Basic Statistics

110 TESTS OF HYPOTHESES ON POPULATION MEANS

Table 8.4 Decisions and Outcomes

Accept Ho Reject HO

HO true Correct decision Type I error

HO false Type I1 error Correct decision

true population mean, and the size of p is larger if the true population mean is close
to the null hypothesis mean than if they are far apart.
In some texts and statistical programs a type I error is called an a-error and type
I1 error is called a p-error.
Decisions and outcomes are illustrated in Table 8.4. Since we either accept or
reject the null hypothesis, we know after making our decision which type of error
we may have made. Ideally, we would like our chance of making an error to be very
small.
If our decision is to reject the Ho, we made either a correct decision or a type I
error. Note that we set the size of a. The chances of our making a correct decision is
1 - a.
If our decision is to accept the null hypothesis, we have made either a correct
decision or a type I1 error. We do not set the size of p before we make a statistical test
and do not know its size after we accept the null hypothesis. We can use a desired
size of ,!? in estimating the needed sample size as seen in Section 8.5.

8.4.3 An Illustration of p

To illustrate the type I1 error, we will examine a simple hypothetical example. We

have taken a simple random sample of n = 4 observations and we know that 0 = 1

so that = .5. The Ho : p = 0 is to be tested and we set a = .05. First, we

will find the values of 5?; that separate the rejection from the acceptance region. The

value of x corresponding to z = +1.96 is obtained by solving

• x-0

1.96 = -

.5

forz;. Thus, x = 1.96(.5) = .98 corresponds to z = 1.96 and, similarly, x = -.98
corresponds to z = -1.96. Any value of x between -.98 and +.98 would cause
us to accept the null hypothesis that p = 0; thus /3 equals the probability that x
lies between -.98 and +.98 given that we have actually taken the sample from a
population whose p has a true value not equal to p = 0. Let us suppose that the true
value of p is .6.
Figure 8.4(a) represents the normal distribution of the x’s about p = 0 with the
acceptance region in the center and the rejection regions in both tails. Figure 8.4(b)
represents the normal distribution of x’s about the true value of p = .6. The two
normal curves are lined up vertically so that zero on the upper curve is directly above
zero on the lower curve. In Figure 8.4(b), the shaded area is p and the unshaded area