I've been struggling with this question for a week and don't know how to set it up:
The sum of the values obtained in a random sample of size n=5 is to be used to test the null hypothesis that on the average there are more than two accidents per week at a certain intersection (λ>2 for this Poisson population) against the alternative hypothesis that on average the number of accidents is two or less. In this exercise, the researchers have decided to reject the null hypothesis if and only if the sum of the number of accidents is five or less. To recap, H0: λ>2.0; H1 ≤ 2.0; ad the null will be rejected if ∑x≥5.
a) Find the probabilities of Type I errors when λ is 2.2, 2.4, 2.6, 2.8, and 3.0
b) Find the probabilities of Type II errors when λ is 2.0, 1.5, 1.0, and 0.5.
I've got the answers to both, a) is .0375, .0203, .0107, .0055, and .0027, and b) is .9329, .7585, .3840, and .0420, but I don't know how to set up the problem to get to those answers.
Best Answer
Let $Y = \sum_{i=1}^{5} X_{i}$. Note that $Y \sim Poisson (5 \lambda)$.
a) $\lambda = 2.2$
$$ \begin{array}{lcl} P (\,\mbox{Type I Error}\,) &=& P(\,\mbox{Reject $H_{0}$ when $\lambda = 2.2$}\,)\\ &=& P ( Y \leq 5 \,\, \mbox{when $\lambda=2.2$})\\ &=& \sum_{k=0}^{5} \frac{e^{-5(2.2)}[(5)(2.2)]^{k}}{k!} \approx 0.0375 \end{array} $$ The other values of $\lambda$ are similar.
b) $\lambda = 2.0$ $$ \begin{array}{lcl} P (\,\mbox{Type II Error}\,) &=& P(\,\mbox{Fail to reject $H_{0}$ when $\lambda = 2.0$}\,)\\ &=& P ( Y >5 \,\, \mbox{when $\lambda=2.0$})\\ &=& 1- \sum_{k=0}^{5} \frac{e^{-5(2.0)}[(5)(2.0)]^{k}}{k!} \approx 0.9329. \end{array} $$ Again, the other values of $\lambda$ are similar.