In American football, it is important for the quarterback to throw the ball accurately to the receiver. The completion rate of a quarterback is the proportion of throws successfully caught by the receivers on the team; that is, it is the number of completed passes divided by the total number of attempted throws. One of the best quarterbacks of the 2010 season was Drew Brees of the New Orlean Saints. Let π represent Brees’ true completion rate for the entire 2010 season, and suppose we want to estimate it using data from the first five games that he played during that season, using Bayesian methodology. After the first five games, Brees had completed 142 passes in 199 attempts. Suppose we assign a beta(60, 38) prior distribution for Brees’ completion rate π (see Exercise 13 for insight into the values for the prior parameters).
a. Assuming a binomial likelihood function for the 142 completed passes in 199 attempts and a beta(60, 38) prior distribution for π, compute the posterior distribution for π and the posterior estimate of π.
b. Construct and interpret a 95% Bayesian credible interval for π.
c. Construct a standard 95% confidence interval for π. Which interval for π is narrower?
d. At the end of the 2010 season, Brees’ completion rate of 68.1% was highest among all starting quarterbacks. How did your posterior estimate compare to his 2010 completion rate? Did his completion rate fall within the confidence interval and credible intervals?