16.3 Exercise 2

A small random sample of size $n = 12$ is taken to determine the proportion of the student body which favors a proposal to expand the student Honor Council by adding two additional members “at large.” Prior information indicates that this proportion is about 0.6 = 3/5. From a Bayesian point of view, the population proportion is taken to be the value of a random variable H. It seems reasonable to assume a prior distribution $H \sim$ beta $\left(4 , 3\right)$, giving a maximum of the density at $\left(4 - 1\right) / \left(4 + 3 - 2\right) = 3 / 5$. Seven of the twelve interviewed favor the proposition. What is the best mean-square estimate of the proportion, given this result? What is the conditional distribution of H, given this result?