Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / O / O4 / b / iv

At level $0.05$ the two-sided critical values are $\pm 1.96$. Since $|-1.89|<1.96$, do not reject the two-sided null. The approximate two-sided $p$-value is $2\Phi (-1.89)\approx 0.059$.

For the specified alternative $q>p$, the statistic $\hat{p}-\hat{q}$ should be negative. Reject in the lower tail, $Z<-1.64$. Since $-1.89<-1.64$, reject for this one-sided alternative, with approximate $p$-value $\Phi (-1.89)\approx 0.029$. The direction of the tail follows the order of subtraction in the statistic.