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300 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

(c) What is the -error if the true mean is 150? (a) Test the hypotheses H 0 : 100 versus H 1 : 100 us-
(d) What value of n would be required if we want 0.1 ing 0.05.
when 150? Assume that 0.01. (b) Compute the power of the test if the true mean speed is as
9-24. The life in hours of a battery is known to be approxi- low as 95 meters per second.
mately normally distributed, with standard deviation 1.25 (c) What sample size would be required to detect a true mean
hours. A random sample of 10 batteries has a mean life of speed as low as 95 meters per second if we wanted the
x 40.5 hours. power of the test to be at least 0.85?
(a) Is there evidence to support the claim that battery life (d) Explain how the question in part (a) could be answered by
exceeds 40 hours? Use 0.05. constructing a one-sided confidence bound on the mean
(b) What is the P-value for the test in part (a)? speed.
(c) What is the -error for the test in part (a) if the true mean 9-28. A bearing used in an automotive application is suppose
life is 42 hours? to have a nominal inside diameter of 1.5 inches. A random sam-
(d) What sample size would be required to ensure that does ple of 25 bearings is selected and the average inside diameter of
not exceed 0.10 if the true mean life is 44 hours? these bearings is 1.4975 inches. Bearing diameter is known to be
(e) Explain how you could answer the question in part (a) normally distributed with standard deviation 0.01 inch.
by calculating an appropriate confidence bound on life. (a) Test the hypotheses H 0 : 1.5 versus H 1 : 1.5 using
9-25. An engineer who is studying the tensile strength of a 0.01.
steel alloy intended for use in golf club shafts knows that (b) Compute the power of the test if the true mean diameter is
tensile strength is approximately normally distributed with 1.495 inches.
60 psi. A random sample of 12 specimens has a mean (c) What sample size would be required to detect a true mean
tensile strength of x 3250 psi. diameter as low as 1.495 inches if we wanted the power of
(a) Test the hypothesis that mean strength is 3500 psi. Use the test to be at least 0.9?
0.01. (d) Explain how the question in part (a) could be answered by
(b) What is the smallest level of significance at which you constructing a two-sided confidence interval on the mean
would be willing to reject the null hypothesis? diameter.
(c) Explain how you could answer the question in part (a) 9-29. Medical researchers have developed a new artificial
with a two-sided confidence interval on mean tensile heart constructed primarily of titanium and plastic. The heart
strength. will last and operate almost indefinitely once it is implanted in
9-26. Suppose that in Exercise 9-25 we wanted to reject the the patient’s body, but the battery pack needs to be recharged
null hypothesis with probability at least 0.8 if mean strength about every four hours. A random sample of 50 battery packs
3500. What sample size should be used? is selected and subjected to a life test. The average life of these
batteries is 4.05 hours. Assume that battery life is normally
9-27. Supercavitation is a propulsion technology for under-
distributed with standard deviation 0.2 hour.
sea vehicles that can greatly increase their speed. It occurs
(a) Is there evidence to support the claim that mean battery
above approximately 50 meters per second, when pressure
life exceeds 4 hours? Use 0.05.
drops sufficiently to allow the water to dissociate into water
(b) Compute the power of the test if the true mean battery life
vapor, forming a gas bubble behind the vehicle. When the gas
is 4.5 hours.
bubble completely encloses the vehicle, supercavitation is
(c) What sample size would be required to detect a true mean
said to occur. Eight tests were conducted on a scale model of
battery life of 4.5 hours if we wanted the power of the test
an undersea vehicle in a towing basin with the average ob-
to be at least 0.9?
served speed x 102.2 meters per second. Assume that speed
is normally distributed with known standard deviation (d) Explain how the question in part (a) could be answered by
constructing a one-sided confidence bound on the mean life.
4 meters per second.
9-3 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION,
VARIANCE UNKNOWN
9-3.1 Hypothesis Tests on the Mean

We now consider the case of hypothesis testing on the mean of a population with unknown
2
variance . The situation is analogous to Section 8-3, where we considered a confidence
interval on the mean for the same situation. As in that section, the validity of the test procedure
we will describe rests on the assumption that the population distribution is at least approximately