Working with an unknown distribution

Document Type

Conference Proceeding

Publication Date

4-27-2019

Abstract

Many statistics and data science programs include calculus, probability, and linear algebra because these concepts provide the basis for almost all statistical methods. This paper, however, presents an application of these math ideas directly to the solution of a problem, which provides additional motivation for students to master these fields. The problem explored is how to compare a sample distribution to a population whose mean, median, standard deviation, and size are known, but whose distribution is both unknown and unlikely to be normal. The data come from student course evaluations at a small liberal arts college, so claims about whether an instructor’s results are significantly above or below the department or college mean are of more than passing interest. Due to the finite precision of the reported population means and standard deviations, a brute force approach is infeasible. However, linear algebra techniques can be used to reduce the search space to a manageable size. Even then, there are multiple solutions for the population distribution that fits the known parameters, so an approach for choosing among those solutions is developed. Additionally, Lagrange multipliers can be used to produce an answer. Sample results will be presented and limitations of the methods will be discussed.

Comments

Presented at the Eighth Annual Conference of the Upstate Chapters of the American Statistical Association in Rochester, New York, April 27, 2019.

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