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PRIMUS: Problems,Resources, and Issues inMathematics UndergraduateStudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20

MAKING STATISTICS A FULLPARTNER IN MATHEMATICALSCIENCES PROGRAMSBarbara A. Wainwright a , Robert M. Tardiff a &Homer W. Austin aa Department of Mathematics and ComputerScience , Salisbury University , Salisbury, MD,21801, USAPublished online: 13 Aug 2007.

To cite this article: Barbara A. Wainwright , Robert M. Tardiff & HomerW. Austin (2002) MAKING STATISTICS A FULL PARTNER IN MATHEMATICALSCIENCES PROGRAMS, PRIMUS: Problems, Resources, and Issues in MathematicsUndergraduate Studies, 12:1, 61-74, DOI: 10.1080/10511970208984018

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MAKING STATISTICS A FULLPARTNER IN MATHEMATICAL

SCIENCES PROGRAMS

Barbara A. Wainwright, Robert M. Tardiff,and Homer W. Austin

ADDRESS: Department of Mathematics and Computer Science, SalisburyUnivers ity, Salisbury MD 21801 USA.

ABSTRACT: We have developed an introductory course that exposesmathematical sciences majors to a full range of issues that most prac­ticing statisticians may face including data acquisition, design of ex­periments, use of theory, and formal writing.

This new course developed for mathematics majors not only requiresthe use of the computer in and out of the classroom for demonstrationsand assignments, but also has a two hour computer laboratory once aweek.

In this paper, we provide a summary of the labs, and share studentperspectives on the course. This laboratory-based course can easilybe transported to other locations.

KEYWORDS: Statistics, computer laboratory, math major, data-driven,simulation, inference.

1 INTRODUCTION

Most would agree that formal course work in statistics is essential for stu­dents majoring in mathematical sciences. Moreover, our department hascome to the position that majors should have a four-pronged experienceearly in their careers ; namely, pure mathematics, applied mathematics,computer science, and statistics. This four-pronged approach at the fresh­man/sophomore level makes statistics a full partner from the beginning.

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To meet this need for an early statistics experience for mathematicalsciences majors, we developed a laboratory-based freshman/sophomore levelcourse. The course has three hours of lecture and a two-hour lab each week.Our course introduces students to statistics soon enough in their careers topursue a concentration in statistics if they so desire.

Over the past few years, several articles have appeared in the literatureaddressing the needs of making the first course in statistics more engagingfor students [1, 3, 5, 6, 7, 8] . Some of these papers address introductorycourses for non-majors; some address the traditional mathematical statisticscourse for majors. These papers support (to varying degrees) active involve­ment of students with data, use of real data, use of computing technology,and writing results. About ten years ago our department developed an in­troductory course in parametric and nonparametric statistics [10] designedprimarily for nursing and geography majors. The course is data driven, in­tegrates parametric and nonparametric procedures, and uses the computerto allow students to shift from calculation to interpretation. Our experiencewith this course over a ten year period gave us a firm basis for designing theintroductory course for the majors. The primary differences between thisnew course and the existing one are the level of mathematical presentationand the addition of a two-hour formal laboratory experience each week.

This introductory course for majors is designed to expose students to thefull range of issues that most practicing statisticians might face. Studentsdevelop a sense for how experiments are designed, how data is collectedand analyzed, how computing and especially graphics enter into an analy­sis, and how mathematics and probability allow statisticians to assess theperformance of procedures. Moreover, since most statisticians are part ofa team and teams produce written reports, students learn to address sta­tistical problems as part of a team. They learn to write reports that areaccessible to a professional who mayor may not be a statistician.

Courses like the one we developed demand much from both instructorsand students. A formal weekly computer laboratory experience necessitatesan active learning environment which engages everyone. In this paper wewill share our laboratory-based course and some of the insights we havegained from designing this course.

