Advanced High-School Mathematics

SECTION 6.5 Hypothesis Testing 395

to test the accuracy of this claim. This claim can be regarded as a
hypothesis and it is up to us as statisticians to decide whether or
not to reject this hypothesis. The above hypothesis is usually called
thenull hypothesisand is an assertion about the meanμ about the
population of manufactured bolts. This is often written

H 0 : μ= 8. 1.

We have no a priori reason to believe otherwise, unless, of course, we
can find asignificantreason to reject this hypothesis. In hypothesis
testing, one typically doesn’t accept a null hypothesis, one usually
rejects(or doesn’t reject) it on the basis of statistical evidence.

We can see there are four different outcomes regarding the hypothesis
and its rejection. Atype I erroroccurs when a true null hypothesis
is rejected, and atype II erroroccurs when we fail to reject a false
null hypothesis. These possibilities are outlined in the table below.

Reject H 0

Do not reject H 0

H 0 is true H 0 is false

Type I error Correct decision

Correct decision Type II error

Perhaps a useful comparison can be made with the U.S. system of
criminal justice. In a court of law a defendent ispresumed inno-
cent(the null hypothesis), unless proved guilty (“beyond a shadow of
doubt”). Convicting an innocent person is then tantamount to making
a type I error. Failing to convict an guilty person is a type II error.
Furthermore, the language used is strikingly similar to that used in
statistics: the defendent is never found “innocent,” rather, he is merely
found “not guilty.”

It is typical to define the following conditional probabilities:

α = P(rejectingH 0 |H 0 is true),

β = P(not rejectingH 0 |H 0 is false).

Notice that asαbecomes smaller,βbecomes larger, and vice versa.