408 Chapter 9 Hypothesis Testing 6. A test is made of H0 : �� = 3 versus H1: �� < 3. The true value of �� is 3, and H0 is rejected. Is this a Type I error, a Type II error, or a correct decision? Type I error Answers are on page 409. SECTION 9.1 Exercises Exercises 1– 6 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 7 and 8, fill in each blank with the appropriate word or phrase. 7. The hypothesis states that a parameter is equal to a certain value while the hypothesis states that the parameter differs from this value. null, alternate 8. Rejecting H0 when it is true is called a error, and failing to reject H0 when it is false is called a error. Type I, Type II In Exercises 9–12, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 9. H1: �� > 50 is an example of a left-tailed alternate hypothesis. False 10. If we reject H0, we conclude that H0 is false. True 11. If we do not reject H0, then we conclude that H1 is false. False 12. If we do not reject H0, we conclude that H0 is true. False Practicing the Skills In Exercises 13–16, determine whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed. 13. H0 : �� = 5 H1: �� < 5 Left-tailed 14. H0 : �� = 10 H1: �� > 10 Right-tailed 15. H0 : �� = 1 H1: �� ≠ 1 Two-tailed 16. H0 : �� = 26 H1: �� ≠ 26 Two-tailed In Exercises 17–20, determine whether the outcome is a Type I error, a Type II error, or a correct decision. 17. A test is made of H0 : �� = 20 versus H1: �� ≠ 20. The true value of �� is 25, and H0 is rejected. Correct decision 18. A test is made of H0 : �� = 5 versus H1: �� < 5. The true value of �� is 5, and H0 is rejected. Type I error 19. A test is made of H0 : �� = 63 versus H1: �� > 63. The true value of �� is 75, and H0 is not rejected. Type II error 20. A test is made of H0 : �� = 45 versus H1: �� < 45. The true value of �� is 40, and H0 is rejected. Correct decision Working with the Concepts 21. Fertilizer: A new type of fertilizer is being tested on a plot of land in an orange grove, to see whether it increases the amount of fruit produced. The mean number of pounds of fruit on this plot of land with the old fertilizer was 400 pounds. Agriculture scientists believe that the new fertilizer may increase the yield. State the appropriate null and alternate hypotheses. H0 : �� = 400, H1: �� > 400 22. Big fish: A sample of 100 flounder of a certain species have sample mean weight 21.5 grams. Scientists want to perform a hypothesis test to determine how strong the evidence is that the mean weight differs from 20 grams. State the appropriate null and alternate hypotheses. H0 : �� = 20, H1: �� ≠ 20 23. Check, please: A restaurant owner claims that the mean amount spent by diners at his restaurant is more than $30. A test is made of H0 : �� = 30 versus H1: �� > 30. The null hypothesis is rejected. State an appropriate conclusion. 24. Coffee: The mean caffeine content per cup of regular coffee served at a certain coffee shop is supposed to be 100 milligrams. A test is made of H0 : �� = 100 versus H1: �� ≠ 100. The null hypothesis is rejected. State an appropriate conclusion. 25. Big dogs: A veterinarian claims that the mean weight of adult German shepherd dogs is 75 pounds. A test is made of H0 : �� = 75 versus H1: �� ≠ 75. The null hypothesis is not rejected. State an appropriate conclusion. 26. Business trips: A sales manager believes that the mean number of days per year her company’s sales representatives spend traveling is less than 50. A test is made of H0 : �� = 50 versus H1: �� < 50. The null hypothesis is not rejected. State an appropriate conclusion. 27. Type I error: A company that manufactures steel wires guarantees that the mean breaking strength (in kilonewtons) of the wires is greater than 50. They measure the strengths for a sample of wires and test H0 : �� = 50 versus H1: �� > 50. a. If a Type I error is made, what conclusion will be drawn regarding the mean breaking strength? b. If a Type II error is made, what conclusion will be drawn regarding the mean breaking strength? c. This test uses a one-tailed alternate hypothesis. Explain why a one-tailed hypothesis is more appropriate than a two-tailed hypothesis in this situation. 28. Type I error: Washers used in a certain application are supposed to have a thickness of 2 millimeters. A quality control engineer measures the thicknesses for a sample of washers and tests H0 : �� = 2 versus H1: �� ≠ 2. a. If a Type I error is made, what conclusion will be drawn regarding the mean washer thickness? b. If a Type II error is made, what conclusion will be drawn regarding the mean washer thickness? c. This test uses a two-tailed hypothesis. Explain why a two-tailed hypothesis is more appropriate than a one-tailed hypothesis in this situation. 29. Scales: It is desired to check the calibration of a scale by weighing a standard 10-gram weight 100 times. Let �� be the population mean reading on the scale, so that the scale is in calibration if �� = 10 and out of calibration if �� ≠ 10. A test is made of the hypotheses H0 : �� = 10 versus H1: �� ≠ 10. Consider three possible conclusions: (i) The scale is in
navidi_monk_elementary_statistics_2e_ch7-9
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