You are the assistant coach of the women's softball team at a Midwestern college. The head coach has asked you to assess a new first year player who is joining the team. As a high school student, she was at bat 120 times and got 40 hits. You wish to estimate , her underlying true probability of getting a hit in any at bat as a college-level player.

1. Specify a beta prior that seems appropriate to capture your knowledge or uncertainty about before the new player plays in any college-level games. Use any information you have that seems importanther high school record, anything you know about college-level women's softball, etc. Use R functions as needed. Explain in a few sentences (supplemented with plots and/or R output) how you chose the values of and .

There is no one right answer hereI want to see how you think about this and what procedure you use.

2. Specify a beta prior that you think might reflect the player's mother's beliefs about . This may be similar to, or quite different from, your prior. Again, justify your choice with graphical or numeric R output.

3. Suppose the player now plays eight college-level games, has thirty at bats, and gets 5 hits. Thus, the data are

y=5, n=30

We will use a binomial likelihood for these data. This requires the assumption that, conditional on , each at bat is an independent Bernoulli trial with success probability . There are several reasons why independence might actually not be a reasonable assumption in this problem. Give one.

Note: For our present purposes, we'll use a binomial likelihood anyway. We'll come up with a better model when we talk about hierarchical models later in the semester.

4. Obtain the following characteristics of the posterior distribution p( |y)

a. Name of posterior distribution and its parameter values

b. Posterior density plot

c. A plot showing the prior density, the likelihood, and the posterior density, all

on the same axes

based on the data from the college-level games under each of three priors:

a. Your prior from part 1

b. The mother's prior from part 2

c. A uniform (noninformative) prior