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

8. Conclusion: Since z 0 2.52 1.645, 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 12.522 0.0059

Therefore, H 0 : 1 2 would be rejected at any significance level 0.0059.
2
2
When the population variances are unknown, the sample variances and can be substituted
s 1
s 2
into the test statistic Equation 10-2 to produce a large-sample test for 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 VId may 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 VIa and VIb is d, where

ƒ ƒ ƒ ƒ
2
0
1
0
d (10-3)
2 2 2 2
2 1
2 2 1
2
and one must choose equal sample sizes, say, n n 1 n 2 . The one-sided alternative hypothe-
ses require the use of Charts VIc and VId. For the one-sided alternatives H 1 : 1 2 0 or
H 1 : 1 2 0 , the abscissa scale is also given by
ƒ ƒ ƒ ƒ
0
0
2
1
d
2
2
2
2 2 2
2 2
1
1
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
n , the operating characteristic curves
2
1
may be entered with an equivalent value of n computed from
2 2
1
2
n 2 2 (10-4)
1 n
2 n 2
1
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 d to obtain . If we are
given d and 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 n in Equation 10-4, and enter the curves with the specified
value of d to 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.