In the nineteenth century many people tried to seek a value for the most famous irrational number, [pi], by means of an experiment known as Buffon's needle, consisting of throwing randomly a needle onto a surface ruled with straight parallel lines. Here we propose to extend this experiment in order to evaluate other irrational numbers, such as…

Contingency tables, and their associated statistical tests, are frequently used in educational and social research. Popular statistical tests used in contingency table analyses include the Pearson chi-square test and the likelihood ratio chi-square test. These two tests are chi-square distributed under large sample conditions. However, when a…

For much of the twentieth century, Markov theory and Markov processes have been widely accepted as valid ways to view statistical variables and parameters. In the complex realm of online searching, where researchers are always seeking the route to the best search strategies and the most powerful query terms and sequences, Markov process analysis…

With the Seeing and Thinking Mathematically materials, students learn mathematics by doing mathematics, by using and connecting mathematical ideas, and by actively constructing their own understandings. In this unit students learn to see probability through a mathematical lens by exploring and creating games and simulations and by applying the…

In a complex sampling scheme it is desirable to select the primary sampling units (PSUs) without replacement to prevent duplications in the sample. Since the estimation of the sampling variances is more complicated when the PSUs are selected without replacement, L. Kish (1965) recommends that the variance be calculated using the formulas…

Arguing that the teaching of stochastic processes should reflect the experience and reality of the student, the author urges concentration on fundamental ideas. The development of intuition should be encouraged, and to this end continuity in teaching and a spiralled curriculum are important. (SD)

Presents a method of solving decision problems in conditions of incomplete knowledge of the probabilities of the states of nature. It uses all available information in order to delimit the region of ignorance as closely as possible. (Author)

A probabilistic approach is introduced to classify and diagnose erroneous rules of operation resulting from a variety of misconceptions ("bugs") in a procedural domain of arithmetic. The model is contrasted with the deterministic approach which has commonly been used in the field of artificial intelligence, and the advantage of treating the…

The Pearson and likelihood ratio statistics are frequently used for assessing the absolute fit of probability models. Researchers are often interested in comparing fits provided by different models which may have a subsuming or non-subsuming relation. A subsuming relation exists when the parameters of the reduced model form a subset of those…

A mathematical setting based on a statistical sampling probability mechanism is described. In this setting, a mathematical meaning is given for the information function; and it becomes possible to study the relative merits of various ability-estimating procedures. The maximum likelihood estimation procedure under the one-, two-, three-parameter…

This paper discusses the theory and presents examples of free-association norms. Examples from several categories of free-association data are given. Their use in experiments on cultural characteristics of the associative structures of words are also explained. A graph of the relationship between primary and secondary associations and tables of…

Through Monte Carlo procedures, three different techniques for estimating the parameter theta (proportion of the "shocks" remaining in the system) in the Integrated Moving Average (0,1,1) time-series model are compared in terms of (1) the accuracy of the estimates, (2) the independence of the estimates from the true value of theta, and…

The Rasch model for the probability of a person's response to an item is extended to the case where this response depends on a set of scoring or category weights, in addition to person and item parameters. The maximum likelihood approach introduced by Wright for the dichotomous case is applicable here also, and it is shown to yield a unique…

Using the concepts of conditional probability, conditional expectation, and conditional independence, the main results of the classical test theory model can be derived in a very few steps with minimal assumptions. The present effort explores the possibility that present classical test theories can be further condensed. (Author/RC)