The number of accidents in a year in a city is $X \sim Po(\lambda)$. Last year there were $N < \lambda$ accidents in the city. Is it true that the accident rate has increased? Study this question for $\lambda = 15$ and $N = 10$.
I think that the hypothesis testing is $$ H_0 : \lambda \leq 15, \quad H_1 : \lambda > 15. $$ Then for acceptance region $$ 0.05 = \alpha = P(X > k | \lambda = 15) = 1 - P(X \leq k | \lambda = 15). $$ I want to compute $P(X \leq k)$ when $\lambda = 15$ but the expression is a Taylor expansion and I cant obtain $k$ with this equation. Is it correct the previous argue?