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9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 289

9-15. If we plot the probability of accepting H 0 : 0 9-18. The proportion of residents in Phoenix favoring the
versus various values of and connect the points with a building of toll roads to complete the freeway system is be-
smooth curve, we obtain the operating characteristic curve lieved to be p 0.3. If a random sample of 10 residents
(or the OC curve) of the test procedure. These curves are used shows that 1 or fewer favor this proposal, we will conclude
extensively in industrial applications of hypothesis testing to that p
0.3.
display the sensitivity and relative performance of the test. (a) Find the probability of type I error if the true proportion is
When the true mean is really equal to 0 , the probability of ac- p 0.3.
cepting H 0 is 1 . Construct an OC curve for Exercise 9-8, (b) Find the probability of committing a type II error with this
using values of the true mean of 178, 181, 184, 187, 190, procedure if p 0.2.
193, 196, and 199. (c) What is the power of this procedure if the true proportion
9-16. Convert the OC curve in Exercise 9-15 into a plot of is p 0.2?
the power function of the test. 9-19. The proportion of adults living in Tempe, Arizona,
9-17. A random sample of 500 registered voters in Phoenix who are college graduates is estimated to be p 0.4. To test
is asked if they favor the use of oxygenated fuels year-round this hypothesis, a random sample of 15 Tempe adults is
to reduce air pollution. If more than 400 voters respond posi- selected. If the number of college graduates is between 4 and
tively, we will conclude that at least 60% of the voters favor 8, the hypothesis will be accepted; otherwise, we will
the use of these fuels. conclude that p 0.4 .
(a) Find the probability of type I error if exactly 60% of the (a) Find the type I error probability for this procedure, assum-
voters favor the use of these fuels. ing that p 0.4.
(b) What is the type II error probability if 75% of the voters (b) Find the probability of committing a type II error if the
favor this action? true proportion is really p 0.2.
Hint: use the normal approximation to the binomial.

9-2 TESTS ON THE MEAN OF A NORMAL DISTRIBUTION,
VARIANCE KNOWN

In this section, we consider hypothesis testing about the mean of a single, normal population
2
where the variance of the population is known. We will assume that a random sample X 1 ,
X 2 , p , X n has been taken from the population. Based on our previous discussion, the sample
2
mean X is an unbiased point estimator of with variance n .

9-2.1 Hypothesis Tests on the Mean

Suppose that we wish to test the hypotheses

H : 0
0
(9-7)
H : 0
1
where 0 is a specified constant. We have a random sample X 1 , X 2 , p , X n from a normal pop-
ulation. Since X has a normal distribution (i.e., the sampling distribution of X is normal)
with mean 0 and standard deviation 1n if the null hypothesis is true, we could construct
a critical region based on the computed value of the sample mean , as in Section 9-1.2.
X
It is usually more convenient to standardize the sample mean and use a test statistic based on
the standard normal distribution. That is, the test procedure for H 0 : 0 uses the test statistic

X 0
Z
0
1n (9-8)