9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN

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is close to 50.5 centimeters per second, and we would not want this value of x from the sample to result in rejection of H0. The following display shows the P-value for testing H0: 50 when we observe x 50.5 centimeters per second and the power of the test at 0.05 when the true mean is 50.5 for various sample sizes n:

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Sample Size n 10 25 50 100 400 1000

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P-value When x 50.5 0.4295 0.2113 0.0767 0.0124 5.73 10 2.57 10

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Power (at When True 0.1241 0.2396 0.4239 0.7054 0.9988 1.0000

0.05) 50.5

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The P-value column in this display indicates that for large sample sizes the observed sample value of x 50.5 would strongly suggest that H0: 50 should be rejected, even though the observed sample results imply that from a practical viewpoint the true mean does not differ much at all from the hypothesized value 0 50. The power column indicates that if we test a hypothesis at a xed signi cance level and even if there is little practical difference between the true mean and the hypothesized value, a large sample size will almost always lead to rejection of H0. The moral of this demonstration is clear:

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Be careful when interpreting the results from hypothesis testing when the sample size is large, because any small departure from the hypothesized value 0 will probably be detected, even when the difference is of little or no practical signi cance.

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EXERCISES FOR SECTION 9-2

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9-20. The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100 F. Past experience has indicated that the standard deviation of temperature is 2 F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98 F. (a) Should the water temperature be judged acceptable with 0.05 (b) What is the P-value for this test (c) What is the probability of accepting the null hypothesis at 0.05 if the water has a true mean temperature of 104 F 9-21. Reconsider the chemical process yield data from Exercise 8-9. Recall that 3, yield is normally distributed and that n 5 observations on yield are 91.6%, 88.75%, 90.8%, 89.95%, and 91.3%. Use 0.05. (a) Is there evidence that the mean yield is not 90% (b) What is the P-value for this test (c) What sample size would be required to detect a true mean yield of 85% with probability 0.95 (d) What is the type II error probability if the true mean yield is 92% (e) Compare the decision you made in part (c) with the 95% CI on mean yield that you constructed in Exercise 8-7. 9-22. A manufacturer produces crankshafts for an automobile engine. The wear of the crankshaft after 100,000 miles (0.0001 inch) is of interest because it is likely to have an impact on warranty claims. A random sample of n 15 shafts is tested and x 2.78. It is known that 0.9 and that wear is normally distributed. (a) Test H0: 3 versus H0: Z 3 using 0.05. (b) What is the power of this test if 3.25 (c) What sample size would be required to detect a true mean of 3.75 if we wanted the power to be at least 0.9 9-23. A melting point test of n 10 samples of a binder used in manufacturing a rocket propellant resulted in x 154.2 F. Assume that melting point is normally distrib1.5 F. uted with (a) Test H0: 155 versus H0: 155 using 0.01. (b) What is the P-value for this test

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CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

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(c) What is the -error if the true mean is 150 (d) What value of n would be required if we want 0.1 when 150 Assume that 0.01. 9-24. The life in hours of a battery is known to be approximately normally distributed, with standard deviation 1.25 hours. A random sample of 10 batteries has a mean life of x 40.5 hours. (a) Is there evidence to support the claim that battery life exceeds 40 hours Use 0.05. (b) What is the P-value for the test in part (a) (c) What is the -error for the test in part (a) if the true mean life is 42 hours (d) What sample size would be required to ensure that does not exceed 0.10 if the true mean life is 44 hours (e) Explain how you could answer the question in part (a) by calculating an appropriate con dence bound on life. 9-25. An engineer who is studying the tensile strength of a steel alloy intended for use in golf club shafts knows that tensile strength is approximately normally distributed with 60 psi. A random sample of 12 specimens has a mean tensile strength of x 3250 psi. (a) Test the hypothesis that mean strength is 3500 psi. Use 0.01. (b) What is the smallest level of signi cance at which you would be willing to reject the null hypothesis (c) Explain how you could answer the question in part (a) with a two-sided con dence interval on mean tensile strength. 9-26. Suppose that in Exercise 9-25 we wanted to reject the null hypothesis with probability at least 0.8 if mean strength 3500. What sample size should be used 9-27. Supercavitation is a propulsion technology for undersea vehicles that can greatly increase their speed. It occurs above approximately 50 meters per second, when pressure drops suf ciently to allow the water to dissociate into water vapor, forming a gas bubble behind the vehicle. When the gas bubble completely encloses the vehicle, supercavitation is said to occur. Eight tests were conducted on a scale model of an undersea vehicle in a towing basin with the average observed speed x 102.2 meters per second. Assume that speed is normally distributed with known standard deviation 4 meters per second.

(a) Test the hypotheses H0: 100 versus H1: 100 using 0.05. (b) Compute the power of the test if the true mean speed is as low as 95 meters per second. (c) What sample size would be required to detect a true mean speed as low as 95 meters per second if we wanted the power of the test to be at least 0.85 (d) Explain how the question in part (a) could be answered by constructing a one-sided con dence bound on the mean speed. 9-28. A bearing used in an automotive application is suppose to have a nominal inside diameter of 1.5 inches. A random sample of 25 bearings is selected and the average inside diameter of these bearings is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation 0.01 inch. (a) Test the hypotheses H0: 1.5 versus H1: 1.5 using 0.01. (b) Compute the power of the test if the true mean diameter is 1.495 inches. (c) What sample size would be required to detect a true mean diameter as low as 1.495 inches if we wanted the power of the test to be at least 0.9 (d) Explain how the question in part (a) could be answered by constructing a two-sided con dence interval on the mean diameter. 9-29. Medical researchers have developed a new arti cial heart constructed primarily of titanium and plastic. The heart will last and operate almost inde nitely once it is implanted in the patient s body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation 0.2 hour. (a) Is there evidence to support the claim that mean battery life exceeds 4 hours Use 0.05. (b) Compute the power of the test if the true mean battery life is 4.5 hours. (c) What sample size would be required to detect a true mean battery life of 4.5 hours if we wanted the power of the test to be at least 0.9 (d) Explain how the question in part (a) could be answered by constructing a one-sided con dence bound on the mean life.

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