Before a strike prematurely ended the 1994 major league baseball season, Tony Gwynn of the San Diego Padres had 165 hits in 419 at bats, for a .394 batting average. There was discussion about whether Gwynn was a potential .400 hitter that year. This issue can be couched in terms of Gwynn’s probability of getting a hit on a particular at bat, call it θ. Let Yi be the Bernoulli(u) indicator equal to unity if Gwynn gets a hit during his ith at bat, and zero otherwise. Then, Y₁, Y₂,…., Y_n is a random sample from a Bernoulli(u) distribution, where u is the probability of success, and n = 419.

Our best point estimate of u is Gwynn’s batting average, which is just the proportion of successes:

y̅ = .394. Using the fact that se(y̅) = √y̅(1 – y̅)/n, construct an approximate 95% confidence interval for u, using the standard normal distribution. Would you say there is strong evidence against Gwynn’s being a potential .400 hitter? Explain.