Abstract
A measurand θ of interest is the ratio of two other quantities, μp and μq. A measurement experiment is conducted and results P and Q are obtained as estimates of μp and μq. The ratio Y = P/Q is generally reported as the result for the measurand θ. In this paper we consider the problem of computing an uncertainty interval for θ having a prescribed confidence level of 1−α. Although an exact procedure, based on an approach due to Fieller, is available for this problem, it is well known that this procedure can lead to unbounded confidence regions in certain situations. As a result, practitioners often use various non-exact methods. One such non-exact method is based on the propagation-of-errors approach described in the ISO Guide to the Expression of Uncertainty in Measurement to calculate a standard uncertainty uy for Y. A confidence interval for θ with a presumed confidence level of 95% is obtained as [Y−2uy, Y+2uy]. In this paper we develop a highly accurate approximation for the coverage probability associated with the interval [Y−kuy, Y+kuy]. In particular, we demonstrate that, using n−1 degrees of freedom for uy, and the corresponding Student's t coverage factor k = t1−α/2 : n−1 rather than k = 2, leads to uncertainty intervals [Y−t1−α/2 : n−1uy, Y+t1−α/2 : n−1uy], that are nearly identical to Fieller's exact intervals whenever the measurement relative uncertainties are small, as is the case in most metrological applications. In addition, they are easy to compute and may be recommended for routine use in metrological applications. Improved coverage factors k can be derived based on the results of this paper for those exceptional situations where the t-interval may not have coverage probability sufficiently close to the desired value.