2 THE LABS

2.1 Our Goals

The goals for this course are a direct consequence of the department's philos­ophy of undergraduate education using the four-pronged approach discussed

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earlier. The department offers courses which actively involve the studentsin the learning process. We design courses using the constructivist approachto teaching. In all our courses we incorporate the spirit of the NCTM stan­dards; i.e., an active, hands-on learning environment. Thus, courses havegroup work, projects, and writing assignments. There is an emphasis oninterpretation and communication of results and a de-emphasis on mechan­ical calculations without explanations. Although all courses contain somelecture, most courses contain minimum lecture. This new statistics coursereported here has been designed to include these characteristics.

We want students to know first hand that data is confusing and oftenmessy, and that statistical reasoning offers a way to extract informationand begin to assess how reliable the information might be. The content weexplore, in both lecture and lab, is not particularly different from typicalintroductory statistics courses. Our purpose is to have students use thiscontent when actively engaged with data. We want them to develop somesense of where data comes from - whether it be through experiments theyperform, from the World Wide Web, or existing data sets in the MINITABlibrary.

We offer a brief summary of each of the labs in the Appendix. Allof the labs are available on the web at henson2. ssu. edu/"-'rmtardif /StatLabs/LabPage . htm. We envision that instructors who want to makemodifications in their courses will find these labs useful. We believe the labsare easily transportable. Instructors at other universities can easily use ourlabs in their courses.

The laboratory experiences are designed with several objectives in mind.We want the laboratory experiences to have exercises which make the the­ory of statistics more plausible for the students. These exercises help thestudents understand the abstract theorems more clearly. The laboratoryexperiences are designed to provide experiences as close to real-world dataanalysis as possible. In this way, the lab becomes a type of internship simi­lar to internships that students obtain in industry. A summary of the labs,including their purpose, is in the Appendix, except for Lab II which wediscuss later in this paper.

In the first few minutes of the first laboratory experience, students collectdata by completing a questionnaire which is given to them by the instructor.In this lab, they code the data, build a data file, and summarize the databoth using descriptive statistics and graphical representations. They alsoanalyze similar data from a MINITAB data file. This laboratory exerciseenables the students to learn not only statistical treatment of data but alsofile manipulation using the statistical software package.

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The second laboratory experience builds upon the first experience. Inthis lab, students access census data available on the web. In this lab wehave them use data found by accessing the U. S. Census Department's web­site (www. census. gov). The students follow several links to acquire agricul­tural data ultimately from the U. S. Department of Agriculture (ftp: / /www .nass.usda.gov/pub/nass/county/). They download the data file barley1980-2000.zip. To use this data file in Minitab they have some additionalfile processing to do. First, our students need to download and install a copyof winzip which can be found at http://www.winzip.com/.Using winzip ,they extract barley 1995-1999.csv from barley1980-2000.zip. Minitabcannot properly read barley 1995-1999.csv because it is an unsupportedfile type, but Microsoft EXCEL can, and Minitab can read .xls files createdby EXCEL. So students first open the file in EXCEL and save it as an .xlsfile. They then open the .xls file in Minitab. This exercise provides hands­on experience with processing data files for analysis. Seldom can data beprocessed directly. Thus, this lab has empowered students to use computersoftware to access real data in a meaningful way.

More precisely, students, using Minitab, perform stat ist ical analyses onthe barley data. The barley file contains census data by year and by stateon variables such as number of acres planted, number of acres harvested,yield in bushels per acre , and production in bushels. Students summarizesome of the data by obtaining descriptive statistics and graphics. They arealso encouraged to make informal inferences. Questions are given in the labinstructions to guide them in these activities.

As it can be seen in the Appendix, laboratories have been developed foralmost all of the topics covered in a beginning one-semester statistics course.There are labs on probability and sampling, discrete and continuous ran­dom variables, and sampling distributions. In other labs , students exploreconfidence intervals and decision making. For example, students investi­gate the performance of confidence intervals when the assumptions are met.Students also explore what can happen when the assumptions are violated(see appendix for more detail). Tests of hypotheses are conducted usingboth one-sample and two-sample statistical procedures. Students also sim­ulate Type I and Type II errors. Laboratories have been developed to focuson data collection, modeling, regression analysis, Chi-square, and ANOVA.For all labs, probing questions are posed for the students, so that they candevelop their insights into the statistical treatment of the data.

