(a) Under \(H_0\), \(X\sim\operatorname{Po}(6.6)\). \[P(X\le2)=e^{-6.6}\left(1+6.6+\frac{6.6^2}{2}\right)=0.0400,\quad P(X\le3)=0.105.\] The largest lower-tail critical region with probability at most 0.05 is \(X\le2\), so the Type I error probability is 0.0400.

(b) \(H_0:m=3.3,\ H_1:m<3.3\). The observation 2 is in the critical region, so reject \(H_0\). There is sufficient evidence that the mean annual number of accidents has decreased.

(c) Under the true mean, \(X\sim\operatorname{Po}(1.2)\). A Type II error occurs when \(X\ge3\): \[P(\text{Type II})=1-e^{-1.2}\left(1+1.2+\frac{1.2^2}{2}\right)=0.121.\]

(d) The 30-year count has distribution \(\operatorname{Po}(18)\), approximated by \(Y\sim N(18,18)\). With continuity correction, \[P(X>10)\approx P(Y>10.5)=1-\Phi\!\left(\frac{10.5-18}{\sqrt{18}}\right)=0.961.\]