I'm in an intro stats class, and I'm wondering how I can argue or prove the question below regarding sample size and poisson distribution.
Suppose a company which produces fire alarms has claimed that the fire alarms make only one false alarm per year, on average. Let $X$ denote the number of false alarms per year. Assume $X \sim Poisson(\lambda)$. Under the company's claim, the probability of observing $x$ fire alarms per year is
$P(X = x) = \frac{e^{-\lambda}\lambda^x}{x!} = \frac{e^{-1}}{x!}, x = 0,1,...$
A customer had a bad experience with the fire alarm he purchased before. He allows 1% chance for falsely rejecting $H_0$. He gathers one hundred people who observed three or more fire alarms and observed:
$(X_1,X_2,...,X_{100}) = (4,6,...,3)$ with
$\bar{X}_{100} = \frac{1}{100} \sum_{i=1}^{100}X_i = 3.32$
a. Ignoring any flaw of data collection, calculate the test statistic (which is compared to the standard normal distribution) and the approximate p-value for testing
$H_0 : \lambda = 1$ versus $H_1 : \lambda > 1$
Also note that the population mean and the population variance are equal to $\lambda = 1$
b. In two sentences, argue why the sample size of $n = 100$ is not useful for the hypothesis testing.
a. For part a, I am doing this, because I am supposed to use a normal distribution.
$T = \frac{\bar{X} - \mu}{\sqrt{\frac{\sigma^2}{n}}} = \frac{\bar{X} - \lambda}{\sqrt{\frac{\lambda^2}{n}}} = \frac{3.32 - 1}{\sqrt{\frac{1}{100}}} = 23.2$
The p-value I got (using R) was just $qnorm(0.99,0,1) \approx 2.326$
b. Part b is where I get confused. I read that having a large sample size is actually good when testing a hypothesis. However I also realize that as $n$ increases, so does the testing statistic while $\lambda$ remains fixed. I'm just wondering what can I argue mathematically.. and why is it like this for Poisson distributions?