9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 299 9-21. Rework Exercise 9-20 when the sample size is 16 and the boundaries of the acceptance region do not change. What impact does the change in sample size have on the results of parts (a) and (b)? 9-22. In Exercise 9-20, find the boundary of the critical re- gion if the type I error probability is (a) 0.01 and n 8 (c) 0.01 and n 16 (b) 0.05 and n 8 (d) 0.05 and n 16 9-23. In Exercise 9-20, calculate the P-value if the observed statistic is (a) x = 5.2 (b) x = 4.7 (c) x = 5.1 9-24. In Exercise 9-20, calculate the probability of a type II error if the true mean output is 5.05 volts and (a) 0.05 and n 10 (b) 0.05 and n 16 (c) Compare the values of calculated in the previous parts. What conclusion can you draw? 9-25. The proportion of adults living in Tempe, Arizona, who are college graduates is estimated to be p 0.4. To test this hypothesis, a random sample of 15 Tempe adults is selected. If the number of college graduates is between 4 and 8, the hypothesis will be accepted; otherwise, we will conclude that p 0.4 . (a) Find the type I error probability for this procedure, assum- ing that p 0.4. shows that 1 or fewer favor this proposal, we will conclude that p 0.3. (a) Find the probability of type I error if the true proportion is p 0.3. (b) Find the probability of committing a type II error with this procedure if p 0.2. (c) What is the power of this procedure if the true proportion is p 0.2? 9-27. A random sample of 500 registered voters in Phoenix is asked if they favor the use of oxygenated fuels year-round to reduce air pollution. If more than 400 voters respond posi- tively, we will conclude that more than 60% of the voters favor the use of these fuels. (a) Find the probability of type I error if exactly 60% of the voters favor the use of these fuels. (b) What is the type II error probability if 75% of the voters favor this action? Hint: use the normal approximation to the binomial. 9-28. If we plot the probability of accepting H 0 : 0 versus various values of and connect the points with a smooth curve, we obtain the operating characteristic curve (or the OC curve) of the test procedure. These curves are used extensively in industrial applications of hypothesis testing to display the sensitivity and relative performance of the test. When the true mean is really equal to 0 , the probability of accepting H 0 is 1 .