Hypothesis Tests · Challenge problem

An analyst tests $H_0\!: \mu = 50$ against $H_a\!: \mu > 50$ with the statistic $z = \dfrac{\bar x - 50}{10/\sqrt n}$, rejecting when $z > 1.645$ (nominal level $0.05$), because she believes the population standard deviation is $10$. In fact the population is normal with standard deviation $12$, and $H_0$ is true. What is the actual probability of a Type I error? Round to three decimal places.