We can take the approach to selecting a prior distribution for a population proportion p proposed in Exercise 11 one step further. Recall that the variance of a beta random variable, Y, is given by Var(Y) = αβ / (α + β)2(α + β + 1)
If we plan to use a beta prior for π and we have a prior estimate for π, E(π), as well as a prior variance for π, Var(π), then we can find the parameter values for the beta prior by solving two equations for two unknowns.
a. Consider the M& M’s activities. Suppose we believe that the prior mean for the proportion of brown or orange M& M’s is E(π) = 0.33 and we are fairly certain that the variability in the prior distribution of p is quite small, say 0.0001. Using the expressions for the mean and variance for a beta distribution, compute the parameter values for α and β. Round the values to the nearest whole numbers.
b. Use the values of a and b that you found in Part A and your M& M’s data or the MyMMs data to compute the posterior estimate of π.
c. Construct a 90% Bayesian credible interval for π, given that you have observed your data.