Past exam of the options II course of the University of Oxford / 2010 / O / O4 / b / iv / Solution

For a two-sided 5% test, the critical values are $\pm 1.96$. Since $|-1.89|<1.96$, do not reject the null at the 5% level.

For the prespecified directional alternative $q>p$, the numerator $\hat{p}-\hat{q}$ tends to be negative. Use the lower-tail critical region $Z<-1.64$ at the 5% level. Since $-1.89<-1.64$, reject in favor of $q>p$ in the one-sided test. The direction should be chosen from the additional scientific information rather than selected after inspecting the sign of this same statistic.