\fThe Ach tribe of Paraguay are part-time hunter-gatherers and have been in contact with Paraguayan society only since the mid-1970s. Part of Ach life is spent away from the village on extended forest treks. While trekking, the Ach subsist exclusively on foods that they collect on a given day. Armadillos comprise the vast majority of food calories consumed by the Ach and it is of interest to quantify the typical number of armadillos killed in a day. Our data involve observations on n : 38 Ache men. Let 95- be the number of armadillos killed by the i'th man on a given day. The Poisson distribution provides a natural model for the number of events occurring haphazardly over a given amount of time, so we assume iid . y1,...,yn|9 N P015 (6). We refer to 6 as the kill rate. The broader Ach data include many hunting days for each man. For this analysis one day has been randomly selected for each man. Dr. Garnett McMillan, an expert on Ache hunting practices, believes that Ache men typically kill an armadillo every other day. This provides a "best guess" for 0 of 0.5 armadillos. Dr. McMillan is 95% sure that the mean daily number of kills is no greater than 2 armadillos. In fact, our data show that across the randomly selected hunting days for our n = 38 Ache men, a total of 10 armadillos were killed. (a) Using the expert information from Dr. McMillan, find a prior distribution for the kill rate. Assume that Dr. McMillan's "best guess" gives the mode of the prior distribu- tion, and use Table 2.2 and the discussion of prior-finding on page 27 of your textbook. (HINT: Find a as a function of 8, then use computer software to find / such that the resulting Gamma distribution accords with Dr. McMillan's beliefs.) (b) Find the posterior distribution for the kill rate, based on your prior and the reported armadillo hunting data. (c) Find the expected number of armadillos that will be killed if these 38 Ache hunters each hunt for one more day. Explain how you found this. (d) Give a probability interval for how many armadillos will be killed if these 38 Ache hunters each hunt for one more day. Explain how you found this interval. There are many ways you might approach this problem, and I'm interested in seeing how you'll try to do it. Make sure to report the probability (approximate or exact) you've used for defining your interval