An educational consulting firm is trying to decide whether high school students who have never before used a hand-held calculator can solve a certain type of problem more easily with a calculator that uses reverse Polish logic or one that does not use this logic. A sample of 25 students is selected and allowed to practice on both calculators. Then each student is asked to work one problem on the reverse Polish calculator and a similar problem on the other. Let p = P(S), where S indicates that a student worked the problem more quickly using reverse Polish logic than without, and let X = number of S's.
a. If p = .5, what is P(7 < X < 18)?
b. If p = .8, what is P(7 < X < 18)?
c. If the claim that p = .5 is to be rejected when either x < 7 or x > 18, what is the probability of rejecting the claim when it is actually correct?
d. If the decision to reject the claim p = .5 is made as in part (c), what is the probability that the claim is not rejected when p = .6? When p = .8?
e. What decision rule would you choose for rejecting the claim p = .5 if you wanted the probability in part (c) to be at most .01?