Operating Characteristics for the Common Statistical Tests of Significance

Abstract

Methods making possible quick calculation of operating characteristics of power curves of common tests of significance involving the $\chi^2, F, t,$ and normal distributions are presented. In addition, a comprehensive set of curves illustrating graphically the power of each test for the 5% significance level are included. We are interested in the power of: (1) the $\chi^2$-test to determine whether an unknown population standard deviation is greater or less than a standard value, (2) the $F$ test to determine whether one unknown population standard deviation is greater than another (one-sided alternative), and (3) the $t$-test and normal test to determine whether an unknown population mean differs from a standard or two unknown population means differ from each other. Such operating characteristics have application for the quality control engineer and statistician in the design of sampling inspection plans using variables where they may be used to determine the sample size that will guarantee a specified consumer's and producer's risk. On the other hand they are of use in displaying the power of a test if the sample size has already been set. Finally, they are a necessary adjunct to the proper interpretation of the common tests of significance.