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

## Question:

Suppose that x has a binomial distribution B(n, θ) of index n and parameter 0, and that it is desired to test H_{0} : θ = θ_{0} against the alternative hypothesis H_{1}: θ ≠ θ_{0}:

(a) Find lower bounds on the posterior probability of H_{0} and on the Bayes factor for H_{0} versus H_{1}, bounds which are valid for any P_{1} (θ).

(b) If n = 20, θ_{0} = ½ and x = 15 is observed, calculate the (two-tailed) P-value and the lower bound on the posterior probability when the prior probability π_{0} of the null hypothesis is ½.

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