Abstract. Since simple linear regression theory was established at the beginning of the 1900s, it has been used in a variety of fields. Unfortunately, it cannot be used directly for calibration. In practical calibrations, the observed measurements (the inputs) are subject to errors, and hence they vary, thus violating the assumption that the inputs are fixed. Therefore, in the case of calibration, the regression line fitted using the method of least squares is not consistent with the statistical properties of simple linear regression as already established based on this assumption. To resolve this problem, "classical regression" and "inverse regression" have been proposed. However, they do not completely resolve the problem. As a fundamental solution, we introduce "reversed inverse regression" along with a new methodology for deriving its statistical properties. In this study, the statistical properties of this regression are derived using the "error propagation rule" and the "method of simultaneous error equations" and are compared with those of the existing regression approaches. The accuracy of the statistical properties thus derived is investigated in a simulation study. We conclude that the newly proposed regression and methodology constitute the complete regression approach for univariate linear calibrations.Keywords: bias / classical regression / error propagation / mean-data-point-based variance / populationregression-line-based variance / reversed inverse regression / simultaneous error equations / Taylor approximation

This article introduces a new method for generating the exact one-sided and two-sided tolerance limit factors for normal populations. This method does not need to handle the noncentral t-distribution at all, but only needs to do a double integration of a joint probability density function with respect to the two independent variables “s” (standard deviation) and “x” (sample mean). The factors generated by this method are investigated through Monte Carlo simulations and compared with the existing factors. As a result, it is identified that the two-sided tolerance limit factors being currently used in practical applications are inaccurate. For the right understanding, some factors generated by this method are presented in Tables along with a guidance for correct use of them. The AQL (Acceptable Quality Level) is a good, common measure about quality of a product lot which was already produced or will be produced. Therefore, when performing sampling inspection on a given lot using a tolerance limit factor, there is a necessity to know the AQL assigned to the factor. This new double integration method even makes it possible to generate the AQLs corresponding to the one-sided and two-sided tolerance limit factors.