Supplementary Exercise 6.96 of IPS7e
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The exercise wording does not give full details about the specific
situation (because the discussion asked for will be essentially the same
regardless of the specific situation). Still it may be helpful to think
about a more specific situation. Let us assume that a series of
measurements X_1,...,X_n are obtained for the difference in purity
obtained on the same samples by the two methods. This could correspond
to the differences obtained from two paired samples (discussed in
Session 7). We would then assume that the observations are i.i.d. 
(independent and identically distributed) with a mean mu that
corresponds to the mean difference between the methods. Our null
hypothesis would therefore be
  H0: mu = 0
and perhaps most naturally (unless a particular direction of the results
was of interest) we would have a two-sided alternative hypothesis
  Ha: mu <> 0

If a test for H0 against Ha gives us P=0.27, we could say

* there is no significant difference in purity between the two catalysts

* the data provide no evidence of a difference between the two catalysts

* our results could very well have arisen by chance alone if there was
no difference between the catalysts.

Note that the difference addressed here is the difference in purity across a
population of samples the two catalysts could be applied to.

This however, and that is the real point of the exercise, is NOT the
same as saying that there is no difference in purity from the two
catalysts. A non-significant test, or generally lack of evidence, does
not allow us to conclude that the null hypothesis is true. It *could* be
true, but it could certainly also be false, in which case our data we
simply not good enough to show that.

So never say "accept the null hypothesis", because the setup does not allow us
to talk about the likelihood of the null hypothesis being true. We will
came back to discuss this issue later (Session 8) in the context of
equivalence testing.