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I understand that in a two-tailed hypothesis test, we must multiply the p-value by two.

i.e. if z=1.95 and it's a one-tailed hypothesis test, our p-value is 0.0256. But, if it's a two-tailed hypothesis test and z=1.95, we must multiply the p-value of 0.0256 by two. Hence, the correct p-value is 0.0512 for the two-tailed hypothesis test.

I can draw it out on the standard normal curve and I understand that we must multiply the p-value by two. But, my question is why we have to multiply by two. What is the conceptual idea behind it?

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  • $\begingroup$ Because you want both sides. Think about doing a $<$ and $>$ in one. with $p$ for each $\endgroup$
    – Alec Teal
    Oct 23, 2015 at 13:28
  • $\begingroup$ Consider what the p value corresponds to: areas under a PDF. When it's 2-tailed, you are taking both the area user the curve to the right of a certain point, and the same area on the other side of the PDF. By symmetry these are the same so you just times 2, for twice the area. $\endgroup$
    – KR136
    May 22, 2016 at 17:36

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$2$ because it is two-tailed.

The test is the probability of seeing that value or something more extreme if the null hypothesis is true. $2$ is more extreme than $1.95$; $-3$ is also more extreme.

So you want $\Pr(Z \ge 1.95)+\Pr(Z \le -1.95)$ which, by the symmetry of the normal distribution is equal to $2\times \Pr(Z \ge 1.95)$

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