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When the null hypothesis $H_0:\beta=0$ is rejected, there are two possible interpretations: (i) the null hypothesis is true but a rare event occurred, or (ii) the null hypothesis is wrong. If $\beta=0$ is the truth, then the null hypothesis is correct and the interpretation is only (i) a rare event occurred. Nevertheless, the textbook says that when the null hypothesis is rejected, the null hypothesis is judged to be wrong. In other words, depending on the value of the true $\beta$, the interpretation of rejecting the null hypothesis will be different. How should this difference in interpretation be considered?

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    $\begingroup$ A frequentist NHST is not a valid grounds for determining if either hypothesis is wrong, which is why you should just say "we are able to reject the null hypothesis" or "we are unable to reject the null hypothesis" - the clunky wording draws attention to the fact the p-value doesn't mean what you want it to mean. This is mainly because it does not account for the prior information, see the question (and answers) on the (in)famous XKCD cartoon stats.stackexchange.com/questions/43339/… $\endgroup$ Commented May 22 at 10:54

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Your initial sentence is not correct.

When the null is rejected, the correct interpretation is along these lines:

If, in the population from which this sample was randomly drawn, the null was true, then it is very unlikely that we would get a test statistic as extreme as the one we got.

We can quantify and specify this, depending on what test we did, what p value we got, and so on.

Your notion of "but a rare event occurred" is known as type 1 error.

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One makes a decision to reject the null hypothesis, i.e., it is judged to be wrong, as the textbook says. It doesn't mean that it is really wrong - there is still a probability that it was rejected incorrectly, but we neglect this for what follows.

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