I have a problem from a book and I don't know how to justify my answer. I am still confused between the difference of the idea of z test and confidence interval.
I have identified Z and P value as 4 and 6.3E-3, respectively. I am confused as whether to reject the null hypothesis because
(1) $$P_{value}= 6.3E-3<\alpha=0.05$$ OR
(2) because in the confidence interval provided, $\mu_0$ does not lie within it.
On
When the conclusions are the same (as they should be - and as they are in this case), from where could any confusion arise? It amounts to two different ways of doing the same thing.
However, if you find it confusing to think about both, focus on doing the test when you need to do a test (that is, if you're testing, compare $p$ with $\alpha$, as in this case, or a test statistic with a critical region) and only focus on the confidence interval when you need a confidence interval. In normal situations the correspondence between the two needn't intrude into either of those calculations all at all.
Both are correct. The confidence interval essentially describes the range of values of $\mu$ such that you would not reject the null hypothesis.