Let $X_1, ..., X_n$ be a random sample from a Poisson distribution with parameter $\lambda$.
If $X$ is Poisson with parameter $\lambda$, then $\text{E}[X] = \text{Var}[X] = \lambda$; hence $\text{E}[X] = \lambda$ and $\text{Var}[X] = \lambda/n$.
(a) Test $H_0 :\lambda=4$ versus $H_A : \lambda>4$ and that our test procedure is to reject $H_0$ if $x \ge k$. Given $n = 100$, what value of $k$ gives a test with significance level approximately equal to $0.05$?
(b) What is the approximate power of the test at $\lambda = 5$?
Hi all, I'm new to this so any help is appreciated. For (a), I tried setting up a $95%$ confidence interval because I thought that $95%$ confidence and $5%$ significance were about the same.
So first to find test with significance level of at approximately 0.05 we see that we want $$P_{\lambda\in\Theta_{Ho}}(\bar{X}\geq k)=P_{\lambda=4}\left(\sum X\geq 100k \right) = 1-P_{\lambda=4}\left(\sum X< 100k\right)=0.05$$ Where we see that $\sum X\sim \operatorname{Pois}(100\cdot4)=\operatorname{Pois}(400)$ Thus in order to find when this is true it can be a little tedious but you are going to have to calculate a list of possible $100k=t$'s and calculate $1-P(\sum X< t)$ (remember for $\operatorname{Pois}(400)$ and trying doing on excel) and see which is closest to $0.05$ (not above though). after you get $t^{*}$ that does this call it $t^{*}$ then $k=\frac{t^{*}}{100}$.
Now to calculate Power we just calculate $$\operatorname{power}(\lambda=5)=P_{\lambda=5}\left(\sum X\geq t^{*}\right) = 1-P_{\lambda=5}\left(\sum X< t^{*}\right)$$ where now $\sum X\sim \operatorname{Pois}(100\cdot5)=\operatorname{Pois}(500)$
also note $\sum X=\sum_{i=1}^{100} X_i$
Also to your question about confidence interval, it is true you could calculate a $95\%$ confidence interval and that would correspond to $5\%$ significance level but the problem with trying to do this is that I don't really know what CI interval is for mean in poisson and also even if you could derive it, CI intervals still rely on some critical value and since you can't really standardize or have table to look at for that it becomes situation above where you just have calculate probabilities for list of values and see which ones closets to significance level