When a thread-cutting machine is operating properly, only 2% of the units produced are defective. Since the machine was bumped by a forklift truck, however, the quality-control manager is concerned that it may require expensive downtime and a readjustment. The manager would like to set up a right-tail test at the 0.01 level of significance, then identify the proportion of defectives in a sample of 400 units produced. His null hypothesis is H0: π ≤ 0.02, with H1: π > 0.02.
a. What critical value of z will be associated with this test?
b. For the critical z determined in part (a), identify the sample proportion that this z value represents, then use this value of p in stating a decision rule for the test.
c. What is the probability that the quality-control manager will fail to reject a false H0 if the actual population proportion of defectives is π = 0.02? If π = 0.03? If π = 0.04? If π = 0.05? If π = 0.06?
d. Making use of the probabilities calculated in part (c), plot the power and operating characteristic curves for the test.