Tolerance intervals for hypergeometric and negative hypergeometric variables

Tolerance intervals for discrete variables are widely used, especially in industrial applications. However, there is no thorough treatment of tolerance intervals when sampling without replacement. This paper proposes methods for constructing one-sided tolerance limits and two-sided tolerance intervals for hypergeometric and negative hypergeometric variables. Equal-tailed tolerance intervals (i.e., tolerance intervals that control the percentages in both tails) are studied followed by a small adjustment to the nominal coverage level to obtain tolerance intervals that control a specified inner percentage of the sampled distribution. The tolerance interval calculations implicitly use confidence bounds for M, the unknown number of elements possessing a certain attribute in the finite population of size N. Three different methods for obtaining such confidence bounds are suggested: a large sample approach, an approach with a continuity correction, and an exact method based on nonrandomization. The intervals are examined for desirable coverage probabilities and expected widths. The methods are also illustrated using some examples.