# 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.

• 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
• It doesn't matter how many pieces your critical region divides up into. If your test statistic falls into any one of them, your null hypothesis should be rejected. The statement that the test is two-sided here seems to me to be a red herring. I agree with Xabier Dominguez, that the null hypothesis should be rejected if $p\le \alpha$, and that A) and B) are both false and C) and D) are both true. – lonza leggiera Jan 15 '19 at 0:20

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).