Therefore,
b¼Pr zb0 : 20 0 : 15
0 : 036
¼Prðzb 1 : 39 Þ
¼ 0 :082 or 8:2%The results above can be represented graphically as shown in Figure 5.1. It
can be seen thatb depends on a specific alternative (e.g.,b is larger for
HA:p¼ 0 :17 or any alternative hypothesis that specifies a value ofpthat is
farther away from 0.25) and from Figure 5.1, if we change the decision-making
rule by using a smallercut point, we would decreaseabut increaseb.
5.2 ANALOGIES
To reinforce some of the definitions or terms that we have encountered, we
consider in this section two analogies: trials by jury and medical screening tests.
5.2.1 Trials by Jury
Statisticians and statistics users may find a lot in common between a court trial
and a statistical test of significance. In a criminal court, the jury’s duty is to
evaluate the evidence of the prosecution and the defense to determine whether a
defendant is guilty or innocent. By use of the judge’s instructions, which pro-
vide guidelines for their reaching a decision, the members of the jury can arrive
at one of two verdicts: guilty or not guilty. Their decision may be correct or
they could make one of two possible errors: convict an innocent person or free
a criminal. The analogy between statistics and trials by jury goes as follows:
Figure 5.1 Graphical display of type I and type II errors.194 INTRODUCTION TO STATISTICAL TESTS OF SIGNIFICANCE