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4.2 Test Errors and Test Power 115

p = P ( ≤ ( − µ B / ) σ X ) = P ( ≤ ( 1260 − 1300 / ) 77 . 94 ) = . 0 304 .
x
Z
Z

If we are basing our conclusions on a 5% level of significance, and since
p > 0.05, we then have no evidence to reject the null hypothesis.
Note that until now we have assumed that we knew the true value of the
standard deviation. This, however, is seldom the case. As already discussed in the
previous chapter, when using the sample standard deviation – maintaining the
assumption of normality of the random variable − one must use the Student’s t
distribution. This is the usual procedure, also followed by statistical software
products, where these parametric tests of means are called t tests.

4.2 Test Errors and Test Power

As described in the previous section, any decision derived from hypothesis testing
has, in general, a certain degree of uncertainty. For instance, in the drill example
there is always a chance that the null hypothesis is incorrectly rejected. Suppose
that a sample from the good quality of drills has x =1190 hours. Then, as can be
seen in Figure 4.1, we would incorrectly reject the null hypothesis at a 10%
significance level. However, we would not reject the null hypothesis at a 5% level,
as shown in Figure 4.3. In general, by lowering the chosen level of significance,
typically 0.1, 0.05 or 0.01, we decrease the Type I Error:

Type I Error: α = P(H 0 is true and, based on the test, we reject H 0).

The price to be paid for the decrease of the Type I Error is the increase of the
Type II Error, defined as:

Type II Error: β = P(H 0 is false and, based on the test, we accept H 0).

For instance, when in Figures 4.1 and 4.3 we decreased α from 0.10 to 0.05, the
value of β increased from 0.10 to:

β = P ( ≥ (x α − µ A / ) σ X ) = P ( ≥ ( 1172 8 . − 1100 / ) 77 . 94 ) = . 0 177 .
Z
Z

Note that a high value of β indicates that when the observed statistic does not
fall in the critical region there is a good chance that this is due not to the
verification of the null hypothesis itself but, instead, to the verification of a
sufficiently close alternative hypothesis. Figure 4.4 shows that, for the same level
of significance, α, as the alternative hypothesis approaches the null hypothesis, the
value of β increases, reflecting a decreased protection against an alternative
hypothesis.
The degree of protection against alternative hypotheses is usually measured by
the so-called power of the test, 1–β, which measures the probability of rejecting the
null hypothesis when it is false (and thus should be rejected). The values of the
power for several alternative values of µ A, using the computed values of β as