A coin is flipped 25 times. Let x be the number of flips that result in heads (H). Consider the following rule for deciding whether or not the coin is fair: Judge the coin fair if 8 # x # 17. Judge the coin biased if either x # 7 or x $ 18.
a. What is the probability of judging the coin biased when it is actually fair?
b. Suppose that a coin is not fair and that P(H) = 0.9. What is the probability that this coin would be judged fair? What is the probability of judging a coin fair if P(H) = 0.1?
c. What is the probability of judging a coin fair if P(H) = 0.6? if P(H) = 0.4? Why are these probabilities large compared to the probabilities in Part (b)?
d. What happens to the “error probabilities” of Parts (a) and (b) if the decision rule is changed so that the coin is judged fair if 7 # x # 18 and unfair otherwise? Is this a better rule than the one first proposed? Explain.