(i) Let X have binomial distribution b(p, n), and consider testing H : p = p0 at level α against the alternatives K : p/q ≤ 1 2 p0/q0 or ≥ 2p0/q0.

For α = .05 determine the smallest sample size for which there exists a test with power ≥ .8 against K if p0 = .1, .2, .3, .4, .5.

(ii) Let X1, …, Xn be independently distributed as N(ξ , σ2). For testing σ = 1 at level α = .05, determine the smallest sample size for which there exists a test with power ≥ .9 against the alternatives σ2 ≤ 1 2 and σ2 ≥ 2.

[See Problem 4.5.]