Basic Statistics

1 12 TESTS OF HYPOTHESES ON POPULATION MEANS

Figure 8.4
Sampling distribution of x when true ,u = .6 with the shaded area ,B.

(a) Sampling distribution of x for ,u = 0 with the rejection region shaded. (b)

size in the two groups is equal, n1 = n2 = n, and that the u in both groups is the

same. As shown in Chapter 7, the variance of 7, - 72 = 2u2/n. For a one-sided

test, the sample size can be obtained from

2u2(z[l - a] + z[l - p])2

(1-11 - 1-1212

n=

and for a two-sided test the approximate formula for n is given by

2441 - a/2] + z[l - P])2

(Pl - P2Y

n=

Since the two-sided test for testing the Ho : 1-11 = ~2 is such a widely used test, the
formula for obtaining the needed sample size for this case is also widely used. From
examining the formula, it is obvious that we have to choose a value for a and p. As
stated previously, cy = .05 and p = .20 are commonly used values; for these values
zjl - a/2] = 1.96 and z[1 - p] = .84 or, more accurately, ,842. Other values of
cy and /3 can be chosen, depending on the seriousness of making type I and type I1
errors.