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296 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

so one set of operating characteristic curves can be used for all problems regardless of the
values of and . From examining the operating characteristic curves or Equation 9-17 and
0
Fig. 9-7, we note that
1. The further the true value of the mean is from , the smaller the probability of
0
type II error
for a given n and . That is, we see that for a specified sample size and
, large differences in the mean are easier to detect than small ones.
2. For a given and , the probability of type II error
decreases as n increases. That
is, to detect a specified difference in the mean, we may make the test more power-
ful by increasing the sample size.

EXAMPLE 9-4 Consider the propellant problem in Example 9-2. Suppose that the analyst is concerned about
the probability of type II error if the true mean burning rate is 51 centimeters per second.
We may use the operating characteristic curves to find
. Note that 51 50 1, n 25,
2, and 0.05. Then using Equation 9-21 gives

0 0 0 0 1
0
d 2

and from Appendix Chart VIa, with n 25, we find that
0.30. That is, if the true mean
burning rate is 51 centimeters per second, there is approximately a 30% chance that this
will not be detected by the test with n 25.

EXAMPLE 9-5 Once again, consider the propellant problem in Example 9-2. Suppose that the analyst would
like to design the test so that if the true mean burning rate differs from 50 centimeters per sec-
ond by as much as 1 centimeter per second, the test will detect this (i.e., reject H : 50)
0
with a high probability, say, 0.90. This is exactly the same requirement as in Example 9-3,
where we used Equation 9-19 to find the required sample size to be n 42. The operating
characteristic curves can also be used to find the sample size for this test. Since
d 0 0
1
2, 0.05 , and
0.10, we find from Appendix Chart VIa that the
0
required sample size is approximately n 40. This closely agrees with the sample size calcu-
lated from Equation 9-19.

In general, the operating characteristic curves involve three parameters:
, d, and n.
Given any two of these parameters, the value of the third can be determined. There are two
typical applications of these curves:
1. For a given n and d, find
(as illustrated in Example 9-3). This kind of problem is often
encountered when the analyst is concerned about the sensitivity of an experiment
already performed, or when sample size is restricted by economic or other factors.
2. For a given
and d, find n. This was illustrated in Example 9-4. This kind of problem
is usually encountered when the analyst has the opportunity to select the sample size
at the outset of the experiment.
Operating characteristic curves are given in Appendix Charts VIc and VId for the one-
sided alternatives. If the alternative hypothesis is either H : or H : ,the abscissa
0
1
0
1
scale on these charts is
0
0 0
d (9-22)