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I am trying to get prediction intervals around a sampled count variable.

For example, say I want to know the number of letters an apartment building receives every day. Each day I record the count from ten different apartments in the building, out of 100 total apartments, and get for example (toy data) [0, 5, 2, 0, 0, 1, 4, ...], which would represent seven days worth of total letters for those ten apartments. I can collect this over many days. (Note: I realize mail isn't delivered on Sundays, etc, but for now let's just assume it's all the same.) Let's assume this data comes from a Poisson distribution. I know I can get lambda from the data (just take mean). What I'm not sure of is how to calculate those prediction intervals for the full building. In my example, I have sampled only 10% of the building. Do I just multiply the limits of my intervals by 10? I suspect no, but I'm not sure where to make the change.

I want to be able to say, for example, "on this day, 9 letters were delivered to my sample apartments, so I think x many letters were delivered to the whole building, and my 95% prediction interval is (x-y, x+y)". Where my guess is x = 90, but I'm not sure how to get y.

Note: this originally asked about confidence intervals but I was advised to change it being that that isn't really what I was asking.

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  • $\begingroup$ What parameter exactly do you want confidence intervals for? The mean, the total, or perhaps something else? $\endgroup$ – whuber Oct 1 '18 at 16:57
  • $\begingroup$ I want to be able to say, for example, "on this day, 9 letters were delivered to my sample apartments, so I think x many letters were delivered to the whole building, and my 95% confidence interval is (x-y, x+y)". Where my guess is x = 90, but I'm not sure how to get y. $\endgroup$ – lilyrobin Oct 1 '18 at 17:59
  • $\begingroup$ That is a description of a (Poisson) prediction interval. It is not a confidence interval. Please clarify your post accordingly, lest readers answer with confidence interval formulas (which will be incorrect). $\endgroup$ – whuber Oct 1 '18 at 18:03
  • $\begingroup$ I have been searching the Web for "Poisson prediction interval." The first useful hit I investigated (after following many dead ends) is a post I wrote just last year and had totally forgotten :-). See stats.stackexchange.com/questions/260775. I also found a nice example of one wrong way to go about this: stats.stackexchange.com/questions/92443. $\endgroup$ – whuber Oct 1 '18 at 18:18
  • $\begingroup$ Thanks! I'm reading your post and still not exactly sure how to translate it to my case, but I'll keep trying. $\endgroup$ – lilyrobin Oct 1 '18 at 19:03

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