Evaluation of polynomial regression models for the Student t and Fisher F critical values, the best interpolation equations from double and triple natural logarithm transformation of degrees of freedom up to 1000, and their applications to quality control in science and engineering

Serious gaps exist in the present critical value tables for the Student t and Fisher F or ANOVA significance tests. Statistically correct applications of these tests to the experimental data therefore become difficult. A total of 18 different regression models were evaluated for the Student t and 24 for the Fisher F critical values. These models varied from simple polynomial (quadratic to 7(th) order) to the combined single (ln), double (lnln), or triple (lnlnln) natural-logarithm- (ln-) transformed polynomial models. The advantage of ln-, lnln- or lnlnln-transformations of the degrees of freedom for interpolating the Student t and Fisher F critical values has been documented for the first time in the published literature. The use of critical value equations applicable in the range of 1-1000 degrees of freedom for In-transformation, 2-1000 for lnln-transformation, or 3-1000 for lnlnln-transformation, instead of the tables, is proposed as a 21(st) century innovation for the computer programming of these significance tests. A number of application examples are pointed out to illustrate the usefulness of this work.