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In my textbook $p$-values are defined as follows: $$p\text{-value} = P(T \text{ is at least as extreme as } t \text{ given } H_0 \text{ is true}) $$

where $T$ is the test statistic. It goes on to say that

The smaller the $p$-value, the stronger the evidence against $H_0$.

Here is my issue. Surely if the probability that $|T| \geq t$ is smaller then the probability that $|T|<t$ is larger. That is, the values are more likely to congregate about $0$. This would provide greater evidence of the null hypothesis.

User1865345
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HMPtwo
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  • There are already so many posts time and again here that can be used to mark this as a duplicate. Have you checked any one of those? – User1865345 May 29 '23 at 11:11
  • I am having trouble understanding the responses in the other posts. – HMPtwo May 29 '23 at 11:25
  • Maybe it helps to focus on the meaning of small t and capital T: t is what you see in the data. T is what you would see if you do the experiment again and again. If $P(|T|>t)$ is small, then you have observed something rare in your data. – Ute May 29 '23 at 11:53
  • Smaller p provides greater evidence against the null hypothesis. You are writing of "evidence of the null hypothesis" by which you may mean (but I'm not sure) "in its favour!? – Christian Hennig May 29 '23 at 14:32

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