0 13.031 13.691 12.620 11.70 12.390 12.440 12.221 13.651 14.31 12.391 13.510 13.211 13.021 12.891 14.790 11.571 14.261 13.941 13.851 11.340 12.410 13.880 11.980 11.181 13.171 14.620 15.411 14.71 13.450 12.661 12.731 12.360 10.591 12.021 13.520 12.970 12.960 11.421 13.120 13.370 13.111 13.11 11.83

10.2 Nesting success: Younger male sparrows may or may not nest during a mating season, perhaps depending on their physical characteristics. Re- searchers have recorded the nesting success of 43 young male sparrows of the same age, as well as their wingspan, and the data appear in the file msparromest.dat. Let Y1; be the binary indicator that sparrow a" successfully nests, and let 9:,- denote their wingspan. Our model for Y,- is logit Pr'i = 1|u, [3, :5] = a + {ls-.5, where the logit function is given by logit fl = log[3f[1 5)] a) Write out the joint sampling distribution 21:1 plygla, 13, 5] and sim plify as much as possible. b} Formulate a prior probability distribution over a and I} by consid ering the range of Pr{Y = 1|r1,;3,s:) as a: ranges over 10 to 15, the approximate range of the observed wingspans. c] Implement a Metropolis algorithm that approximates a,;'3]y, 2:). Adjust the proposal distribution to achieve a reasonable acceptance rate, and run the algorithm long enough so that the effective sample size is at least 1,000 for each parameter. d} Compare the posterior densities of a and {3 to their prior densities. e] Using output from the Metropolis algorithm, come up with a way to make a confidence band for the following function fastj of wingspan: goth]: 4's\") = W= where a and I3 are the parameters in your sampling model. Make a plot of such a band