Applied Statistics and Probability for Engineers

where denotes the probability to the left of zin the standard normal distribution. Note

that Equation 9-17 was obtained by evaluating the probability that Z 0 falls in the interval

when H 1 is true. Furthermore, note that Equation 9-17 also holds if , due

to the symmetry of the normal distribution. It is also possible to derive an equation similar to

Equation 9-17 for a one-sided alternative hypothesis.

Sample Size Formulas

One may easily obtain formulas that determine the appropriate sample size to obtain a partic-

ular value of for a given and. For the two-sided alternative hypothesis, we know from

Equation 9-17 that

or if  0,

(9-18)

since when is positive. Let zbe the 100upper percentile of the

standard normal distribution. Then,. From Equation 9-18

or

zz    

2

 1 n

 1

 z 2

 1
z

2
 1 n
2  0

 az  

2

 1 n

b

 az  

2

 1 n

b

^ a^ z^2

 1 n

b

3

 z  

2 , z   

24  0

 1 z 2

The distribution of the test statistic Z 0 under both the null hypothesis H 0 and the alternate

hypothesis H 1 is shown in Fig. 9-7. From examining this figure, we note that if H 1 is true, a

type II error will be made only if where. That is, the

probability of the type II error is the probability that Z 0 falls between and given

that H 1 is true. This probability is shown as the shaded portion of Fig. 9-7. Expressed mathe-

matically, this probability is

z   

2 z     

2

z

2 Z 0 z

2 Z 0 N 1  1 n , 1 2

294 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

 az  

2

 (9-17)

 1 n

b

^ a^ z^2

 1 n

b

Under H 0 : = μμ 0 Under H 1 : μ μ≠ (^0)
N(0,1)
–z (^) α/ 2 0 z (^) α / 2 δ√n
σ
δ√n
N(σ, (^1) (
β
Z 0
Figure 9-7 The
distribution of Z 0
under H 0 and H 1.
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