In an interview with a local newspaper, a respected trial lawyer claims that he wins at least 75% of his court cases. Bert, a skeptical statistics student, sets up a one-tail test at the 0.05 level of significance to evaluate the attorney’s claim. The student plans to examine a random sample of 40 cases tried by the attorney and determine the proportion of these cases that were won. The null and alternative hypotheses are H0: π ≥ 0.75 and H1: π < 0.75. Using the techniques of this chapter, Bert sets up a hypothesis test in which the decision rule is “Reject H0 if z < 21.645, otherwise do not reject.” What is the probability that Bert will make a Type II error (fail to reject a false null hypothesis) if the attorney’s true population proportion of wins is actually
a. π = 0.75?
b. π = 0.70?
c. π = 0.65?
d. π = 0.60?
e. π = 0.55?
f. Making use of the probabilities calculated in parts (a) through (e), describe and plot the power curve for Bert’s hypothesis test.