Assume I have an infinite sequence of natural numbers $n_1, \ldots$ where each number comes from a poisson distribution $P(k)$ with $k$ unknown.
I want to estimate $k$: I take the first $N$ numbers $n_1, \ldots, n_N$ and calculate its mean, $k_0$, and I have a 95% probability that the actual $k$ is in the interval $k_0 \pm 1.96\sqrt{k_0/N}$.
If instead of $N$ numbers I take $5*N$ numbers and calculate a new mean $k_1$, shouldn't the probability of $k$ being in the interval $k_1\pm 1.96\sqrt{k_1/(5*N)}$ be higher than 95% since I'm using more data?
I mean, I expect to trust more a mean taken from 10000000 samples than from 10, but it seems the maths above doesn't want to say it.