# Question on $p$-value for two-sided test

I'm doing some revision here and I think one of the answers in my notes is wrong. It says on my notes the answer is D). Here's the question:

A researcher conducted a large sample two-sided test of the null hypothesis that $u = 100$. She reports a $p$-value of $0.034$.

Which one of the following is correct?

A). The null hypothesis is not rejected at $\alpha = 0.05$.

B). The $95\%$ confidence interval for $u$ would contain $100$.

C). The null hypothesis is not rejected at $\alpha = 0.01$.

D). The $99\%$ confidence interval for $u$ would contain $100$.

...actually I'm also wondering if the question itself is worded wrong because I see two statements that are true. Here's my thoughts:

As it is a two sided test $\alpha = 0.05$ means there is a critical region on both sides of $0.025$

A). This looks true to me as the $p$-value of $0.034$ isn't greater than $0.025$

B). There is no guarantee that a $95\%$ confidence interval will contain $u$ so false.

C). Again this looks true to me as the $p$-value of $0.034$ isnt greater than $\frac{\alpha}{2} = 0.005$ so the null won't be rejected.

D). There is no guarantee that a $99\%$ confidence interval will contain the population mean $u$ so this is false.

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A) and C) are identical (??) –  Xabier Domínguez May 1 '12 at 20:37
oops, that should be 0.01, fixed it now. –  Jim_CS May 1 '12 at 20:44
You should compare the $p$-value with the significance level $\alpha$, not $\alpha/2$, regardless of the fact that the test is one-sided or two-sided. I would say that C) and D) are both true, since the confidence interval can be identified with the acceptance region of the test. –  Xabier Domínguez May 1 '12 at 21:27
I dont see why I would compare the p value with $\alpha$ instead of $\frac{\alpha}{2}$. Doesn't the fact that it is a two sided test mean that there is a critical region on each side where we reject $H_0$ but of which have $\frac{\alpha}{2}$ area? –  Jim_CS May 1 '12 at 21:44

The factor of $2$ doesn't enter into comparing the $p$-value and the significance level. A $p$-value of $0.034$ means that if the null hypothesis were true data at least as extreme as the observed data would have been observed with probability $0.034$. That's enough to reject the null hypothesis at a significance level of $0.05$, independent of how "at least as extreme as" has been defined. The fact that it's a two-sided test means that "at least as extreme as" is taken to mean "as least as far away from $u=100$ on either side", so you're right to say that there are two critical regions, one on either side, and they both have area $\alpha/2$, but the $p$-value likewise takes into account both regions, one on each side, that are at least as far away from $u=100$ as the observed value(s).

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