Alternative approach for precise and accurate Student´s t critical values and application in geosciences

Abstract

We applied an alternative Monte Carlo simulation approach to obtain precise and accurate critical values (with 3 to 8 decimal places) for the Student´s t test for normal samples of degrees of freedom up to 6000 and for two-sided confidence levels of 50% to 99.9% and correspondingly for one-sided confidence levels of 75% to 99.95%. As an innovation, unlike the existing literature precision estimates of critical values are also individually reported. Prior to our work, critical values (with 2 or 3 decimal places) were available in tables in the published literature. Twenty-eight regression models were evaluated to obtain the best regression fitting of the tabulated data. All conventional polynomial (from quadratic up to 8th order) regressions failed for this purpose. New equations based on double or triple natural logarithm-transformations are proposed for all not-tabulated degrees of freedom, including fractional values if required as well as for probability computations. More importantly, we suggest that these kinds of log-transformations are likely to be useful for all cases where conventional polynomial regressions fail to perform satisfactorily. To demonstrate the utility of these new critical values and best-fit equations, we provide specific examples of applications in geochemistry, chemistry and medicine. The probability estimates, obtained from the present explicit approach, are consistent with commercial and freely available software. Additionally, a more extensive application of our t values is given for processing inter-laboratory data for geochemical reference materials granites G-1 and G-2 from U.S.A. as well as for evaluating geochemical data for basic rocks from the Canary and Azores Islands