Suppose two brothers named Mario and Luigi like to compete by playing a certain video game. Mario thinks he is better at this game than Luigi and sets out to prove it by keeping track of who wins. Aft er playing the game 30 times, Mario won 18 of them (or 60%). Mario then declares that this proves he is obviously the better player. Luigi, who just finished Chapter 1 in his statistics class, realizes that perhaps he and his brother are evenly matched and just by chance Mario won 60% of the last 30 games. Luigi is going to test this by running a test of significance.
a. Describe what the parameter is that Luigi should be testing.
b. State the appropriate null and alternative hypotheses in the context of this study. (Hint: Luigi likes two-sided tests because he knows that will make it harder to get strong evidence that his brother is better.)
c. Using an appropriate applet, find the p-value using a theory-based test (one-proportion z-test; normal approximation).
d. Summarize your conclusion from your p-value.
e. Th e p-value you found in part (c) should have been greater than 0.05. Suppose the two brothers continue to compete and Mario continues to win 60% of the games. How many games will they have to play until he gets a p-value less than 0.05 aft er retesting?