Let X ~ n(p, 1) and consider the confidence interval Ca(x) = {: min{0, (x - a)} . max{0, (x + a)}}. a.

Let X ~ n(p, 1) and consider the confidence interval
Ca(x) = {μ: min{0, (x - a)} ≤ μ. ≤ max{0, (x + a)}}.
a. For a = 1.645, prove that the coverage probability of Ca{x) is exactly .95 for all μ, with the exception of μ, = 0, where the coverage probability is 1.
b. Now consider the so-called non-in-formative prior π(μ) = 1. Using this prior and again taking a = 1.645, show that the posterior credible probability of Ca(x) is exactly .90 for -1.645 ≤ x ≤ 1.645 and increases to .95 as |x| → ∞.
This type of interval arises in the problem of bioequivalence, where the objective is to decide if two treatments (different formulations of a drug, different delivery systems of a treatment) produce the same effect. The formulation of the problem results in "turning around" the roles of the null and alternative hypotheses (see Exercise 8.47), resulting in some interesting statistics. See Berger and Hsu (1996) for a review of bioequivalence and Brown, Casella, and Hwang (1995) for generalizations of the confidence set.