Biostatistics

278 CHAPTER 7 HYPOTHESIS TESTING
In many statistical inference procedures, the investigator wishes to consider the type
II error as well as the type I error when determining the sample size. To illustrate the
procedure, we refer again to Example 7.9.2.
EXAMPLE 7.10.1
In Example 7.9.2, the hypotheses are
H 0 : m 65; H A : m < 65
The population standard deviation is 15, and the probability of a type I error is set at .01.
Suppose that we want the probability of failing to reject H 0 ðbÞ to be .05 if H 0 is false
because the true mean is 55 rather than the hypothesized 65. How large a sample do we
need in order to realize, simultaneously, the desired levels of a and b?
Solution:
For a ¼ :01 and n ¼ 20; b is equal to .2743. The critical value is 57. Under the
new conditions, the critical value is unknown. Let us call this new critical value
C. Let m 0 be the hypothesized mean and m 1 the mean under the alternative
hypothesis. We can transform each of the relevant sampling distributions of x,
the one with a mean of m 0 and the one with a mean of m 1 to a z distribution.
Therefore, we can convert C to a z value on the horizontal scale of each of the
two standard normal distributions. When we transform the sampling distribution
of x that has a mean of m 0 to the standard normal distribution, we call the z
that results z 0 . When we transform the sampling distribution x that has a
mean of m 1 to the standard normal distribution, we call the z that results z 1 .
Figure 7.10.1 represents the situation described so far.
We can express the critical value C as a function of z 0 and m 0 and also as
a function of z 1 and m 1 . This gives the following equations:
s
C ¼ m 0 z 0 p ffiffiffi
(7.10.1)
n
s
C ¼ m 1 þ z 1 p ffiffiffi
(7.10.2)
n
a
b
m 1 C m 0
x – z
0
z 1
0
z 0
z
FIGURE 7.10.1 Graphic representation of relationships in determination
of sample size to control both type I and type II errors.