All of the laboratory experiences build upon previous laboratory work;i.e., material used in one lab is often required to do the next lab . Thereare key words that capture the essential characteristics of the sequenced

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laboratories: computer, teamwork, graphics, theory, simulation and inter­pretation, robustness, and writing.

Computer:

A typical laboratory period has students using the computer (one studentper machine) to make the statistical software produce results from the data.Computers are close enough to each other to enable students to act as con­sultants with each other. The instructor is a facilitator of this activity. Theinstructor walks around and answers specific questions which the studentspose.

Teamwork:

In all the laboratory experiences, we want students to work with eachother. A formal weekly laboratory experience naturally encourages team­work. Even if students are using their own computer stations, studentsworking side-by-side tend to ask each other questions. Moreover, we haveexperiments where they work in teams in order to collect appropriate data.For example, to measure the period of a pendulum, one student swings thependulum while another measures the time for several oscillations. Afterthe measurements are recorded , the students use the data to discover thatthere is a nonlinear relationship between these two variables. They arriveat their conclusion by investigating scatterplots and transformations of thedata. Because of errors in both measurement and recording, students learnfirst-hand how important it is have accurate data to discover these relation­ships .

Graphics:

We want students to see graphics as a powerful tool for data analysis. Seri­ous graphics cannot be done without computer support. Moreover, findingthe right graphic for a particular application is often a process of trial anderror, an activity which a laboratory supports nicely.

Theory:

We want students to develop an appreciation for the role mathematics andprobability play in statistical analysis. To do this we use simulations. Thestudents use simulation to show that theorems such as the Central LimitTheorem are plausible, and to demonstrate that significance levels, confi­dence coefficients, and power of tests, as predicted theoretically, are indeedobserved.

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Simulation and interpretation:

We use simulation in a way that is a little unusual. Often, simulation isused to make the conclusion of a theorem plausible. We do this in a fewlabs ; however, we also use simulation to develop informal inference. For ex­ample, intuitive probability is used to predict what a histogram of a samplefrom a particular distribution, e.g., uniform , looks like. That is, if randomsamples of a sufficiently large size are taken from some particular distribu­tion, then the shape of the histogram of that sample data should resemblethe shape of the given distribution. Students experiment with simulateddata from various distributions to refine their beliefs of what the histogramshould look like. Finally, we provide histograms from sample data and askthe students to decide which of several possible distributions is the mostplausible. Students see first hand that sometimes the empirical distributionis quite "ragged", a genuine surprise for most people. Part of the ragged­ness is expected due to random sampling and part can be accounted for byusing too many sub-intervals in the graphical display. However, care hasto be taken not to reduce the number of intervals to produce an illusion ofnormality, for example, when it does not exist.

Robustness:

Simulation also allows the students to experiment with notions of robust­ness early on. We are able to explore how departure from assumptionsaffects performance of standard methods. We can also readily explore theperformance trade-off between parametric and non-parametric methods.

Writing:

Each week the labs are written up as formal reports. Because the lab reportscount as ten to fifteen percent (10-15%) of their final grade and topics fromthe labs are reflected on class quizzes and/or exams, students realize the labreports are important. To help students produce refined reports, we devotea substantial portion of the class period following a lab to discussion of whatthe results of the lab might mean and why such results might be observed.This collaboration with students reinforces the notion of teamwork and helpsstudents focus on the statistical issues.

2.2 Student Issues

Sometimes labs contain topics that have not yet been formally covered inthe classroom portion of the course. Students can explore statistical con­cepts first and then ask why it is done this way or why such a proceduredoes or does not work so well. This inductive approach to learning demands

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some versatility and confidence on the part of the instructor, and the stu­dents need much patience. However, as a payoff, students and instructorsoperate in an environment more like the consulting environment. As youmight expect though, students do find this unsettling and need a lot ofencouragement.

