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280 CHAPTER 9 TESTS OF HYPOTHESES FOR A SINGLE SAMPLE

9-1.2 Tests of Statistical Hypotheses

To illustrate the general concepts, consider the propellant burning rate problem introduced
earlier. The null hypothesis is that the mean burning rate is 50 centimeters per second, and the
alternate is that it is not equal to 50 centimeters per second. That is, we wish to test

H : 50 centimeters per second
0
H : 50 centimeters per second
1
Suppose that a sample of n 10 specimens is tested and that the sample mean burning

rate is observed. The sample mean is an estimate of the true population mean . A value of
x
the sample mean that falls close to the hypothesized value of 50x centimeters per second

is evidence that the true mean is really 50 centimeters per second; that is, such evidence sup-
ports the null hypothesis H . On the other hand, a sample mean that is considerably different
0
from 50 centimeters per second is evidence in support of the alternative hypothesis H 1 . Thus,
the sample mean is the test statistic in this case.
The sample mean can take on many different values. Suppose that if 48.5 x 51.5, we
will not reject the null hypothesis H : 50 , and if either x 48.5 or x 51.5 , we will
0
reject the null hypothesis in favor of the alternative hypothesis H : 50 . This is illustrated
1
x
in Fig. 9-1. The values of that are less than 48.5 and greater than 51.5 constitute the critical
region for the test, while all values that are in the interval 48.5 x 51.5 form a region for
which we will fail to reject the null hypothesis. By convention, this is usually called the
acceptance region. The boundaries between the critical regions and the acceptance region are
called the critical values. In our example the critical values are 48.5 and 51.5. It is customary
to state conclusions relative to the null hypothesis H . Therefore, we reject H in favor of H 1
0
0
if the test statistic falls in the critical region and fail to reject H otherwise.
0
This decision procedure can lead to either of two wrong conclusions. For example, the
true mean burning rate of the propellant could be equal to 50 centimeters per second.
However, for the randomly selected propellant specimens that are tested, we could observe a
x
value of the test statistic that falls into the critical region. We would then reject the null
hypothesis H in favor of the alternate H 1 when, in fact, H is really true. This type of wrong
0
0
conclusion is called a type I error.
Definition
Rejecting the null hypothesis H when it is true is defined as a type I error.
0

Now suppose that the true mean burning rate is different from 50 centimeters per second, yet
the sample mean falls in the acceptance region. In this case we would fail to reject H 0 when
x
it is false. This type of wrong conclusion is called a type II error.

Definition
Failing to reject the null hypothesis when it is false is defined as a type II error.

Thus, in testing any statistical hypothesis, four different situations determine whether the final
decision is correct or in error. These situations are presented in Table 9-1.