On the Range-Midrange Test and Some Tests with Bounded Significance Levels

This paper is divided into two parts. The significance tests investigated in Part I concern the population mean and are based on the quantity [(sample midrange)-(hypothetical mean)]/(sample range). The case in which the observations are a sample from a normal population is considered in detail. The tests investigated are summarized. These tests are found to be very efficient for small samples (power efficiency is defined in this paper). An investigation of several extremely non-normal populations using the values of Dα obtained for normality indicates that the significance level of the range-midrange test is not very sensitive to the requirement of normality for small samples. Also the tests can be applied without computation through the use of an easily constructed graph. These properties suggest that the range-midrange test is preferable to the Student t-test and the analogue of the Student t-test using the sample range whenever the sample size is sufficiently small. Use of the range-midrange test for the case of normality was proposed by E. S. Pearson, where properties of the test were experimentally investigated for the normal and certain non-normal populations. In Part II several significance tests for the mean are developed which have a specified significance level for the case of a sample from a normal population but whose significance level is bounded near the specified value under very general conditions, one of which is that the observations are from continuous symmetrical populations. Some of these tests are range-midrange tests. A table contains a summary of the tests and their properties.