Page 358 - Applied Statistics And Probability For Engineers

c09.qxd 4/6/02 2:55 PM Page 306

306 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

(a) Test the hypotheses H 0 : 22.5 versus H 1 : 22.5, follows: 131.15, 130.69, 130.91, 129.54, 129.64, 128.77, 130.72,
using 0.05. Find the P-value. 128.33, 128.24, 129.65, 130.14, 129.29, 128.71, 129.00, 129.39,
(b) Is there evidence to support the assumption that interior 130.42, 129.53, 130.12, 129.78, 130.92, 131.15, 130.69, 130.91,
temperature is normally distributed? 129.54, 129.64, 128.77, 130.72, 128.33, 128.24, and 129.65.
(c) Compute the power of the test if the true mean interior (a) Can you support a claim that mean sodium content of this
temperature is as high as 22.75. brand of cornflakes is 130 milligrams? Use 0.05.
(d) What sample size would be required to detect a true mean (b) Is there evidence that sodium content is normally distrib-
interior temperature as high as 22.75 if we wanted the uted?
power of the test to be at least 0.9? (c) Compute the power of the test if the true mean sodium
(e) Explain how the question in part (a) could be answered by content is 130.5 miligrams.
constructing a two-sided confidence interval on the mean (d) What sample size would be required to detect a true mean
interior temperature. sodium content of 130.1 milligrams if we wanted the
9-31. A 1992 article in the Journal of the American Medical power of the test to be at least 0.75?
Association (“A Critical Appraisal of 98.6 Degrees F, the Upper (e) Explain how the question in part (a) could be answered by
Limit of the Normal Body Temperature, and Other Legacies of constructing a two-sided confidence interval on the mean
Carl Reinhold August Wundrlich”) reported body temperature, sodium content.
gender, and heart rate for a number of subjects. The body tem- 9-34. Reconsider the tire testing experiment described in
peratures for 25 female subjects follow: 97.8, 97.2, 97.4, 97.6, Exercise 8-22.
97.8, 97.9, 98.0, 98.0, 98.0, 98.1, 98.2, 98.3, 98.3, 98.4, 98.4, (a) The engineer would like to demonstrate that the mean life
98.4, 98.5, 98.6, 98.6, 98.7, 98.8, 98.8, 98.9, 98.9, and 99.0. of this new tire is in excess of 60,000 kilometers. Formu-
(a) Test the hypotheses H 0 : 98.6 versus H 1 : 98.6 , late and test appropriate hypotheses, and draw conclu-
using 0.05. Find the P-value. sions using 0.05.
(b) Compute the power of the test if the true mean female (b) Suppose that if the mean life is as long as 61,000 kilome-
body temperature is as low as 98.0. ters, the engineer would like to detect this difference with
(c) What sample size would be required to detect a true mean probability at least 0.90. Was the sample size n 16 used
female body temperature as low as 98.2 if we wanted the in part (a) adequate? Use the sample standard deviation s
power of the test to be at least 0.9? as an estimate of in reaching your decision.
(d) Explain how the question in part (a) could be answered by 9-35. Reconsider the Izod impact test on PVC pipe described
constructing a two-sided confidence interval on the mean in Exercise 8-23. Suppose that you want to use the data from this
female body temperature. experiment to support a claim that the mean impact strength
(e) Is there evidence to support the assumption that female exceeds the ASTM standard (foot-pounds per inch). Formulate
body temperature is normally distributed? and test the appropriate hypotheses using 0.05.
9-32. Cloud seeding has been studied for many decades as 9-36. Reconsider the television tube brightness experiment
a weather modification procedure (for an interesting study of in Exercise 8-24. Suppose that the design engineer believes
this subject, see the article in Technometrics by Simpson, that this tube will require 300 microamps of current to pro-
Alsen, and Eden, “A Bayesian Analysis of a Multiplicative duce the desired brightness level. Formulate and test an
Treatment Effect in Weather Modification”, Vol. 17, pp. 161– appropriate hypothesis using 0.05. Find the P-value for
166). The rainfall in acre-feet from 20 clouds that were se- this test. State any necessary assumptions about the underly-
lected at random and seeded with silver nitrate follows: 18.0, ing distribution of the data.
30.7, 19.8, 27.1, 22.3, 18.8, 31.8, 23.4, 21.2, 27.9, 31.9, 27.1,
9-37. Consider the baseball coefficient of restitution data
25.0, 24.7, 26.9, 21.8, 29.2, 34.8, 26.7, and 31.6.
first presented in Exercise 8-79.
(a) Can you support a claim that mean rainfall from seeded
(a) Does the data support the claim that the mean coefficient
clouds exceeds 25 acre-feet? Use 0.01.
of restitution of baseballs exceeds 0.635? Use 0.05.
(b) Is there evidence that rainfall is normally distributed?
(b) What is the P-value of the test statistic computed in part (a)?
(c) Compute the power of the test if the true mean rainfall is
(c) Compute the power of the test if the true mean coefficient
27 acre-feet.
of restitution is as high as 0.64.
(d) What sample size would be required to detect a true mean
(d) What sample size would be required to detect a true mean
rainfall of 27.5 acre-feet if we wanted the power of the test
coefficient of restitution as high as 0.64 if we wanted the
to be at least 0.9?
power of the test to be at least 0.75?
(e) Explain how the question in part (a) could be answered by
9-38. Consider the dissolved oxygen concentration at TVA
constructing a one-sided confidence bound on the mean
dams first presented in Exercise 8-81.
diameter.
(a) Test the hypotheses H 0 : 4 versus H 1 : 4 . Use
9-33. The sodium content of thirty 300-gram boxes of organic
0.01.
corn flakes was determined. The data (in milligrams) are as