Introductory Biostatistics

5.1.3 Errors

Since a null hypothesisH 0 may be true or false and our possible decisions are
whether to reject or not to reject it, there are four possible outcomes or combi-
nations. Two of the four outcomes are correct decisions:

1. Not rejecting a trueH 0


2. Rejecting a falseH 0

but there are also two possible ways to commit an error:

1.Type I:A trueH 0 is rejected.

2.Type II:A falseH 0 is not rejected.

These possibilities are shown in Table 5.1.
The general aim in hypothesis testing is to keepaandb, the probabilities—
in the context of repeated sampling—of types I and II respectively, as small as
possible. However, if resources are limited, this goal requires a compromise
because these actions are contradictory (e.g., a decision to decrease the size ofa
will increase the size ofb, and vice versa). Conventionally, we fixaat some
specific conventional level—say, 0.05 or 0.01—andbis controlled through the
use of sample size(s).

Example 5.1 Suppose that the national smoking rate among men is 25% and
we want to study the smoking rate among men in the New England states. Let
pbe the proportion of New England men who smoke. The null hypothesis that
the smoking prevalence in New England is the same as the national rate is
expressed as

H 0 :p¼ 0 : 25

Suppose that we plan to take a sample of sizen¼100 and use this decision-
making rule:

Ifpa 0 : 20 ;H 0 is rejected;

wherepis the proportion obtained from the sample.

TABLE 5.1

Decision

Truth H 0 Is Not Rejected H 0 Is Rejected

H 0 is true Correct decision Type I error
H 0 is false Type II error Correct decision

192 INTRODUCTION TO STATISTICAL TESTS OF SIGNIFICANCE