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 an
unknown population. The tolerance interval is an interval estimator based on the results
of 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, of
the items in the population under consideration. Typically, the limits of the tolerance
intervals functionally depend on the tolerance factors. In contrast to other statistical
intervals commonly used for statistical inference, the tolerance intervals are used relatively
rarely. One reason is that the theoretical concept and computational complexity of the
tolerance intervals is significantly more difficult than that of the standard confidence and
prediction intervals.
In this paper we present a brief overview of the theoretical background and approaches
for computing the tolerance factors based on samples from one or several univariate normal
(Gaussian) populations, as well as the tolerance factors for the non-simultaneous
and simultaneous two-sided tolerance intervals for univariate linear regression. Such tolerance
intervals are well motivated by their applicability in the multiple-use calibration
problem and in construction of the calibration confidence intervals. For illustration, we
present examples of computing selected tolerance factors by the implemented algorithm
in MATLAB.