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The resulting $t$ statistic from a t-test follows a t-distribution with parameter $k = $ freedom-degrees. For us to compute the probability of $t$ to occur, we use the cumulative distribution function of the t-distribution ($CDF_t$).

For a left-sided test, this is equal to finding the probability on the y-axis of the CDF, which intersects with the given $t$ value.

For a right-sided test, this is equal to 1 - the left-sided p-value.

What about the two-sided test? I am not sure how this p-value is computed using the CDF.

So far I have only seen this method: (from https://stackoverflow.com/questions/45045802/how-to-do-a-one-tail-pvalue-calculate-in-python)

$ p = CDF_t(t , df) * 2$

i.e. the left-sided p-value multiplied by two.. which strikes me as odd, as the p-value ranges from 0 to 1, this equation generates a p-value in the domain [0-2].

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For an easy way of understanding, the two sided $p$ value turns out to be equal to double the smallest one-sided $p$ value. Since these sum to 1, the smallest is in the domain [0-0.5], so the two sided is in the domain [0,1].

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    $\begingroup$ This is easy and (crucially) correct, but why do you say it’s intuitive? Perhaps you can elaborate on that. $\endgroup$
    – Dave
    Apr 30, 2020 at 21:29
  • $\begingroup$ I guess intuitive wasn't the right word to use. Kind of meant easy to remember. Edited that out of the answer $\endgroup$
    – E. Rei
    Apr 30, 2020 at 21:36
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    $\begingroup$ @H.Green that makes sense. But it also means that the equation I wrote in the question is incomplete right? $\endgroup$
    – hirschme
    Apr 30, 2020 at 22:22

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