CAS Exam MAS-II Spring 2019, Question 22
Study guide: Exam MAS-II
Data are collected on 10 players on a baseball team. For each individual player, i, let the number of hits be denoted as hits_i and the number of at-bats be denoted n_i. The batting average is the ratio of hits to at-bats (hits_i/n_i).
The following three models are proposed:
Model X
\begin{align*}
hits_i &\sim \text{Binomial}(n_i, p_i) \\
\text{logit}(p_i) &= \alpha_i \\
\alpha_i &\sim \text{Normal}(0, 25)
\end{align*}Model Y
\begin{align*}
hits_i &\sim \text{Binomial}(n_i, p_i) \\
\text{logit}(p_i) &= \alpha_i \\
\alpha_i &\sim \text{Normal}(\alpha, \sigma^2) \\
\alpha &\sim \text{Normal}(0, 1) \\
\sigma &\sim \text{HalfCauchy}(0, 1)
\end{align*}Model Z
\begin{align*}
hits_i &\sim \text{Binomial}(n_i, p_i) \\
\text{logit}(p_i) &= \alpha \\
\alpha &\sim \text{Normal}(0, 25)
\end{align*}The models are fit and the posterior means of batting average for each player are plotted below:
Determine which of the following are true.
- A) Model X produced Output 1, Model Y produced Output 2, and Model Z produced Output 3.
- B) Model X produced Output 2, Model Y produced Output 1, and Model Z produced Output 3.
- C) Model X produced Output 1, Model Y produced Output 3, and Model Z produced Output 2.
- D) Model X produced Output 3, Model Y produced Output 1, and Model Z produced Output 2.
- E) None of the above statements are true.
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