Suppose that x has a binomial distribution B(n, ) of index n and parameter 0, and that

Suppose that x has a binomial distribution B(n, θ) of index n and parameter 0, and that it is desired to test H₀ : θ = θ₀ against the alternative hypothesis H₁: θ ≠ θ₀:

(a) Find lower bounds on the posterior probability of H₀ and on the Bayes factor for H₀ versus H₁, bounds which are valid for any P₁ (θ).

(b) If n = 20, θ₀ = ½ and x = 15 is observed, calculate the (two-tailed) P-value and the lower bound on the posterior probability when the prior probability π₀ of the null hypothesis is ½.