What conclusions can we draw if p>α? Does not rejecting the H0 mean anything?
Statistical hypothesis testing is in some way similar to the technique ‘proof by contradiction’ in mathematics, i.e. if you want to prove something then assume the opposite and derive a contradiction, i.e. something that is impossible.
In statistics ‘impossible’ does not exist, but some events are very ‘improbable’. So in statistics, if you want to ‘prove’ something (i.e. H1) then you assume the opposite (i.e. H0) and if H0 is true you try to derive something improbable. ‘Improbable’ is defined by the confidence level that you choose.
If, assuming H0 is true, you can find something very improbable, then H0 can not be true because it leads to a ‘statistical contradiction’. Therefore H1 must be true.
This implies that in statistical hypothesis testing you can only find evidence for H1. If one can not reject H0 then the only conclusion you can draw is ‘We can not prove H1‘ or ‘we do not find evidence that H0 is false and so we accept H0 (as long as we do not find evidence against it)’.
But there is more … it is also about power.
Obviously, as nothing is impossible, one can draw wrong conclusions; we might find ‘false evidence’ for H1 meaning that we conclude that H0 is false while in reality it is true. This is a type I error and the probability of making a type I error is equal to the signficance level that you have choosen.
One may also accept H0 while in reality it is false, this is a type II error and the probability of making one is denoted by β.
The power of the test is defined as 1−β so 1 minus the probability of making a type II error. This is the same as the probability of not making a type II error.
So β is the probability of accepting H0 when H0 is false, therefore 1−β is the probability of rejecting H0 when H0 is false which is the same as the probability of rejecting H0 when H1 is true.
By the above, rejecting H0 is finding evidence for H1, so the power is 1−β is the probability of finding evidence for H1 when H1 is true.
If you have a test with very high power (close to 1), then this means that if H1 is true, the test would have found evidence for H1 (almost surely) so if we do not find evidence for H1 (i.e. we do not reject H0) and the test has a very high power, then probably H1 is not true (and thus probably H0 is true).
So what we can say is that if your test has very high power , then not rejecting H0 is ”almost as good as” finding evidence for H0.