Suppose a researcher obtains a grant to study the health and nutrition (or lack thereof) at a university with a large undergraduate population. To begin the study, a random sample of n = 100 undergrads is drawn and their current weights are taken and recorded, with a recorded average weight of y 145 pounds. Let's assume that the weights of the undergrad population follow a normal distribution with unknown mean and known standard deviation 20 pounds. Further suppose that, before taking the observed data, a prior distribution for was specified to be normal with mean 175 and a conservative standard deviation of 35 pounds. = (a) Write down the posterior distribution for (you may use results given in the lecture slides). Then, simulate 10,000 draws from the posterior distribution and make a histogram of the draws. (4 points) (b) Using your simulated draws from part (a), estimate using the posterior median and give an equal-tailed 95% posterior interval for 0. (2 points) (c) Generally, calculating HPD intervals/regions can be complicated. For this example, what can we say about the HPD interval in relation to the equal-tailed interval given above? (1 point)