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Statement: The lower the p-value corresponding to some test statistic, the greater the evidence against the null hypothesis.

I see this statement everywhere, I would argue that this statement is false for the simple reason that "evidence against the null hypothesis" can only translate into its probabilistic equivalent $P(\text{null hypothesis being true| the data})$. Ofcourse this probability must be unknown because p-values deal with the reverse conditional probability--- the test statistic exceeding the observed value assuming that the null is true.

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