Applied Statistics and Probability for Engineers

10-2 INFERENCE FOR A DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES KNOWN 331

8. Conclusion: Since z 0 2.521.645, we reject H 0 :  1  2 at the 0.05 level
   and conclude that adding the new ingredient to the paint significantly reduces the
   drying time. Alternatively, we can find the P-value for this test as

P-value 1 

Therefore, H 0 :  1  2 would be rejected at any significance level 0.0059.

When the population variances are unknown, the sample variances and can be substituted

into the test statistic Equation 10-2 to produce a large-sample testfor the difference in means.

This procedure will also work well when the populations are not necessarily normally distrib-

uted. However, both n 1 and n 2 should exceed 40 for this large-sample test to be valid.

10-2.2 Choice of Sample Size

Use of Operating Characteristic Curves

The operating characteristic curves in Appendix Charts VIa, VIb, VIc, and VIdmay be used

to evaluate the type II error probability for the hypotheses in the display (10-2). These curves

are also useful in determining sample size. Curves are provided for 0.05 and 0.01.

For the two-sided alternative hypothesis, the abscissa scale of the operating characteristic

curve in charts VIaand VIbis d, where

(10-3)

and one must choose equal sample sizes, say, nn 1 n 2. The one-sided alternative hypothe-

ses require the use of Charts VIcand VId. For the one-sided alternatives H 1 :  1  2  0 or

H 1 :  1  2   0 , the abscissa scale is also given by

It is not unusual to encounter problems where the costs of collecting data differ substantially

between the two populations, or where one population variance is much greater than the other.

In those cases, we often use unequal sample sizes. If n 1

 n 2 , the operating characteristic curves

may be entered with an equivalentvalue of ncomputed from

(10-4)

If n 1

 n 2 , and their values are fixed in advance, Equation 10-4 is used directly to calculate n,

and the operating characteristic curves are entered with a specified dto obtain. If we are

given dand it is necessary to determine n 1 and n 2 to obtain a specified , say, *, we guess at

trial values of n 1 and n 2 , calculate nin Equation 10-4, and enter the curves with the specified

value of dto find. If *, the trial values of n 1 and n 2 are satisfactory. If *,

adjustments to n 1 and n 2 are made and the process is repeated.

n

(^21
22)
(^21) n 1
22 n 2
d
ƒ 1  2  0 ƒ
212
22

ƒ 0 ƒ
212
22
d
ƒ 1  2  0 ƒ
(^221
22)

ƒ 0 ƒ
(^221
22)
s^21 s 22
 1 2.52 2 0.0059
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