The Bayesian model updating strategy used for the HIV testing example discussed in Section 11.6 can also be used to update the posterior estimate of the proportion of brown or orange M&M’s. In Section 11.8, we found that the posterior distribution for p(using the author’s data) was beta(24, 33), assuming a uniform prior on [0, 1] and a binomial likelihood of observing x = 23 brown or orange M& M’s in a sample of n = 55 candies. The posterior estimate was p* = 0.421. Now suppose we take another sample of 51 M& M’s and find 19 brown or orange candies.
a. Treat the posterior distribution as the new prior distribution for π— that is, the prior for π is beta(24, 33). If 19 brown or orange candies are observed in 51 M&M’s, find the updated posterior distribution for π.
b. Find the revised posterior estimate for π. How does this value compare to the first posterior estimate p* = 0.421?
c. The 95% Bayesian credible interval for p found in Section 11.8 was (0.297, 0.550). Using the updated posterior distribution, construct a new 95% credible interval for π. What happened to the width of the interval when the new data were incorporated?