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274 CHAPTER 7 HYPOTHESIS TESTING
Solution:
When we study the power of a test, we locate the rejection and nonrejection
regions on the x scale rather than the z scale. We find the critical values
of x for a two-sided test using the following formulas:
x U = m 0 + z s 1n
(7.9.1)
and
x L = m 0 - z s 1n
(7.9.2)
where x U and x L are the upper and lower critical values, respectively, of x;
+z and -z are the critical values of z; and m 0 is the hypothesized value of
m. For our example, we have
and
x U = 17.50 + 1.96 13.62
1102
= 17.50 + .7056 = 18.21
x L = 17.50 - 1.961.362 = 17.50 - .7056 = 16.79
Suppose that H 0 is false, that is, that m is not equal to 17.5. In that case,
m is equal to some value other than 17.5. We do not know the actual value of
m. But if H 0 is false, m is one of the many values that are greater than or
smaller than 17.5. Suppose that the true population mean is m 1 = 16.5. Then
the sampling distribution of x 1 is also approximately normal, with
m x = m = 16.5. We call this sampling distribution f1x 1 2, and we call the sampling
distribution under the null hypothesis f1x 0 2.
b, the probability of the type II error of failing to reject a false null
hypothesis, is the area under the curve of f1x 1 2 that overlaps the nonrejection
region specified under H 0 . To determine the value of b, we find the
area under f1x 1 2, above the x axis, and between x = 16.79 and x = 18.21.
The value of b is equal to P116.79 … x … 18.212 when m = 16.5. This is
the same as
16.79 - 16.5
Pa
.36
… z …
= 17.50 + 1.961.362
18.21 - 16.5
b = Pa .29
.36
.36 … z … 1.71
.36 b
= P1.81 … z … 4.752
L 1 - .7910 = .2090
Thus, the probability of taking an appropriate action (that is, rejecting
H 0 ) when the null hypothesis states that m = 17.5, but in fact m = 16.5, is