The Behrens-Fisher problem arises when one seeks to make inferences about the means of two normal populations without assuming the variances are equal. This paper presents a review of fundamental concepts and applications used to address the Behrens-Fisher problem under fiducial, Bayesian, and frequentist approaches. Methods of approximations to…

The sampling distribution is a common source of misuse and misunderstanding in the study of statistics. The sampling distribution, underlying distribution, and the Central Limit Theorem are all interconnected in defining and explaining the proper use of the sampling distribution of various statistics. The sampling distribution of a statistic is…

D. Robinson and J. Levin (1997) proposed what they called a two-step procedure for analyzing statistical data in which researchers first evaluate the probability of an observed effect statistically (i.e., statistical significance), and, if and only if, it can be concluded that the underlying finding is too improbable to be due to chance, then they…

Many research-related classes in social sciences present probability as a unified approach based upon mathematical axioms, but neglect the diversity of various probability theories and their associated philosophical assumptions. Although currently the dominant statistical and probabilistic approach is the Fisherian tradition, the use of Fisherian…

The probabilities of attaining varying magnitudes of standardized effect sizes by chance and when protected by a 0.05 level statistical test were studied. Monte Carlo procedures were used to generate standardized effect sizes in a one-way analysis of variance situation with 2 through 5, 6, 8, and 10 groups with selected sample sizes from 5 to 500.…

The random variable p and its functions figure in several "tests of combined significance," meta-analysis summaries based on sample significance values, and ps have been used singly, as well as in other tests for evaluating the outcomes of individual research studies. In this work, asymptotic distributions of the sample one-sided…

In the educational research literature alpha, the a priori level of significance, and p, the a posteriori probability of obtaining a test statistic of at least a certain value when the null hypothesis is true, are often confused. Explanations for this confusion are offered. Paradoxically, alpha retains a prominent place in textbook discussions of…

Replication is important to viable scientific inquiry; results that will not replicate or generalize are of very limited value. Statistical significance enables the researcher to reject or not reject the null hypothesis according to the sample results obtained, but statistical significance does not indicate the probability that results will be…

A test of statistical significance is a procedure for determining how likely a result is assuming a null hypothesis to be true with randomization and a sample of size n (the given size in the study). Randomization, which refers to random sampling and random assignment, is important because it ensures the independence of observations, but it does…

The real problems of contextual analysis concern the conceptualization of contextual effects, the kinds of data with which to estimate them, and the selection and implementation of appropriate statistical techniques. This paper focuses on detection; specifically, an approach to contextual analysis based on the estimation and interpretation of a…

A linear classification rule (used with equal covariance matrices) was contrasted with a quadratic rule (used with unequal covariance matrices) for accuracy of internal and external classification. The comparisons were made for seven situations which resulted from combining three data conditions (equal and unequal covariance matrices, minimal and…

Researchers should investigate statistical models that can help counselors decide how to treat individual clients. This research investigated the questions, "Given an effect size (ES) from a counseling outcome study, what is the probability that a client would have a negative response to the treatment, and what is the probability that the client…

Power refers to the probability that a statistical test will yield statistically significant results. In spite of the close relationship between power and statistical significance, there is a consistent overemphasis in the literature on statistical significance. This paper discusses statistical significance and its limitations and also includes a…