Disagreement about rejection region for upper-tailed hypothesis test If we look at the solutions for the first problem here and the first problem here, we see both problems are one-tailed tests for the upper-tail. However, in the first paper the rejection region is Z > 1.65, but in the second paper the rejection region is Z > 1.96.
How can this be if both problems have a significance level of 0.05 and both are upper-tailed tests? Shouldn't they both have a rejection region of Z > 1.65? Is the second paper incorrect?
 A: Yes, the second one is incorrect. They have a diagram which clearly shows the area under the standard normal distribution to the right of $z=1.96$ is 0.05, i.e. $P(Z>1.96)=0.05$. It isn't: $P(Z>1.96)=0.025$, and $P(Z>1.65)=0.05$
Here's a bit of background to explain where the 1.96 came from. Typically in science, you perform a hypothesis test against the null hypothesis that there is no change (example 1 true cost of books isn't different from what the bookstore says it is, example 2 true mean of insurance claims isn't different from what the insurance company says it is). Your alternative hypothesis is that it might be different in either direction, so your rejection region is at either end. For a 5% test this would be 2.5% at the left and 2.5% at the right, and that's where the 1.96 comes into play.
The two-sided test is typically viewed as a better way to perform a hypothesis test, because the one-sided test is easier to get significance with if you choose which side to go for (which you can once you get the data). So the more complicated answer is that in neither case should the one-sided test have been used. The first problem used numbers suitable for a one-sided test, whereas the second paper used a numbers for the upper tail rejection region suitable for a two-sided test (but in the process the total rejection region was only 2.5%).
