Applied Statistics and Probability for Engineers

332 CHAPTER 10 STATISTICAL INFERENCE FOR TWO SAMPLES

EXAMPLE 10-2 Consider the paint drying time experiment from Example 10-1. If the true difference in mean

drying times is as much as 10 minutes, find the sample sizes required to detect this difference

with probability at least 0.90.

The appropriate value of the abscissa parameter is (since  0 0, and 10)

and since the detection probability or power of the test must be at least 0.9, with 0.05, we

find from Appendix Chart VIcthat nn 1 n 2 11.

Sample Size Formulas

It is also possible to obtain formulas for calculating the sample sizes directly. Suppose that the null

hypothesisH 0 :  1  2  0 is false and that the true difference in means is 1  2 ,

where  0. One may find formulas for the sample size required to obtain a specific value

of the type II error probability for a given difference in means and level of significance .



d

ƒ 1  2 ƒ

(^221
22)

10
282
82
0.88
For the two-sided alternative hypothesis with significance level , the sample size
n 1 n 2 nrequired to detect a true difference in means of with power at least
1 is
n (10-5)
1 z 2
z 22121
222
1  022
This approximation is valid when 1 z 2  1  021 n 121
222 is small compared to.
For a one-sided alternative hypothesis with significance level , the sample size
n 1 n 2 nrequired to detect a true difference in means of ( 0 ) with power
at least 1is
n (10-6)
1 z z 22121
222
1  022
The derivation of Equations 10-5 and 10-6 closely follows the single-sample case in Section
9-2.3. For example, to obtain Equation 10-6, we first write the expression for the -error for
the two-sided alternate, which is
 ±z 2 
 0
B
(^21)
n 1
(^22)
n 2
≤ ±z 2 
 0
B
(^21)
n 1
(^22)
n 2
≤
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