perfect remedy factory. Medical tests are expected to diagnose correctly any
disease (and physicians are expected to treat and cure all diseases e¤ectively!).
Another common misconception is the assumption that all tests, regardless of
the disease being tested for, are equally accurate. People are shocked to learn
that a test result is wrong (of course, the psychological e¤ects could be devas-
tating). Another analogy between tests of significance and screening tests exists
here: Statistical tests are also expected to provide a correct decision!
In some medical cases such as infections, the presence or absence of bacteria
and viruses is easier to confirm correctly. In other cases, such as the diagnosis
of diabetes by a blood sugar test, the story is di¤erent. One very simple model
for these situations would be to assume that the variableX(e.g., the sugar level
in blood) on which the test is based is distributed with di¤erent means for the
healthy and diseased subpopulations (Figure 5.2).
It can be seen from Figure 5.2 that errors are unavoidable, especially when
the two means,mHandmD, are close. The same is true for statistical tests of
significance; when the null hypothesisH 0 is not true, it could be a little wrong
or very wrong. For example, for
H 0 :m¼ 10the truth could bem¼12 orm¼50. Ifm¼50, type II errors would be less
likely, and ifm¼12, type II errors are more likely.
5.3 SUMMARIES AND CONCLUSIONS
To perform a hypothesis test, we take the following steps:
- Formulate a null hypothesis and an alternative hypothesis. This would
follow our research question, providing an explanation of what we want
to prove in terms of chance variation; the statement resulting from our
research question forms our alternative hypothesis. - Design the experiment and obtain data.
Figure 5.2 Graphical display of a translational model of diseases.196 INTRODUCTION TO STATISTICAL TESTS OF SIGNIFICANCE