Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 44 / 1 / c / Solution

Under the null, $Y$ has the standard normal distribution, so the Type I error is $\alpha ={\mathbb{P}}_0(Y>1.65)=1-\Phi (1.65)\simeq 0.05$. Under the alternative, the Type II error is $\beta ={\mathbb{P}}_1(Y\le 1.65)=\Phi (1.65-2.49)=\Phi (-0.84)\simeq 0.20$. Hence $\boxed{\alpha \simeq 0.05,\beta \simeq 0.20}$, using the supplied rounded normal probabilities.