Find the probability of the type I error and type II error

Question

For a Poisson arrival process with parameter $\lambda$ per hour. Let $Y$ be the number of arrivals in a ten-hour period. Find (i) the probability of the type I error ($\alpha$); (ii) the probability of type II error ($\beta$) at $\lambda = 3.2$ for a test with rejection region $RR = \{Y : Y > 24 \}$, in testing $H_0$: $\lambda = 2$ vs $H_1$: $\lambda > 2$. Then (iii) adjust the $RR$ so the significance level is the closest to $0.05$.

what I have already done:

1) I think the way that I to have solve for $\alpha$ is by calculating the $1-P(Y \le 24)$ with $\lambda =20$. which I get $\alpha = .15677$.

2) Then I solved for $\beta$ using $P(Y \le 24)$ with $\lambda = 32$. I get $\beta = .0881$.

3) For the last part I used $1-P(Y \le c)$ guessing which value of $c$ would give me $\alpha =.05$. I got that $c =27$ because it gives me $\alpha = .05248$.

Am I understanding this correctly?

Answer 1

Yes, I think you are correct. For (c) it might be more convincing if you showed that $26$ and $28$ led to values further from $0.05$

As a check, this R code gives your results

> 1 - ppois(24, 20)
[1] 0.1567726
> ppois(24, 32)
[1] 0.08810069
> 1 - ppois(26:28, 20) 
[1] 0.07788678 0.05248071 0.03433352