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I calculated the proportion ($\approx$0.12) of people who had a binary variable = true at a specific calendar date in a large geographically-defined population (N $\approx$ 2,000,000).

I used data for the entire population (i.e., sample size / population size = 1).

A reviewer requested that I also report the 95% confidence interval for this proportion.

My understanding is that a confidence interval is needed when estimating a population statistic using a small sample.

However, when the sample is the entire finite population, do I still need to report a confidence interval? If so, why? What would such a confidence interval exactly mean?

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  • $\begingroup$ Is it possible that your measurement (or observation) of this variable could have been erroneous in some cases? That is, are you absolutely sure about all two million of the values? And that every one of them was correctly transcribed and summarized? That every person observed must belong to the population? That not a single person was overlooked? That all observations were made on the correct date? If not, it can still make sense to compute a confidence interval to account for those other uncertainties. $\endgroup$
    – whuber
    Commented Jun 8, 2019 at 18:45
  • $\begingroup$ @whuber The data is from a county-wide administrative registry which has been used by several studies to study the local population. Assumptions about data quality are needed of course. But how would you compute the CI? As Glen_b has shown, the FPC would be zero. $\endgroup$
    – Orion
    Commented Jun 8, 2019 at 20:48
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    $\begingroup$ I was trying to find a constructive way of responding to the reviewer's suggestion. For assessing measurement errors, there is no such thing as a finite population nor would you apply any FPC: you model the errors as random variables. Doing this with full technical rigor is likely not necessary, but what the reviewer (and readers) might welcome would be (1) a statement saying you view this as the entire population, whence all your conclusions apply only to these people at this time; and (2) a brief assessment of the kinds and magnitudes of errors in the results, like those I listed. $\endgroup$
    – whuber
    Commented Jun 8, 2019 at 21:12
  • $\begingroup$ @whuber Thanks! I'll include these in the discussion and the rebuttal letter. $\endgroup$
    – Orion
    Commented Jun 8, 2019 at 21:20

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You're right, if the population you have is the population of interest then you know the population proportion and the error is $0$; there's no sampling variation (every time you draw a sample of size n=N without replacement you get exactly the same value for the proportion -- it's the population proportion).

Tests and intervals are for when you don't have the full population but want to infer information about it on the basis of a sample.

For some reason, some people have difficulty with the concept even when it's explained and still want an interval. One potential strategy if that happens is just to apply the finite population correction to the usual binomial standard error (FPC = $\sqrt\frac{N-n}{N-1}$) which will give 0 when $n=N$.

For some reason, some people find that FPC more compelling than the simpler, more basic explanation.

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