David Jones wrote:
> As said in another post, the null hypothesis is the one which one is
> prepared to accept unless there is strong evidence to the contrary.
No. The null hypothesis is the one which you may be unable to reject,
even probabilistically, unless (a) it is false AND (b) there is enough
data. By De Morgan's law, the negative of this is (a') it is true OR
(b') there is not enough data. As you will be unable to rule out (b'),
(a') cannot be concluded. This means that *no* outcome of a hypothesis
test can be taken as a confirmation of the null hypothesis.
(There are unusual circumstances, such as a parameter with a discrete
range of possible values, in which a null hypothesis can actually be
confirmed. In the standard sort of problem, however, there is only one
_definite_ outcome to a hypothesis test. Either you reject the null or
it hangs around like Banquo's ghost. The best that can be said for the
phrase "accept the null hypothesis" is that _probably_ most of the
professionals using it know that it is a figure of speech that cannot be
taken literally. There is no excuse for using it in front of students.
Even "retain" is dubious - I would suggest the deliberately indirect
"fail to reject".)
The null hypothesis is thus the one that you are prepared not to ever
have a positive confirmation of. Therefore, for instance, in a
safety-related issue, safety should be confirmed by rejecting a null
hypothesis that the strength, combustion temperature, or whatever is
"barely adequate" in favor of an alternative that "there is a margin of
safety". When this cannot be confirmed, the product's safety shold be
considered "unproven" and the product not approved. The one-sided alpha
value is then a bound on the (more important) consumer's risk, not the
producer's risk. Introductory textbooks often advocate doing the
opposite - perhaps by confusion with quality-control, in which
producer's risk is the defining probability, and the "action" is to stop
the production line for recalibration, not to approve the use of what
comes off it.
-Robert Dawson
.
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