Inferring the Central Limit Theorem (ARC)

Brett Elliott, Southeastern Oklahoma State University
Michael Hardy, Southeastern Oklahoma State University

The Central Limit Theorem is one of the foundational theorems in statistics. It justifies use of many of the fundamental analytical tools from approximating measures of typical performance and spread to hypothesis testing and construction of confidence. Unfortunately, the theorem is too frequently taught as facts to be memorized. This activity is intended to provide an alternative to the traditional expository approach. The investigation begins with having students consider a problem set in a realistic context that the Central Limit Theorem would help them solve and provide justification for their solution. After reflecting on the problem, students will realize that even if they are able to provide an informed answer to the question posed, they have no statistically sound method of justifying that answer. Out of that realization stems a necessity to learn about the nature of the sampling distribution for the mean. Construction of that knowledge is facilitated by affording students the opportunity to use an online simulation tool to generate sampling distributions and identify relationships and trends that will allow them to infer both the assertions and prerequisite conditions of the Central Limit Theorem, and then apply what they have learned to determine and justify an answer to a problem set in a realistic context. In the process, students will produce sampling distributions for each of at least three sample sizes, with the sample sizes becoming progressively smaller, and students will explore how changing the parent population’s distribution impacts the inferences drawn, if at all. Upon completion of all of the preceding, students then return to the original problem, solve it and justify their solution. Thus, the lesson can be used in an introductory statistics or quantitative reasoning course, and it interrelates virtually all of the content explored in such courses, making it easily worth the time investment.

Inferring the Central Limit Theorem

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