We will use the Minitab macro BinoMixP, or function binomixp in R, to nd the posterior distribution of π given an observation y from the binomial(n,π) distribution when we use a mixture prior for π. Suppose our prior experience leads us to believe a beta(7, 13) prior would be appropriate. However we have a nagging suspicion that our experience was under different circumstances, so our prior belief may be quite incorrect and we need a fallback position. We decide to use a mixture prior where g₀(π) is the beta(7, 13) and g₁(π) is the beta(1, 1) distribution, and the prior probability P(I = 0) = :95. Suppose we take a random sample of n = 100 and observe y = 76 successes.

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(a) [Minitab:] Use BinoMixp to find the posterior distribution g(|Y). [R:] Use the function binomixp to find the posterior distribution g(Ty). (b) Find a 95% Bayesian credible interval for T. (c) Test the hypothesis Ho.5 vs. H > .5 at the 5% level of significance.