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If I have to test a hypothesis like:

\begin{align} H_0:\mu\leq10\\H_1:\mu>10\\ Z_{1-p}=\dfrac{\bar{X}-\mu}{\sigma/\sqrt n} \end{align} From which I get p-value and I make a comment about whether it makes sense to reject $H_0$.

What happens for double sided tests? $\mu=10,\mu\neq10$.

Do I simply take $2\times p$ from the above and then make comments?

If yes, why are we doing that?

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closed as off-topic by Austin Mohr, Sami Ben Romdhane, TooTone, Shuchang, 6005 Feb 28 '14 at 1:14

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Yes. Ours is not to question why, ours is but to do or die. – Code-Guru Dec 14 '12 at 2:26
up vote 0 down vote accepted

Now for a more serious answer.

You are correct that you take $2p$ where $p$ is the value from a table of normal distributions. This value represents the area under the curve of one tail of the normal distribution. With a double sided test, we need the area under the curve of both tails and so multiply by 2.

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But why assume that the area of the tails will be the same? – Inquest Dec 14 '12 at 2:34
@Inquest Because a normal distribution is symmetric about the y-axis. That is we can show that the tails are the same from the definition of the normal distribution; it's not an assumption. – Code-Guru Dec 14 '12 at 2:35
If we were doing a two-tailed test on non-normal data (I've heard such a thing exists :P ), the calculation would be less simple, right? – Ben Millwood Dec 14 '12 at 2:42
@BenMillwood yes. – Code-Guru Dec 14 '12 at 2:50
So, how would I test $\sigma=10, \sigma\neq 10$ using chi-squared. (Two sided of course). – Inquest Dec 14 '12 at 3:18

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