SECTION 7.5 Hypothesis Testing for Variance and Standard Deviation 401 Using and Interpreting Concepts Hypothesis Testing Using Rejection Regions In Exercises 23–30, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic x 2, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed. 23. Tires A tire manufacturer claims that the variance of the diameters in a tire model is 8.6. A random sample of 10 tires has a variance of 4.3. At a = 0.01, is there enough evidence to reject the claim? 24. Gas Mileage An auto manufacturer claims that the variance of the gas mileages in a model of hybrid vehicle is 0.16. A random sample of 30 vehicles has a variance of 0.26. At a = 0.05, is there enough evidence to reject the claim? 25. Mathematics Assessment Tests A school administrator claims that the standard deviation for grade 12 students on a mathematics assessment test is less than 37 points. A random sample of 28 grade 12 test scores has a standard deviation of 34 points. At a = 0.10, is there enough evidence to support the claim? (Adapted from National Center for Educational Statistics) 26. Reading Assessment Tests A school administrator claims that the standard deviation for grade 12 students on a reading assessment test is greater than 41 points. A random sample of 25 grade 12 test scores has a standard deviation of 46 points. At a = 0.01, is there enough evidence to support the claim? (Adapted from National Center for Educational Statistics) 27. Waiting Times A hospital claims that the standard deviation of the waiting times for patients in its emergency department is no more than 0.5 minute. A random sample of 25 waiting times has a standard deviation of 0.7 minute. At a = 0.10, is there enough evidence to reject the claim? 28. Hotel Room Rates A travel analyst claims that the standard deviation of the room rates for two adults at three-star hotels in Denver is at least $68. A random sample of 18 three-star hotels has a standard deviation of $40. At a = 0.01, is there enough evidence to reject the claim? (Adapted from Expedia) 29. Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At a = 0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from $13,056? (Adapted from Salary.com) 30. Salaries The annual salaries (in dollars) of 12 randomly chosen nursing supervisors are shown in the table at the left. At a = 0.10, is there enough evidence to reject the claim that the standard deviation of the annual salaries is $18,630? (Adapted from Salary.com) Extending Concepts P-Values You can calculate the P-value for a chi-square test using technology. After calculating the standardized test statistic, use the cumulative distribution function (CDF) to calculate the area under the curve. From Example 4 on page 397, x 2 = 43.2. Using a TI-84 Plus (choose 8 from the DISTR menu), enter 0 for the lower bound, 43.2 for the upper bound, and 40 for the degrees of freedom, as shown at the left. Because it is a right-tailed test, the P-value is approximately 1 - 0.6638 = 0.3362. Because P 7 a = 0.05, fail to reject H0. In Exercises 31–34, use the P-value method to perform the hypothesis test for the indicated exercise. 31. Exercise 25 32. Exercise 26 33. Exercise 27 34. Exercise 28 TI-84 PLUS χ2cdf(0,43.2,40) 0.6637768667 Annual salaries 55,060 75,140 89,050 73,200 67,400 86,220 96,000 59,900 111,700 99,750 52,250 98,100 74,700 56,000 77,900 TABLE FOR EXERCISE 29 Annual salaries 68,700 108,300 107,000 98,900 73,900 87,800 83,400 116,600 94,300 96,000 99,300 79,900 TABLE FOR EXERCISE 30
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