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310 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

EXERCISES FOR SECTION 9-4
9-43. Consider the rivet holes from Exercise 8-35. If the 9-46. Consider the Izod impact test data in Exercise 8-23.
standard deviation of hole diameter exceeds 0.01 millimeters, (a) Test the hypothesis that 0.10 against an alternative
there is an unacceptably high probability that the rivet will not specifying that 0.10, using 0.01, and draw a
fit. Recall that n 15 and s 0.008 millimeters. conclusion. State any necessary assumptions about the
(a) Is there strong evidence to indicate that the standard devi- underlying distribution of the data.
ation of hole diameter exceeds 0.01 millimeters? Use (b) What is the P-value for this test?
0.01. State any necessary assumptions about the underly- (c) Could the question in part (a) have been answered by
2
ing distribution of the data. constructing a 99% two-sided confidence interval for ?
(b) Find the P-value for this test. 9-47. Reconsider the percentage of titanium in an alloy used
(c) If is really as large as 0.0125 millimeters, what sample size in aerospace castings from Exercise 8-39. Recall that s 0.37
will be required to defect this with power of at least 0.8? and n 51.
9-44. Recall the sugar content of the syrup in canned peaches (a) Test the hypothesis H 0 : 0.25 versus H 1 : 0.25
from Exercise 8-36. Suppose that the variance is thought to be using 0.05. State any necessary assumptions about
2
2
18 (milligrams) . A random sample of n 10 cans yields the underlying distribution of the data.
a sample standard deviation of s 4.8 milligrams. (b) Explain how you could answer the question in part (a) by
2
2
(a) Test the hypothesis H 0 : 18 versus H 1 : 18 using constructing a 95% two-sided confidence interval for .
0.05. 9-48. Consider the hole diameter data in Exercise 8-35.
(b) What is the P-value for this test? Suppose that the actual standard deviation of hole diameter
(c) Discuss how part (a) could be answered by constructing a exceeds the hypothesized value by 50%. What is the probabil-
95% two-sided confidence interval for . ity that this difference will be detected by the test described in
9-45. Consider the tire life data in Exercise 8-22. Exercise 9-43?
(a) Can you conclude, using 0.05, that the standard devia- 9-49. Consider the sugar content in Exercise 9-44. Suppose
2
tion of tire life exceeds 200 kilometers? State any necessary that the true variance is 40. How large a sample would be
assumptions about the underlying distribution of the data. required to detect this difference with probability at least 0.90?
(b) Find the P-value for this test.

9-5 TESTS ON A POPULATION PROPORTION

It is often necessary to test hypotheses on a population proportion. For example, suppose that
a random sample of size n has been taken from a large (possibly infinite) population and that
ˆ
X( n) observations in this sample belong to a class of interest. Then P X
n is a point esti-
mator of the proportion of the population p that belongs to this class. Note that n and p are the
parameters of a binomial distribution. Furthermore, from Chapter 7 we know that the sam-
pling distribution of P ˆ is approximately normal with mean p and variance p(1
p) n, if p is
not too close to either 0 or 1 and if n is relatively large. Typically, to apply this approximation
we require that np and n(1
p) be greater than or equal to 5. We will give a large-sample test
that makes use of the normal approximation to the binomial distribution.
9-5.1 Large-Sample Tests on a Proportion

In many engineering problems, we are concerned with a random variable that follows the
binomial distribution. For example, consider a production process that manufactures items
that are classified as either acceptable or defective. It is usually reasonable to model the oc-
currence of defectives with the binomial distribution, where the binomial parameter p repre-
sents the proportion of defective items produced. Consequently, many engineering decision
problems include hypothesis testing about p.
We will consider testing
H : p p 0 (9-31)
0
H : p p 0
1