Stat 6623 – Statistical Methods with SAS Programming
More on hypothesis testing: adapted from “Intro Stats” by Richard DeVeaux and Paul
Velleman (Addison Wesley).
Hypothesis testing is very much like a court trial. Suppose the defendant has been
accused of robbery. We wonder whether he is guilty or not. In U.S. law, the null
hypothesis is that the defendant is innocent. Instructions to juries are explicit about
this. In the trial the prosecutor gathers and presents evidence. This evidence takes the
form of facts that seem to contradict the presumption of innocence. (“If the defendant
were innocent, wouldn’t it be remarkable that the police found him at the scene of the
crime with a big bag of money in his hand, a mask on his face, and a getaway car parked
outside?”)
The next step is to judge the evidence. The jury considers the evidence in light of the
presumption of innocence and judges whether the evidence against him would be plausible
if the defendant were in fact innocent. We ask, “Could these data plausibly have happened
by chance if the null hypothesis were true?” If they were very unlikely to have occurred,
then the evidence raises reasonable doubt in our minds about the null hypothesis.
If the evidence is not strong enough to reject the defendant’s presumption of innocense,
what verdict does the jury return? They say “not guilty.” They do not say the defendant
is innocent. All they say is they have not seen sufficient evidence to convict (reject
innocence).
o
The jury’s null hypothesis is H : innocent defendant. If the evidence is too unlikely given
this assumption, the jury rejects the null hypothesis and finds the defendant guilty. If
o
there is insufficient evidence to convict the defendant, the jury does not decide that H
is true and declare him innocent. Rather, juries can only fail to reject the null hypothesis
and declare the defendant “not guilty.”
In the same way, if the data are not particularly unlikely under the assumption that the
null hypothesis is true, then the most we can do is to “fail to reject” the null hypothesis.
We never declare the null hypothesis to be true, because we simply do not know whether
it’s true or not. Sometimes in this case we say the null hypothesis has been retained.
