I'd like to try out an explanation of something on you all: A p-value evaluates an argument that the data proves some claim, by applying the same argument to a fake model where we know the claim is false. The p-value is how often we would erroneously conclude it was true anyway.
Ah. Well, I don’t think this is right: that’s not what the null hypothesis does, and you don’t generally know that the null hypothesis is false. But I’m not expert on this, so I will shut up now!
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No I'm pretty sure it is, and that is what the null hypothesis does. Indeed you don't generally know that the null hypothesis is false (though it almost always is in the sense that all models are false), but the point is you can generate fake data conditional on it being true.
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The null hypothesis is some statistical model under which the alternative hypothesis is false, and the p-value is the probability that if you were to generate a dataset from the null hypothesis then you would get a test statistic at least as extreme as the one you observed.
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We know the claim is false for the null hypothesis; we don't know if the null hypothesis is, but we apply some excluded middle (danger!) between the null and our claim.
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Right, this is what I'm trying to get at with my explanation (and clearly still missing the boat): The null hypothesis gives us a way of evaluating our purported evidence for the claim by saying "Under this model where the claim is false, we'd still claim it was true this often".
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