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.