Students also comment on the level of detail in the lab instructions;some want more, some feel the instructions are overly detailed. In the firstfew labs a lot of detail with explicit instructions for completing them isessential. As the semester progresses, we reduce the level of detail. Somestudents appreciate the reduction of detail while others are lost and insecure.A balance is needed between detail and discovery when it comes to software.As detail is left out of the later labs, students with questions are first askedto refer to previous labs. Usually they figure it out themselves or with thehelp of a classmate; again another opportunity is provided to encourageteamwork. This leads to greater confidence when using software, which isessential.

Special attention needs to be given to older students because they arenot as comfortable with computers as today's students who are fresh outof high school. We had such a student the first semester and we allowedhim to turn the first few labs in late. He had the background to develop agood statistical sense, and we didn't want him to drop the course becauseof his lack of computer experience. Indeed this paid off; by the end of thecourse, he had become quite proficient. During the second semester, wehad another such student. He had taken statistics courses before, but hadlittle exposure to the computer and needed much guidance. After only afew labs he became much more comfortable. Attention must be given tosuch students or you lose them.

Finally, you will have students who are quite computer literate. One suchstudent became our resident expert in Minitab commands and graphics. Wecertainly learned from him, but we also learned to be careful that our stu­dents don't become so enamored with software at the expense of developingtheir statistical reasoning. This is a very real problem in a course such asthis where critical thinking as well as analysis and synthesis of informationare vital.

3 CONCLUSION

We find that in five hours per week (3 hours lecture and 2 hours lab) andwith small classes, we are able to produce the consulting/data-driven envi­ronment we are seeking. The labs generate many questions and help moststudents develop a good statistical sense.

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As we observed earlier, some lab topics surface before they are discussedin the lectures, a situation which makes some students very uncomfortable.It isn't unusual to experience such uneasiness in a consulting environment.Topics introduced in a laboratory before being discussed in lecture, buildsstudents' confidence in their ability to pose their own questions about theproblem and consequently students begin to do statistics as a practicingstatistical consultant does. In the lab, students address "messy" problemsin a safe environment and do indeed generate many questions. The lecturethen becomes a natural follow up to developing an understanding of thestatistical issues involved.

Discussing the lab in the next class period gives students an opportunityto discuss what they saw or should have seen and how they might interpretthe results. They even begin to ask questions about the inferential issuessuch as how an inference would be made without ever seeing a formal discus­sion in this context. This also generates many questions because studentsstill must write up the lab . We will continue to have one class period fordiscussion between the lab and the due date for the written report. This isnot unlike the analysis a sports team does a couple of days after a game.

The goals of the course are to instill inquiry, reasoning, and communica­tion which we have found we can do in the laboratory setting. The successof the academic experiences is supported by many students through their fi­nal evaluations of the course. Many students comment that the labs are themost valuable aspect of the course. Others say that the labs help them tobetter understand the material. Most students say the book is used merelyfor the tables and occasional reference.

After only two labs , one student goes out of his way to tell us how sur­prised he is at his ability to do statistical analysis. This same student, whois older and very adept with both computers and mathematics, commentson how the lab has changed his view of statistics. Going into the course, hetells us he saw statistics essentially as Disraeli did [2] : "lies, damned lies,and statistics." By the end of the course, this student tells us that becauseof the labs he not only sees statistics as a real science, but also that it canbe important, informative, illuminating, and fun and wishes he had takenmore.

Why should anyone make this type of experience happen? We findthat this type of course with the laboratory experience is good for boththe students and for us. We learn from each other and from the students.This mode of teaching is rejuvenating and if organized properly, is notoverwhelming. We continue to have fun with this course and hope you do,too!

