50. (i) For testing Ho : (J = 0 against HI : (J = (JI when X is N((J, 1), given any o < a < 1 and any 0 < 'IT < 1 (in the notation of the preceding problem), there exists (JI and x such that

(a) Ho is rejected when X = x but

(b) P(Holx) is arbitrarily close to 1. (ii) The paradox of part (i) is due to the fact that a is held constant while the power against (JI is permitted to get arbitrarily close to 1. The paradox disappears if a is determined so that the probabilities of type I and type II error are equal [but see Berger and Sellke (1984)]. [For a discussion of such paradoxes, see Lindley (1957), Bartlett (1957) and Schafer (1982).]