Question: Professor Hansen believes that some people have the ability to predict in advance the outcome of a spin of a roulette wheel. He takes 100 student volunteers to a casino. The roulette wheel has 38 numbers, each of which is equally likely to occur. Of these 38 numbers, 18 are red, 18 are black, and 2 are green. Each student is to place a series of five bets, choosing either a red or a black number before each spin of the wheel. Thus, a student who bets on red has an 18/38 chance of winning that bet. The same is true of betting on black.
a. Assuming random guessing, what is the probability that a particular student will win all five of his or her bets?
b. Suppose for each student we formulate the hypothesis test
H0: The student is guessing
H1: The student has some predictive ability. Suppose we reject H0 only if the student wins all five bets. What is the significance level?
c. Suppose that 2 of the 100 students win all five of their bets. Professor Hansen says, "For these two students we can reject H0 and conclude that we have found two students with some ability to predict." What do you make of Professor Hansen's conclusion?