Section 9.7 Power 467 SECTION 9.7 Exercises Exercises 1 and 2 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 3–5, fill in each blank with the appropriate word or phrase. 3. The greater the power, the less likely we are to make a error. Type II 4. The power is the probability of rejecting H0 when it is . false 5. If the sample size is increased, the power will . increase In Exercises 6–8, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 6. Once the significance level �� has been chosen, the only way to increase the power of a test is to decrease the sample size. False 7. The power is the probability of making a Type I error. False 8. For a test about the population mean, the power of a test depends on a specified value that satisfies the alternate hypothesis. True Practicing the Skills 9. A test has power 0.90 when ��1 = 15. True or false: a. The probability of rejecting H0 when ��1 = 15 is 0.90. True b. The probability of making a correct decision when ��1 = 15 is 0.90. True c. The probability of making a correct decision when ��1 = 15 is 0.10. False d. The probability that H0 is true when ��1 = 15 is 0.10. False 10. A test has power 0.80 when ��1 = 3.5. True or false: a. The probability of rejecting H0 when ��1 = 3.5 is 0.80. True b. The probability of making a Type I error when ��1 = 3.5 is 0.80. False c. The probability of making a Type I error when ��1 = 3.5 is 0.20. False d. The probability of making a Type II error when ��1 = 3.5 is 0.80. False e. The probability of making a Type II error when ��1 = 3.5 is 0.20. True f. The probability that H0 is false when ��1 = 3.5 is 0.80. False Working with the Concepts 11. Tire lifetimes: A tire company claims that the lifetimes of its tires average 50,000 miles. The standard deviation of tire lifetimes is known to be �� = 5000 miles. You sample 100 tires and will test the hypotheses H0 : �� = 50,000 versus H1: �� < 50,000 at the �� = 0.05 level of significance. a. Find the power of the test against the alternative ��1 = 49,500. 0.2578 Tech: 0.2595 b. Find the power of the test against the alternative ��1 = 49,000. 0.6406 Tech: 0.6388 c. Find the power of the test against the alternative ��1 = 49,500 if the test is made at level �� = 0.01. 0.0918 Tech: 0.0924 d. Find the power of the test against the alternative ��1 = 49,000 if the test is made at level �� = 0.01. 0.3707 Tech: 0.3721 12. Coffee beans: Shipments of coffee beans are checked for moisture content. A high moisture content indicates water contamination and will result in the shipment being rejected. Let �� represent the mean water content (in percent by weight) in a shipment. Fifty moisture measurements will be made on beans chosen at random from the shipment. A test of the hypotheses H0 : �� = 10 versus H1: �� > 10 will be made at the �� = 0.05 level of significance. Assume the standard deviation of moisture content is �� = 5.0. a. Find the power of the test against the alternative ��1 = 11. 0.4090 Tech: 0.4088 b. Find the power of the test against the alternative ��1 = 12. 0.8810 Tech: 0.8817 c. Find the power of the test against the alternative ��1 = 11 if the test is made at level �� = 0.01. 0.1814 Tech: 0.1808 d. Find the power of the test against the alternative ��1 = 12 if the test is made at level �� = 0.01. 0.6915 Tech: 0.6922 13. SAT scores: A college admissions officer will draw a simple random sample of 100 mathematics SAT scores from the entering freshman class. The admissions officer will perform a test of the hypotheses H0 : �� = 500 versus H1: �� > 500 at the �� = 0.05 level of significance. Assume the population standard deviation is �� = 116. a. Find the power of the test against the alternative ��1 = 520. 0.5319 Tech: 0.5316 b. Find the power of the test against the alternative ��1 = 550. 0.9962 c. Find the power of the test against the alternative ��1 = 520 if the test is made at level �� = 0.01. 0.2743 Tech: 0.2735 d. Find the power of the test against the alternative ��1 = 550 if the test is made at level �� = 0.01. 0.9761 Tech: 0.9764 14. Watch your cholesterol: An article in the International Journal of Clinical Practice described a study in which a sample of 314 patients took a combination of the drugs ezetimbe and simvastatin in order to reduce their total blood cholesterol levels. A test of the hypotheses H0 : �� = 1 versus H1: �� > 1 will be made at the �� = 0.05 level of significance. Assume the population standard deviation is �� = 0.2. a. Find the power of the test against the alternative ��1 = 1.02. 0.5517 Tech: 0.5506
navidi_monk_elementary_statistics_2e_ch7-9
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