Through roughly the first week of the 2011 season, Miguel Montero of the Arizona Diamondbacks was the leading hitter in Major League Baseball. In his first 26 at-bats, he had 13 hits to his credit for a batting average of .500 (the batting average for a player is the number of hits divided by the total tries, called at-bats). We would like to estimate π, Montero’s true batting average for the entire 2011 season, using Bayesian methods; that is, we want to compute the posterior estimate of π.
a. Suppose you know nothing about baseball or a batting average beyond the fact that it is a number between 0 and 1. What prior distribution should you assign to π? Briefly explain. What is the prior estimate for π?
b. Using the prior distribution you defined in Part A and assuming that the likelihood of observing 13 hits in 26 at-bats is binomial with the probability of a successful hit equal to π, find the posterior distribution of π. Calculate the posterior estimate of π and compare it to the prior estimate.
c. Suppose you know that batting averages typically fall between .200 and .300 during a regular season. Use software to determine an appropriate beta prior distribution for π. Recompute the posterior distribution and posterior estimate for π after observing 13 hits in 26 at-bats. How does your posterior estimate compare to that found in Part B?
d. Visit a sports website to find Montero’s true 2011 season batting average, and compare it to your posterior estimates found in Parts B and C. Which posterior estimate was closer to his true batting average?