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    $\begingroup$ This answer is inferior to better procedures that apply a finite population correction -- which is the entire point of the question! A prediction interval is not a correct solution at all, BTW. With a finite population correction, the confidence interval width will shrink by a factor of $\sqrt{1-1/3}=0.82.$ $\endgroup$ Commented Aug 13, 2020 at 14:21
  • $\begingroup$ @whuber What about the method using the Hypergeometric Distribution? See WRIGHT, Tommy. Exact confidence bounds when sampling from small finite universes: an easy reference based on the hypergeometric distribution. Springer Science & Business Media, 1991 $\endgroup$ Commented Aug 31, 2020 at 14:54