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APPENDIX: LABORATORIES

I. Getting Started with Minitab:

Students survey the class to obtain data and enter the data in the work­sheet. They also obtain a data file from the Minitab library and use Minitabto summarize data, using both descriptive statistics and graphical displays.The goal is not only to have students become comfortable with the sys­tem, but also to have them begin to understand the different measurementscales for variables and the associated ramifications for graphical displaysand calculations. Students begin to do informal inference by exploring re­lationships between such variables as height and weight, and by comparingheights of males and females.

II. Accessing Data from the Web and Formatting it forMinitab:

Students experience and solve problems which can occur when you linksoftware systems. Students access census data on the web. Students gainexperience in finding, retrieving, extracting, and cleaning data so that itcan be used for a specific task. For a more complete summary see section2.1 of this paper.

III. Probability and Sampling:

Using the Pulse Minitab worksheet students construct 2x2 contingency ta­bles with nominal variables. Students see probability as relative frequencyand explore the notion of independence and dependence. They are askedhow 2x 2 tables should appear if variables are independent and if they aredependent. Treating the Pulse data set as a population, students also ex­plore how a random sample from a population can produce statistics forestimating population parameters. Students generate samples of size 10and size 50 from this population with known mean and observe how thisaffects the quality of the estimation.

IV. Discrete Random Variables:

In this lab students are comparing empirical distributions to their theoret­ical counterparts. Thus, students compare graphs of sample data selectedrandomly from various discrete distributions to the graphs of the theoreti­cal distributions. Students learn that empirical distributions might not look

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like the theoretical distributions when sample sizes are small. However, asthe sample size increases, the empirical distributions usually begin to lookmore like the parent distributions. To conclude the lab, we provide stu­dents with sample data sets from various distributions (Poisson, Binomial,discrete Uniform) and ask them to infer what distribution seems most plau­sible. Once again, students are using informal inference, but they are alsobeing introduced to the notion that the parent population being sampledplays a central role in parametric statistics.

V. Continuous Random Variables:

Students repeat similar activities as in Lab IV, except continuous distribu­tions (Exponential, Normal, Uniform) are used. While the activities in thislab are virtually identical to those in Lab IV, continuous distributions intro­duce a wrinkle not found in the discrete case. In particular, to create graphsof the theoretical distributions for these continuous random variables, thestudents have to partition the domain into subintervals. But there is moreto the story; the choice of size of subintervals affects the appearance of thegraph. Students see first-hand that computer graphics cannot be acceptedat face value; computer graphics must be interpreted with an understandingof the theory. Surprisingly, some students were puzzled to find that decreas­ing the width of subintervals produced a continuous graph in the case ofthe normal distribution, but failed to do so with the Poisson.

VI. Sampling Distributions:

Repeated random sampling is done using normal, exponential, and Cauchydistributions. Students simulate sampling distributions of both mean andmedian. They observe that the sampling distribution of the mean tendsto be normal for both the normal and exponential distributions, but theexponential distribution requires a larger sample size to achieve this. Theyalso observe that the Cauchy distribution is a pathological case in that theCentral Limit Theorem (CLT) does not work at all. This leads to a discus­sion as to why not. So this lab uses simulations to illustrate consequencesof the most important theorem in inferential statistics; namely the CLT.

VII. Confidence Intervals:

Through simulations with normal, exponential, and Cauchy distributions,students construct confidence intervals for the mean using repeated randomsampling. Students construct a graph of 100 confidence intervals displayed

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in a format so that they can see whether or not the confidence intervalcontains the parameter being estimated. Students gain an appreciation ofwhat a 95% confidence interval is. Once again , they see that in the normalcase the procedure works as advertised; in the exponential, it does not workquite as well. Increasing sample size improves performance as expected; andin the case of the Cauchy distribution, the procedure fails dismally. Thisleads to a discussion as to why it does not work.

