Applications of the Noncentral t–Distribution

The noncentral t–distribution is intimately tied to statistical inference procedures for samples from normal populations. For simple random samples from a normal population the usage of the noncentral t–distribution includes basic power calculations, variables acceptance sampling plans (MIL– STD–414) and confidence bounds for percentiles, tail probabilities, statistical process control parameters CL, CU and Cpk and for coefficients of variation. The purpose of these notes is to describe these applications in some detail, giving sufficient theoretical derivation so that these procedures may easily be extended to more complex normal data structures, that occur, for example, in multiple regression and analysis of variance settings. We begin by giving a working definition of the noncentral t–distribution, i.e., a definition that ties directly into all the applications. This is demonstrated upfront by exhibiting the basic probabilistic relationship underlying all these applications. Separate sections deal with each of the applications outlined above. The individual sections contain no references. However, a short list is provided at the end to give an entry into the literature on the noncentral t–distribution. For many of the computations we use the R functions qnct and del.nct. They represent the quantile function and the inverse δ-function of the noncentral t-distribution. They do not yet exist in the standard distribution of R. These functions and all other R code used here are provided as part of an R work space at the class web site