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9-1 HYPOTHESIS TESTING 285

The results in boxes were not calculated in the text but can easily be verified by the
reader. This display and the discussion above reveal four important points:

1. The size of the critical region, and consequently the probability of a type I error ,
can always be reduced by appropriate selection of the critical values.
2. Type I and type II errors are related. A decrease in the probability of one type of error
always results in an increase in the probability of the other, provided that the sample
size n does not change.
3. An increase in sample size will generally reduce both and , provided that the
critical values are held constant.
4. When the null hypothesis is false, increases as the true value of the parameter
approaches the value hypothesized in the null hypothesis. The value of decreases
as the difference between the true mean and the hypothesized value increases.
Generally, the analyst controls the type I error probability when he or she selects the
critical values. Thus, it is usually easy for the analyst to set the type I error probability at
(or near) any desired value. Since the analyst can directly control the probability of
wrongly rejecting H 0 , we always think of rejection of the null hypothesis H 0 as a strong
conclusion.
On the other hand, the probability of type II error is not a constant, but depends on
the true value of the parameter. It also depends on the sample size that we have selected.
Because the type II error probability is a function of both the sample size and the extent to
which the null hypothesis H 0 is false, it is customary to think of the decision to accept H 0 as a
weak conclusion, unless we know that is acceptably small. Therefore, rather than saying we
“accept H 0 ”, we prefer the terminology “fail to reject H 0 ”. Failing to reject H 0 implies that we
have not found sufficient evidence to reject H 0 , that is, to make a strong statement. Failing to
reject H 0 does not necessarily mean that there is a high probability that H 0 is true. It may
simply mean that more data are required to reach a strong conclusion. This can have impor-
tant implications for the formulation of hypotheses.
An important concept that we will make use of is the power of a statistical test.

Definition
The power of a statistical test is the probability of rejecting the null hypothesis H 0
when the alternative hypothesis is true.

The power is computed as 1
, and power can be interpreted as the probability of
correctly rejecting a false null hypothesis. We often compare statistical tests by comparing
their power properties. For example, consider the propellant burning rate problem when we
are testing H : 50 centimeters per second against H : 50 centimeters per second.
1
0
Suppose that the true value of the mean is 52 . When n 10, we found that 0.2643,
so the power of this test is 1
1
0.2643 0.7357 when 52 .
Power is a very descriptive and concise measure of the sensitivity of a statistical test,
where by sensitivity we mean the ability of the test to detect differences. In this case, the
sensitivity of the test for detecting the difference between a mean burning rate of 50 centime-
ters per second and 52 centimeters per second is 0.7357. That is, if the true mean is really
: 50 and “detect” this differ-
52 centimeters per second, this test will correctly reject H 0
ence 73.57% of the time. If this value of power is judged to be too low, the analyst can increase
either or the sample size n.