Question

3. A cancer laboratory is estimating the rate of tumorigenesis in two strains of mice, A and B. They have tumor counts for 10 mice in strain A and 13 mice in strain B. Type A mice have been well studied, and information from other laboratories suggests that type A mice have tumor counts that are approximately Poisson-distributed with mean 12. Tumor count rates for type B mice are unknown, but type B mice are related to type A mice. The observed tumor counts for the two populations are (12,9, 12, 14, 13, 13, 15, 8, 15,6); (11, 11, 10, 8, 8, 8, 7, 10, 6, 8, 8, 9, 7) yA ув (a) Find the posterior distributions, means, variances, and 95% quantile-based confidence intervals for and θΒ, assuming a Poisson sampling distribution for each group and the prior distribution: e. ~ gamma( 120, 10), θΒ ~ gamma( 12, 1), pa,Mg)-P(%) x p(%) b) Compute and plot the posterior expectation of OB under the prior distribution B gamma(12 x no,o) for each value of no E (1,2,...,50. Describe what sort of prior beliefs about θΒ would be necessary in order for the posterior expectation ofbp to be close to that of θΑ c Should knowledge of population A tell us anything about population B? Discuss whether or not it makes sense to have p(9A+%) p(%) p(%)

Answer 1