Suppose that for data taken from a Normal distribution with unknown mean M and known standard deviation $1,$ we hypothesize that H$0$: M$=0$ versus H$1$: M $!=0.$ Suppose that we conduct a hypothesis test at the $0\mathrm{.}05$ significance level and with sample size N$=100.$

$1)$ Calculate the probability of a Type I error if the true value of M is $0.$

$2)$ Calculate the probability of a Type I error if the true value of M is $1.$