Before a strike prematurely ended the 1994 major league baseball season, Tony Gwynn of the...

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(θ) indicator equal to unity if Gwynn gets a hit during his ith at bat, and zero otherwise. Then, Y1, Y2, …, Yn is a random sample from a Bernoulli(θ) distribution, where θ is the probability of success, and n = 419. Our best point estimate of θ is Gwynn’s batting average, which is just the proportion of successes: = .394. Using the fact that se() = , construct an approximate 95% confidence interval for θ, using the standard normal distribution. Would you say there is strong evidence against Gwynn’s being a potential .400 hitter? Explain.