Applied Statistics and Probability for Engineers

in many practical situations we would not be as concerned with making a type II error if the mean

were “close” to the hypothesized value. We would be much more interested in detecting large

differences between the true mean and the value specified in the null hypothesis.

The type II error probability also depends on the sample size n. Suppose that the null

hypothesis is centimeters per second and that the true value of the mean is

If the sample size is increased from n10 to n16, the situation of Fig. 9-5 results.

The normal distribution on the left is the distribution of when the mean , and the

normal distribution on the right is the distribution of when. As shown in Fig. 9-5,

the type II error probability is

When , the standard deviation of is , and the z-values

corresponding to 48.5 and 51.5 when are

Therefore

0.2119 0.00000.2119

Recall that when and , we found that ; therefore, increasing the

sample size results in a decrease in the probability of type II error.

The results from this section and a few other similar calculations are summarized in the

following table:

n 10  52 0.2643

P 1

 5.60Z

0.80 2 P 1 Z

0.80 2

 P 1 Z

5.60 2

z 1 

48.5 

52

0.625



5.60 and z 2 

51.5 

52

0.625



0.80

 52

n 16 X 
1 n2.5
116 0.625

P 1 48.5X51.5 when  522

X  52

X  50

52.

H 0 :  50

284 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

Acceptance Sample

Region Size     at  52 at 50.5

10 0.0576 0.2643 0.8923

10 0.0114 0.5000 0.9705

16 0.0164 0.2119 0.9445

48 x 52 16 0.0014 0.5000 0.9918

48.5x51.5

48 x 52

48.5x51.5

46

0.6

0.8

0.4

0.2

0

48 50 52 54 56

Under H 0 : = 50 Under H 1 : = 52

Probability density

μ μ

x–

Figure 9-5 The

probability of type II

error when

and n16.

 52

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