 = p2 = p. By symmetry this is given by 2β(p, p) = P{|Y − X| > C} + γP{|Y − X| = C}.

Let Xi be 1 or 0 as the ith trial in the first series is a success or failure, and let Y1, be defined analogously with respect to the second series. Then Y − X = n i−1(Yi −

Xi), and the fact that 2β(p, p) attains its maximum for p = 1 2 can be proved by induction over n.

(ii): Since β(p, p) < α for p = 1, the power β(p1, p2) is < α for alternatives p1 < p2 sufficiently close to the line p1 = p2. That the test is not UMP now follows from a comparison with φ(x, y) ≡ α.]