Page 334 - Applied Statistics And Probability For Engineers

c09.qxd 5/15/02 8:02 PM Page 282 RK UL 9 RK UL 9:Desktop Folder:

282 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

From inspection of Fig. 9-2, notice that we can reduce by widening the acceptance

region. For example, if we make the critical values 48 and 52, the value of is
48
50 52
50
P aZ b P aZ b P1Z
2.532 P1Z 2.532
0.79 0.79
0.0057 0.0057 0.0114

We could also reduce by increasing the sample size. If n 16,
1n 2.5
116
0.625, and using the original critical region from Fig. 9-1, we find

48.5
50 51.5
50

2.40 and z 2.40
z 1 2
0.625 0.625
Therefore

P1Z
2.402 P1Z 2.402 0.0082 0.0082 0.0164

In evaluating a hypothesis-testing procedure, it is also important to examine the proba-
bility of a type II error, which we will denote by . That is,

P(type II error) P(fail to reject H 0 when H 0 is false) (9-4)

To calculate (sometimes called the -error), we must have a specific alternative hypothe-
sis; that is, we must have a particular value of . For example, suppose that it is important to
reject the null hypothesis H 0 : 50 whenever the mean burning rate is greater than 52
centimeters per second or less than 48 centimeters per second. We could calculate the proba-
bility of a type II error for the values 52 and 48 and use this result to tell us some-
thing about how the test procedure would perform. Specifically, how will the test procedure
work if we wish to detect, that is, reject H 0 , for a mean value of 52 or 48? Because
of symmetry, it is necessary only to evaluate one of the two cases—say, find the probability of
accepting the null hypothesis H 0 : 50 centimeters per second when the true mean is
52 centimeters per second.
Figure 9-3 will help us calculate the probability of type II error . The normal distribution
on the left in Fig. 9-3 is the distribution of the test statistic X when the null hypothesis
H 0 : 50 is true (this is what is meant by the expression “under H 0 : 50”), and the nor-
mal distribution on the right is the distribution of X when the alternative hypothesis is true and
the value of the mean is 52 (or “under H 1 : 52”). Now a type II error will be committed if
the sample mean X falls between 48.5 and 51.5 (the critical region boundaries) when 52.
As seen in Fig. 9-3, this is just the probability that 48.5 X 51.5 when the true mean is
52, or the shaded area under the normal distribution on the right. Therefore, referring to
Fig. 9-3, we find that

P148.5 X 51.5 when 522