Solved: The mean water temperature downstream from a power plant cooling tower discharge

The mean water temperature downstream from a power plant cooling tower discharge pipe should be no more than 100F. Past experience has indicated that the standard deviation of temperature is 2F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 98F. (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?

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  2.78. It is known that  0.9 and that wear is normally distributed. (a) Test H0:  3 versus 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 Assume that melting point is normally distributed with . (a) Test H0:  155 versus H0:  155 using  0.01. (b) What is the P-value for this test?

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 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 confidence bound on life.

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 psi. (a) Test the hypothesis that mean strength is 3500 psi. Use  0.01. (b) What is the smallest level of significance 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 confidence interval on mean tensile strength. 9-26. Sup

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?

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 sufficiently 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 meters per second. Assume that speed is normally distributed with known standard deviation  4 meters per second.

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 confidence interval on the mean diameter

9-29. Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patients 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 confidence bound on the mean life.