Suppose that a baseball player has been a .250 hitter for his career, which means that his probability of a hit (success) has been 0.250. Then during one winter the player genuinely improves to the point that his probability of success improves to 0.333. You will investigate how likely the player is, in a sample of at-bats, to convince the manager that he has improved.
a. Open the Power Simulation applet. Enter 0.250 for the hypothesized value, 0.333 for the alternative value, and 20 for the sample size. Enter 200 for the number of samples, and click on Draw Samples. Repeat four more times to produce a total of 1,000 simulated samples. Comment on how much overlap you see between the two distributions. Does this overlap suggest that it’s easy or difficult to distinguish between a .250 hitter and a .333 hitter in a sample of 20 at-bats?
b. Choose the Level of Significance option, and enter 0.05 for the significance level. Click on Count. The top graph indicates how many hits the player needs in 20 at-bats in order to produce a p-value less than 0.05 to provide convincing evidence to reject the null hypothesis that his success probability is 0.250. How many hits are needed?

c. Based on the bottom graph, in what proportion of the 1,000 samples will the player with a 0.333 probability of success obtain enough hits to reject the hypothesis that he has a 0.250 probability of success? This is the approximate power of the test.

d. Would you say that the player has a good chance of convincing the manager that he has improved in a sample of 20 at-bats? Write a summary, as if to the player.

e. What factors could be changed in order to increase the power of the test?