Using tolerance bounds in scientific investigations

Assessment of the variability in population values plays an important role in the analysis of scientific data. Analysis of scientific data often involves developing a bound on a proportion of a population. Sometimes simple probability bounds are obtained using formulas involving known mean and variance parameters and replacing the parameters by sample estimates. The resulting bounds are only approximate and fail to account for the variability in the estimated parameters. Tolerance bounds provide bounds on population proportions which account for the variation resulting from the estimated mean and variance parameters. A beta content, gamma confidence tolerance interval is constructed so that a proportion beta of the population lies within the region bounded by the interval with confidence gamma. An application involving corrosion measurements is used to illustrate the use of tolerance bounds for different situations. Extensions of standard tolerance intervals are applied to generate regression tolerance bounds, tolerance bounds for more general models of measurements collected over time, and tolerance intervals for varying precision data. Tolerance bounds also provide useful information for designing the collection of future data