IN THE CLASSROOM

RICK HESSE, Feature Editor, Mercer University

Resampling and Lotus:
Together they make teaching the ideas
behind the pooled t-test a lot easier!

by Richard L. Morris and Barbara A. Price,
School of Business Administration, Winthrop University

Those of us involved in the teaching of the introductory business statistics courses are always looking for ways to increase the learning/understanding of the students in our classes. Granted, many of these undergraduates manage to successfully "make it through" the course; but we often wonder how much of the real meaning and use of statistical reasoning stays with them as they continue through the program and out into the "real world." At the 1994 Meeting of Making Statistics More Effective in Schools of Business, a presentation on the use of resampling as a method for teaching basic statistics started the authors thinking of ways to incorporate this methodology into our two-course sequence without making a complete conversion. A major strength of resampling as a pedagogical tool in the undergraduate course is that it encourages the class to learn through doing and seeing--this "hands-on" approach is intuitively appealing and often leads to a better understanding of the concepts than simply plugging into a formula.

The Data

As an illustration of how resampling differs from the classical parametric approach, we will use data collected by an MBA student (Jones, 1991) for a course project. The data consists of two sets of observations of downtime on a paper making machine. The first set is of 22 observations where the speed is controlled by analog tachometers. The second set is of 23 observations of the same machine using digital tachometers to control the speed. It was felt that there would be less downtime with the digital tachs than with the analog ones.

The Classical Parametric Solution

The case described in the previous section may be analyzed using a test on the difference between two means where the set of hypotheses can be expressed as follows:

<= mu sub 2>

<= mu sub 2>

= mean downtime for analog tachometers
= mean downtime for digital tachometers

The standard pooled t-test may be performed in Lotus 4+ using the @function:

@@TTEST(DATA1,DATA2,0,1)

where @TTEST performs a t-test on DATA1, the observations on the analog tachs, and DATA2, the observations on the digital tachs, 0 is the type of test (pooled t), and 1 is the number of tails in the test. This function returns a p-value, in this case, 3.154%, so that for an of 5% we would reject the null and conclude that the digital tachs produced a lower mean downtime.

Classroom "Hands-on" Resampling Analysis

To begin the analysis, have the students think about the implications of believing that the null hypothesis is true, that is, that the average downtime using the analog tachs is equal to (or less than) the average downtime using the digital tachs. Hopefully, they will quickly pick up on the fact that it implies that the individual values could have been observed from either population. Furthermore, the class can be led to understand that the population of observations is infinite and thus resampling should be with replacement. This leads to resampling with a bootstrap-type test which can be done using the procedure given in Table 1 and provides understanding through seeing and reasoning together. Plus, the involvement keeps the students interested!

Using the Computer and Lotus to Replicate

At least one software package has already been developed for resampling. It is Resampling Stats, developed by Resampling Stats Software (Rosa-Hatko, 1995). Since most students are already familiar with spreadsheets, they don't have to learn a new software package. And, since a spreadsheet approach offers certain advantages, we have developed a series of macros in Lotus 4+ to resample for several of the more common applications of hypothesis testing and confidence intervals.

The data for our example is entered in the input worksheet shown in Figure 1. The number of resamples desired is set at 1,000, and H0, the value for the null hypothesis, is set at zero. We have also selected U for the upper-tailed option and 95% for the confidence level.

When through with the input, the user clicks the RESAMPLE macro button with the mouse arrow. The macro will first automatically clear the results of any previous resampling. Then it will combine the observations from the two samples into one set of 45 observations, and from this will construct a probability distribution. The Lotus @VLOOKUP function is then used to randomly select two samples from this distribution. The macro computes the mean of each sample, subtracts the second from the first, and stores the differences, in our case, for 1,000 pairs of samples. This collection of differences is converted into a frequency distribution using the Range-Analyze-Distribution command and then graphed as the distribution of sample differences.

To obtain p-values, the @PRANK function is used to find the percentile of the difference between the two original sample means in the resampled differences. The @PRANK function is also used to compute the lower and upper limits of the confidence interval.

All of the output will appear in a separate RESULTS worksheet, in three sections. The first section to appear is shown in Figure 2, which lists the p-values and confidence intervals from resampling. As checks, the theoretical p-values and confidence intervals are also shown. As can be seen, the resampling p-value of 3.505% agrees fairly well with the theoretical p-value of 3.154%. Likewise, the resampled confidence interval is close to the theoretical one.

To view the graph, the user should scroll down one screen and the graph will appear as in Figure 3. The graph can be seen to approximate the normal fairly well, which is not too surprising.

When finished, the user may run another set of resamples by clicking the REPEAT macro button. This is faster than the RESAMPLE macro because the original setup of the probability distribution and graph have already been made and don't have to be done from scratch.

At present, we have developed resampling macros for hypothesis testing and confidence intervals for one-mean, two-means and paired difference situations. We plan to do one or two more special cases such as these, and then to develop a more generalized macro in which the user (who should be reasonably proficient in Lotus) can write his or her own formulas and lookup functions to adapt to any application encountered.

Conclusion

We think the macro package discussed here, or one like it, offers a good way of introducing students to resampling. It is spreadsheet-based, which means no time spent learning a new package, it's free (we would be happy to share it) and it allows the student and instructor to see what is going on by looking at the macros, the resamples themselves and the graphs. The spreadsheet, in other words, is transparent. We hope it allows interested persons to be introduced to this exciting new topic of resampling.

References

Jones, R.W. (1991) "Digital Technology vs. Analog Technology." Unpublished class project.

Peterson, I. (1991) "Pick a Sample." Science News, v 140, pp 56-58. Ricketts, C. and Berry, J. (1994) "Teaching Statistics Through Resampling." Teaching Statistics, v 16, no 2, pp 41-44.

Rosa-Hatko, W. (1995) "Resampling Stats." ORMS Today, v 22, no 2, pp 72-74.

Simon, J. L. (1993) Resampling: The New Statistics. Duxbury.

Simon, J. L. (1994) "The Resampling Method for Statistical Inference." Basics chapter of Philosophy document of Internet manuscript.

Step 1:	Put disks with the observed values into an urn (or bag or box),, mix them up,, and select one. Record the value as observation 1 for the first sample. Replace the disk and repeat the process until both samples have been selected.
Step 2:	Compute the sample means and their difference.
Step 3:	Replicate Steps 1 and 2 at least 10 times.
Step 4:	List the values of obtained from the replications,, and discuss why they vary and the concept of sampling error.
Step 5:	Compare the value of for the actual data to the list obtained in Step 4. How does it compare?
Step 6:	Discuss the need to perform many more replications, produce a histogram of the observed differences, and compare the original sample difference to the values observed via resampling.

RICHARD L. MORRIS is Professor of Quantitative Methods in the School of Business Administration at Winthrop University. He received a B.S. in mechanical engineering from West Virginia University, an M.B.A. from the College of William and Mary, and a Ph.D. in management science from Virginia Tech. His published work has been in the areas of multiobjective decision analysis and finance.

BARBARA A. PRICE is a Professor of Quantitative Methods in the School of Business at Winthrop University. Her degrees are from Grove City College (B.S. - mathematics) and Virginia Tech (M.S. and Ph.D - statistics). Price is a member of the Decision Sciences Institute, INFORMS and IIF; regularly participates in meetings as a presenter, reviewer, discussant and session chair; and has been an officer in SEDSI since 1990.

Dr. Rick Hesse
Industrial and Systems Engineering Department
Mercer University
Macon, GA 31207