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Can we interpret p-value: $$P(T(x)>t\mid H_0:\mu=\mu_0)$$ in terms of usual conditional probability, which some random variable (T - test statistics) is conditioned on an event that some population parameter ($\mu$, which is a random variable) takes a certain value ($\mu_0$ in case of one sample hypothesis)? The, it could be re-written (in this, one sample t-test case) to this form: $$P(T(X)>t\mid \mu=\mu_0)$$

The question is: is it legitimate?

Lil'Lobster
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    The p-value is the probability of observing a value for the test statistic as extreme or more extreme when the null hypothesis is assumed. To the extent that your notation says that it is correct. Keep in mind that there are one-sided p-values as in your case and also two-sided p-values. So if the null hypothesis is two-sided P(Tt) is considered. – Michael R. Chernick Jan 16 '17 at 13:12
  • @MichaelChernick Right, I should explicitly show that I meant two-sided test by taking an absolute value of t (like $|t|$) – Lil'Lobster Jan 16 '17 at 13:18
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    p-values arise in a setting where no assumptions are made about whether a parameter $\mu$ even has a probability. Thus it makes no sense to describe them as conditional probabilities. If you're unsure about this, please read http://stats.stackexchange.com/questions/31/what-is-the-meaning-of-p-values-and-t-values-in-statistical-tests – whuber Jan 16 '17 at 17:13

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