Solved – Confusion about confidence interval around mean of poisson distribution

This question is based on this question: How to calculate a confidence level for a Poisson distribution? and its answers.

In that post, the question is "How do I calculate the confidence interval of a poisson distribution with $n = 88$ and $lambda =47.18$?"

The answer came as

$$ lambda pm 1.96sqrt{dfrac{lambda}{n}}, $$

for the upper and lower bounds. It might be noted that this is an approximation which is okay when $nlambda$ is big enough — whatever big enough might be, apparently $4152$ is big enough.

Now, as far as I know — this might be completely wrong since I haven't really properly studied statistics yet — the confidence interval gives you an interval such that the probability that the mean is in this interval is $95%$. So, I'd think the probability of being outside this interval would be $2.5%$? But it's not. So I'm confused.

  • "the confidence interval gives you an interval such that the probability that the mean is in this interval is 95%"

This is not quite true. Indeed, in this framework, the unknown mean is fixed (either it lies inside the interval or it lies outside, there is no probability attached to it). Instead, you should say that the probability that the interval (which is random) covers the true mean equals 95%. Or that there is a 5% chance that the interval does not cover the true mean.

  • "Ok, but the confidence interval with using the approximate technique above gives (45.7467, 48.617). Yet for λ=47.18 I get P(X≥49)=0.41469. That I don't get." (cf. comment below question)

$(45.7467, 48.617)$ is meant to provide information concerning the mean of the Poisson distribution, i.e. $lambda$. Not about an observation from the Poisson distribution.

Perhaps you are interested in a prediction interval

  • my answer to your comment below

$Pr(X > 330)$ is the probability that the first Poisson variable (with mean 326) returns a count larger than the mean of the second Poisson variable. It does not answer your question (which I can rephrase: are the two means equivalent?).

To convince yourself, you can compute Pr(Pois(326) > 326), i.e. just as if the second Poisson has the same mean as the first ("equivalence").

What you might want to do instead is to construct a (90%) confidence interval for the mean difference and check that it lies within some equivalence margins. This is called an equivalence test (for means) and this approach is known as TOST.

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