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    $\begingroup$ Given the null hypothesis, you want to know the probability of the null hypothesis? $\endgroup$
    – Dave
    Commented Jul 5 at 16:55
  • $\begingroup$ "Have alternative statistics been suggested, the probability of which given the null hypothesis is the value to be considered for rejecting (or not) the null hypothesis?" Given what you acknowledge in your third dotpoint, the meaning of your question is unclear. Please clarify what you are requesting. $\endgroup$
    – Graham Bornholt
    Commented Jul 5 at 20:53
  • $\begingroup$ I think that all of your bulleted items are false. It is a statistical model that needs to be introduced for the first, not 'a random sample'. Think of resampling tests that yield p-values from an assumption of random treatment allocation instead of sampling from a population. $\endgroup$
    – Michael Lew
    Commented Jul 5 at 21:16
  • $\begingroup$ ...cont. For the second bullet, the probability of exactly the experimenter's results is often (in theory) exactly zero. If you are interested in the probability of the results then you should be interested in the likelihood function, not a p-value. $\endgroup$
    – Michael Lew
    Commented Jul 5 at 21:17
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    $\begingroup$ P-values are not difficult to understand, but most attempts to introduce them to students are almost unbelievably bad, and they are usually contaminated with non-p-value stuff about error rate accounting of the all-or-none significant/not significant error rate stuff. $\endgroup$
    – Michael Lew
    Commented Jul 5 at 21:22