Applied Statistics and Probability for Engineers

so one set of operating characteristic curves can be used for all problems regardless of the

values of  0 and. From examining the operating characteristic curves or Equation 9-17 and

Fig. 9-7, we note that

1. The further the true value of the mean is from  0 , the smaller the probability of
   type II error for a given nand . 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 nincreases. 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

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 0 :  50)

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

, and 

 0.10, we find from Appendix Chart VIathat the

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 nand 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 VIcand VIdfor the one-
   sided alternatives. If the alternative hypothesis is either H 1 :   0 or H 1 :   0 , the abscissa
   scale on these charts is

d (9-22)

0  00

d 0  00
 1
2, 0.05

d

0  00



00



1

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296 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

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