So I have a question regarding hypothesis tests where i have to justify a claim with statistical evidence. It is as follows:
The average number of accidents in previous years in a city has been 15 per year, but this year it was only 10. It is assumed that the number of accidents follows a Poisson distribution. Is it justified to claim that the accident rate has dropped?
So my thinking is to test the hypotheses $H_0 : \lambda = 15$ against $H_a : \lambda <15$ using the pdf of a poison distribution, i.e. $f(x) = \frac{\lambda^xe^{-\lambda}}{x!}$. But not sure how to proceed as there is no critical function given. Should i formulate my own with my own chosen significance level? I could then work out the probabilities of type 1 and 2 errors but don't know how this would help. I'm thinking i'd need a p-value or something but not sure how i'd calculate one.
Suggestions and hints only please as this is homework and i'd like to get there myself, thanks.
You have chosen to test the hypothesis $$H_0 : \lambda = 15 \quad \text{vs.} \quad H_a : \lambda < 15.$$ So, under the assumption of the null hypothesis, what is the probability of observing a result at least as extreme as $X = 10$? That is to say, what is $$\Pr[X \le 10 \mid H_0] = \sum_{k=0}^{10} e^{-15} \frac{15^k}{k!}?$$ Consequently, what would you conclude about this observation? If we set our Type I error to be $\alpha = 0.05$, would we reject or fail to reject $H_0$? Is there sufficient evidence to suggest that the true rate is less than $15$?