EXERCISES FOR SECTION 9-4310 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE9-43. Consider the rivet holes from Exercise 8-35. If the
standard deviation of hole diameter exceeds 0.01 millimeters,
there is an unacceptably high probability that the rivet will not
fit. Recall that n15 and s0.008 millimeters.
(a) Is there strong evidence to indicate that the standard devi-
ation of hole diameter exceeds 0.01 millimeters? Use
0.01. State any necessary assumptions about the underly-
ing distribution of the data.
(b) Find the P-value for this test.
(c) If is really as large as 0.0125 millimeters, what sample size
will be required to defect this with power of at least 0.8?
9-44. Recall the sugar content of the syrup in canned peaches
from Exercise 8-36. Suppose that the variance is thought to be
2 18 (milligrams)^2. A random sample of n10 cans yields
a sample standard deviation of s4.8 milligrams.
(a) Test the hypothesis H 0 : 2 18 versus H 1 : 2 18 using
0.05.
(b) What is the P-value for this test?
(c) Discuss how part (a) could be answered by constructing a
95% two-sided confidence interval for.
9-45. Consider the tire life data in Exercise 8-22.
(a) Can you conclude, using 0.05, that the standard devia-
tion of tire life exceeds 200 kilometers? State any necessary
assumptions about the underlying distribution of the data.
(b) Find the P-value for this test.9-46. Consider the Izod impact test data in Exercise 8-23.
(a) Test the hypothesis that 0.10 against an alternative
specifying that 0.10, using 0.01, and draw a
conclusion. State any necessary assumptions about the
underlying distribution of the data.
(b) What is the P-value for this test?
(c) Could the question in part (a) have been answered by
constructing a 99% two-sided confidence interval for 2?
9-47. Reconsider the percentage of titanium in an alloy used
in aerospace castings from Exercise 8-39. Recall that s0.37
and n51.
(a) Test the hypothesis H 0 : 0.25 versus H 1 : 0.25
using 0.05. State any necessary assumptions about
the underlying distribution of the data.
(b) Explain how you could answer the question in part (a) by
constructing a 95% two-sided confidence interval for.
9-48. Consider the hole diameter data in Exercise 8-35.
Suppose that the actual standard deviation of hole diameter
exceeds the hypothesized value by 50%. What is the probabil-
ity that this difference will be detected by the test described in
Exercise 9-43?
9-49. Consider the sugar content in Exercise 9-44. Suppose
that the true variance is 2 40. How large a sample would be
required to detect this difference with probability at least 0.90?9-5 TESTS ON A POPULATION PROPORTIONIt is often necessary to test hypotheses on a population proportion. For example, suppose that
a random sample of size nhas been taken from a large (possibly infinite) population and that
X(n) observations in this sample belong to a class of interest. Then is a point esti-
mator of the proportion of the population pthat belongs to this class. Note that nand pare the
parameters of a binomial distribution. Furthermore, from Chapter 7 we know that the sam-
pling distribution of is approximately normal with mean pand variance p(1 p)n, if pis
not too close to either 0 or 1 and if nis relatively large. Typically, to apply this approximation
we require that npand 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 ProportionIn 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 prepre-
sents the proportion of defective items produced. Consequently, many engineering decision
problems include hypothesis testing about p.
We will consider testing(9-31)
H 1 : pp 0H 0 : pp 0PˆPˆX n
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