On the Exact Two-Sided Tolerance Intervals for Univariate Normal Distribution and Linear Regression

Abstract

Statistical tolerance intervals are another tool for making statistical inference on anunknown population. The tolerance interval is an interval estimator based on the resultsof a calibration experiment, which can be asserted with stated confidence level 1 ? ,for example 0.95, to contain at least a specified proportion 1 ? , for example 0.99, ofthe items in the population under consideration. Typically, the limits of the toleranceintervals functionally depend on the tolerance factors. In contrast to other statisticalintervals commonly used for statistical inference, the tolerance intervals are used relativelyrarely. One reason is that the theoretical concept and computational complexity of thetolerance intervals is significantly more difficult than that of the standard confidence andprediction intervals.In this paper we present a brief overview of the theoretical background and approachesfor computing the tolerance factors based on samples from one or several univariate normal(Gaussian) populations, as well as the tolerance factors for the non-simultaneousand simultaneous two-sided tolerance intervals for univariate linear regression. Such toleranceintervals are well motivated by their applicability in the multiple-use calibrationproblem and in construction of the calibration confidence intervals. For illustration, wepresent examples of computing selected tolerance factors by the implemented algorithmin MATLAB.