In the traditional "math stat" sequence, we often begin with ideas about probability.

     Randomness is present everywhere. Students are familiar with randomness from, e.g., games of chance and athletic contests, experiences in the laboratory, and simply reading newspapers and listening to the news. We can use the interest of students in randomness to "sell" them on studying probability and statistics.

     When observing random phenomena, we don't know what will happen in a particular situation but we can use an understanding of probability to construct models for various situations. When constructing these models and characteristics of the models, students are able to apply mathematical techniques that they have learned in mathematics courses.

     Many students have also learned how to use a Computer Algebra System (CAS) for solving problems symbolically. There are many opportunities to reinforce some of these ideas.

     Whenever possible, we should use examples that are easy for students to understand. "A Police Log Example" illustrates this.

      Students in a "math stat" course should understand the importance of hypotheses in theorem. Simulation is an excellent vehicle to illustrate what can go wrong if hypotheses are not satisfied.  "Functions of samples from N(0, 1)" illustrates this.

     There are many interesting applets illustrating the Central Limit Theorem. In a "math stat" class, the students should be able to actually write a simulation program and for some distributions find the theoretical p.d.f.s. They should also understand the importance of the hypotheses, e.g., that the variance should be finite.  "Sums of Continuous Random Variables " uses a U -shaped p.d.f. and the Cauchy distribution to show this.

     When finding confidence intervals, simulation is an excellent tool to help students understand the meaning of, e.g., a 90% confidence interval for the mean. Simulation can also help students understand the difference between z and t  confidence intervals. Is there a difference in the lengths of these intervals? See  "Comparison of Z and T   Confidence Intervals" for an example.

    Changes can and should be made in the "math stat" sequence. More real examples must be incorporated into this sequence. Meaningful data sets should be used. (If you have such examples and are willing to share them, please send them to tanis@hope.edu.) Students must learn how to interpret data. Almost all students have had a lot of experience with computers. We must not ignore this ability. Computer programs for data analysis are becoming very easy to use and we should help the students learn how to use this tool. A Computer Algebra System (CAS) should be used when appropriate.

    The statistics package that comes with MAPLE is not complete. However, Zaven Karian has written more than 130 additional procedures to support instruction in probability and statistics. These procedures are available at no cost and several will be incorporated into these programs.

   The following command line loads the supplementary statistics package as well as some other procedures that are used.

>	restart: read `e:stat.m`: with(plots): randomize(): with(student): read `e:ProbHistFill.txt`: read `e:EmpCDF.txt`:  read `e:HistogramFill.txt`:read `e:ProbHistB.txt`:  read `e:ProbHistFillY.txt`:  read `e:ScatPlotCirc.txt`:  read `e:ScatPlotPoint.txt`: read `e:BoxPlot.txt`:

  A   Police Log Example

     Several times a week, a POLICE LOG lists times of dispatched calls to the Department with the disclaimer that all calls are not necessarily listed. Looking at, for example, a 36 hour period, do calls arrive randomly in accordance with an approximate Poisson distribution? And if they do, do the times between calls have an approximate exponential distribution?

     Find the means and variances of the Poisson and exponential distribution, either using Maple or known mathematical techniques.    Solution

Functions of samples from N(0, 1)

Let       be a random sample of size 3 from a standard normal distribution. This example illustrates four functions of this random sample.    Solution

  Sums of Continuous Random Variables

      Let   ,  , ...,    be a random sample of size n  from a continuous distribution with p.d.f.   f(x) ,   where f(x)  > 0 for a  < x  < b   and       < a  < x < b  <   . This section illustrates how MAPLE can be used to find the p.d.f. of the sum of these random variables .   Example

1. Let   f(x)  =    ,    < x < 1.   Show how to find the p.d.f. of   ,   ,   .   Solution

2. Let   f(x)  =   ,    <    < 1.   Find the p.d.f. of     + . . . + .

Empirical Evidence
Theoretical Solution

3.   Simulation is used to illustrate what happens when taking a random sample from a Cauchy distribution. In particular, how is the sample mean distributed? A CAS can be used to show that the weighted average of two Cauchy random variables is Cauchy. This in turn can be used to prove that the distribution of   X - bar  is Cauchy when sampling from a Cauchy distribution.   Solution

  Comparison of Z   and T   Confidence Intervals

When sampling from a normal distribution, what are 90% confidence intervals for the mean ? Is there any difference between z and t  confidence intervals? How do the lengths of the two types of intervals compare? In particular, which are shorter?   Solution