4-39. The yield of a chemical process is being studied.

From previous experience with this process the standard devi-

ation of yield is known to be 3. The past 5 days of plant oper-
ation have resulted in the following yields: 91.6, 88.75, 90.8,
89.95, and 91.3%. Use a== 0.05.
(a) Is there evidence that the mean yield is not 90%? Use the
P-value approach.
(b) What sample size would be required to detect a true mean
yield of 85% with probability 0.95?
(c) What is the type II error probability if the true mean yield
is 92%?
(d) Find a 95% two-sided CI on the true mean yield.
(e) Use the CI found in part (d) to test the hypothesis.
1 EXERCISES FOR SECTION 4--4
4--42. The life in hours of a thermocouple used in a furnace
is known to be approximately normally distributed, with stan-
dard deviation er == 20 hours. A random sample of 15 thermo-
couples resulted in the following data: 553, 552, 567, 579,
550,541,537,553,552,546,538,553,581,539,529.
(a) Is there evidence to support the claim that mean life
exceeds 540 hours? Use a fixed-level test with a == 0.05.
(b) What is the P-value for this test?
(c) What is the ~-value for this test if the true mean life is 560
hours?
(d) What sample size would be required to ensure that ~ does
not exceed 0.10 if the true mean life is 560 hours?
(e) Construct a 95% one-sided lower CI on the mean life.
(f) Use the CI found in part (e) to test the hypothesis.
2 EXERCISES FOR SECTION 4--4
4.. 44. Suppose that in Exercise 4-42 we wanted to be 95%
confident that the error in estimating the mean life is less than
5 hours. What sample size should we use?

3 EXERCISES FOR SECTION 4--4

4-59. In building electronic circuitry, the breakdown volt-
age of diodes is an important quality characteristic. The break-
down voltage of 12 diodes was recorded as follows: 9.099,
9.174, 9.327, 9.377, 8.471, 9.575, 9.514, 8.928, 8.800, 8.920,
9.913, and 8.306.
(a) Check the normality assumption for the data.
(b) Test the claim that the mean breakdown voltage is less
than 9 volts with a significance level of0.05.
(c) Construct a 95% one-sided upper confidence bound on
the mean breakdown voltage.
(d) Use the bound found in part (c) to test the hypothesis.
(e) Suppose that the true breakdown voltage is 8.8 volts; it is
important to detect this with a probability of at least 0.95.
Using the sample standard deviation to estimate the popu-
lation standard deviation and a significance level of 0.05,
determine the necessary sample size.
4 EXERCISES FOR SECTION 4-5
4.-65. Cloud seeding has been studied for many decades as a
weather modification procedure ( for an interesting study of
this subject, see the article in Technometrics , "A Bayesian
Analysis of a Multiplicative Treatment Effect in Weather
Modification," Vol. 17, 1975, pp. 161-166). The rainfall in
acre-feet from 20 clouds that were selected at random and
seeded with silver nitrate follows: 18.0, 30.7, 19.8, 27.1 , 22.3,
18.8, 31.8, 23.4, 21.2, 27.9, 31.9, 27.1 , 25.0, 24.7, 26.9, 21.8,
29 .2, 34.8, 26. 7, and 31.6.
(a) Can you support a claim that mean rainfall from seeded
clouds exceeds 25 acre-feet? Use a == 0. 01. Find the
P-value.
(b) Check that rainfall is normally distributed.
(c) Compute the power of the test if the true mean rainfall is
2 7 acre-feet.
(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
to be at least O. 9?
(e) Explain how the question in part (a) could be answered by
constructing a one-sided confidence bound on the mean
diameter.
5 EXERCISES FOR SECTION 4 .. 5
4. .68. The sugar content of the syrup in canned peaches is nor-
2 2
mally distributed, and the variance is thought to be a = 18 (mg) .
(a) Test the hypothesis that the variance is not 18 (mg) 2 if a
random sample of n = 10 cans yields a sample standard
deviation of s = 4 mg, using a fixed-level test with a =
0.05. State any necessary assumptions about the underly-
ing distribution of the data.
(b) What is the P-value for this test?
(c) Find a 95% two-sided CI for a.
(d) Use the CI in part ( c) to test the hypothesis.
6 EXERCISES FOR SECTION 4--6
4 . . 7 5. Large passenger vans are thought to have a high
propensity of rollover accidents when fully loaded. Thirty
accidents of these vans were examined, and 11 vans had rolled
over.
(a) Test the claim that the proportion of rollovers exceeds
0.25 with a == 0.10.
(b) Suppose that the true p == 0.35 and a == 0.10. What is the
~-error for this test?
(c) Suppose that the true p == 0.35 and a == 0.10. How large a
sample would be required ifwe want~ == 0.10?
(d) Find a 90% traditional lower confidence bound on the
rollover rate of these vans.
(e) Use the confidence bound found in part (d) to test the hy-
pothesis.
(f) How large a sample would be required to be at least 95%
confident that the error onp is less than 0.02? Use an ini-
tial estimate of p from this problem.
7 EXERCISES FOR SECTION 4--7
4 . . 76. A random sample of 50 suspension helmets used by
motorcycle riders and automobile race-car drivers was sub-
jected to an impact test, and on 18 of these helmets some dam-
age was observed.
(a) Test the hypotheses H 0 : p == 0.3 versus H 1 : p -=I=- 0.3 with
a == 0.05.
(b) Find the P-value for this test.
(c) Find a 95% two-sided traditional CI on the true proportion
of helmets of this type that would show damage from this
test. Explain how this confidence interval can be used to
test the hypothesis in part (a).
(d) Using the point estimate of p obtained from the prelimi-
nary sample of 50 helmets, how many helmets must be
tested to be 95% confident that the error in estimating the
true value ofpis less than 0.02?
(e) How large must the sample be if we wish to be at least
95% confident that the error in estimating p is less than
0. 02, regardless of the true value of p?
8 EXERCISES FOR SECTION 4 . . 7
4--89. Consider the helmet data given in Exercise 4-76.
Calculate the 95% Agresti-Coull two-sided CI from equa-
tion 4-76 and compare it to the traditional CI in the original
•
exercise.

9 EXERCISES FOR SECTION 4--7

4 .. 95. Consider the breakdown voltage of diodes described
in Exercise 4-59.
(a) Construct a 99% PI for the breakdown voltage of a single
diode.
(b) Find a tolerance interval for the breakdown voltage that
includes 99% of the diodes with 99% confidence.

10 EXERCISES FOR SECTION 4--8