It is known that a certain disease affects 10% of a population. In a random sample of 50 patients in the disease group who are exposed to a new treatment, we observe that 12 patients were hospitalized in a year. Let m be the population rate that needs hospitalization in a year. Assume m has a Gamma (0.1, 2) prior. Let μ ~ Gamma (0.1,2) and f(x | μ) ~ Poi (50μ). Given that x = 0.24 is an observation of X, find 95% credible internal for m. Obtain a Bayesian credible interval for m. (If X is the number of patients admitted in a year, assume X ~ Poi (50μ), the Poisson approximation of the binomial.) How can we improve on this estimate?