Computer Simulations to Clarify Key Ideas of Statistics.

Abstract

Hand calculators have come into common use for performing calculations in high school and undergraduate courses in elementary statistics. Computers have also been used to do standard statistical computations and tests. However, computers can be used in elementary statistics courses not only to do standard statistical tests, but to perform simulations which clarify concepts and theorems of statistics. These simulations allow the nonmathematically oriented student in elementary statistics to have inductive experiences with statistical concepts in a time-efficient manner. A basic concept in statistics is that of a sample distribution. The sampling distribution of a sample statistic is defined to be the theoretical probability distribution of values for that sample statistic obtained from all possible samples of a certain fixed size from the population. Typically elementary texts illustrate this concept by using a very small finite population which has a small number of possible samples of a given size. Although large or infinite populations are of interest since they have a large or infinite number of samples of a certain size it is difficult to develop a sample distribution. Computer simulation, however, makes this easy to explore. The computer can simulate taking many samples of a certain size from a population. By taking many samples the law of large numbers guarantees that the observed frequency distribution of the sample statistic is close to the theoretical sample distribution. The sample distribution of the mean is the most important sample distribution. The computer simulations in Figure 1 explore a sample distribution of the mean. These simulations employ an elementary statistical package called MINITAB developed at Pennsylvania State University. The MINITAB command URANDOM simulates random observations from a population where numbers between 0 and 1 all have an equal probability of occurring. The first simulation consists of drawing 200 samples of size 2 and computing the means of the samples. A histogram of the results is given. Continuing with samples of size 2, drawing 750 samples is simulated and then 1500 samples. As the number of samples increases the student can see the isosceles triangle-shaped theoretical sample distribution emerging. Note that, because of the small sample size and the nonnormal population, the sample distribution should not be expected to be normal.