VIII. Decision Making:

Students create Type I and Type II errors by adjusting the mean in theirparent population when sampling. When the hypothesized mean of thepopulation is close to the parameter (mean), the students see first-hand thata Type II error is likely. They also are able to conceptualize these errors andsee that these errors do occur in data analysis, but control over the size ofthe errors can be gained by adjusting sample size. This is very different froma typical classroom discussion of hypothesis testing, because there all youcan do in that setting is talk about errors. In our experience, the instructorplays a vital role in that he/she needs to be questioning students as theyproduce results to ensure they are interpreting their output properly.

IX. Small Sample Tests:

Assumptions underlying the Wilcoxon Signed Rank test, Sign test, and t­test are explored, and students are given a sample data set to use to performtests of hypothesis. They justify their choice of procedure. For example, ifthey choose the t-test , they should supply a normality test. Students seethat power of a test becomes an issue when selecting a statistical proce­dure. It is important that students understand the conditions under whichhigh power is achieved because data collection and analysis are both time­consuming and expensive. So using a less powerful test may cause one tolose valuable information in the data. On the other hand, if assumptions arenot satisfied, then using a less powerful test may be the only choice. In somesituations the selection of an inappropriate test could lead to nonsense.

X. Hypothesis Testing: Two Samples:

Students learn to distinguish data from paired samples versus independentsamples. They implement the two-sample independent t-test and the cor­related (paired) t-test. They also compare the two-sample t-test with theMann-Whitney test. Students learn under which conditions a particular

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test should be used. These tests are performed on the pulse data set sup­plied by Minitab. Using the clotting and the health data sets, students areasked questions about the two sampled populations, and they then deter­mine whether to use a paired or independent design and which procedureto use.

XI. Data Collection:

The goal of this lab is to have the students grapple with the vagaries ofreal measurements. Students are instructed to bring various round items,string, a heavy object such as a fishing weight, a measuring device and a stopwatch to the lab. The round items can be paper cups, coins, jars, etc. (Onestudent brought a hubcap!) Students measure radii and circumferences ofthese round items, enter the data, do a scatter plot , and should observe alinear relationship. Because of their familiarity with circles, students real­ize measurement errors occur regardless of measurement techniques. Somestudents experience unit problems; e.g. diameter measured in one scale,circumference in another. The students use the string and heavy object tomake pendulums. They then measure the length of the pendulum and thetime it takes to make five swings. They vary the length and repeat theprocess. The instructor needs to ensure that each group has at least onependulum with a short length and another one with a long length. Onceagain, the students enter this data into Minitab and produce scatter plots.The pendulum experiment introduces students to curvature in the data.

XII. Correlation and Regression:

Through simulations students explore correlation and linear regression us­ing perfect linear , linear with additive normal errors, additive errors withoutliers, quadratic relationships with additive errors, and finally multiplica­tive errors. Thus students see a range of what might happen in correla­tional studies. The purpose of this lab is to sensitize the students to theassumptions for using linear regression models. For example, students seehow correlation coefficients, F-ratios, R-squares, etc. are changed when theassumptions change.

XIII. Regression Analysis and Modeling:

This lab is an application of data collected in Lab XI and XII. Students per­form a linear regression analysis on both the circle data and the pendulumdata. Students give an interpretation of their results. In particular, they

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discover that the circumference vs. radius for the circle is linear , whereaswith the pendulum, a linear model does not seem to be the best fit to thedata. Then after a transformation on the data, they discover that, for thependulum, the time to swing five swings vs. its length is quadratic.

XIV. Chi-Square and ANOVA:

In this lab we come full circle. Recall, in our first lab we wanted students torecognize different measurement scales and select the appropriate tool foranalysis. Here, we look at real data (Furnace data from Minitab library) toexplore a multivariate data set. Some of these variables are nominal, someare ordinal, and some are interval/ratio. Students perform cross-tabulationsand make tables for several pairs of nominal variables. For these variables,Chi-square is an appropriate test for looking at whether the variables arerelated. Students are instructed to perform several Chi-square tests. Forthe interval/ratio variables, students perform several ANOVA's in orderto ascertain whether there are differences among the means across severalpopulations.

REFERENCES

1. Bradstreet, Thomas E. 1996. Teaching Introductory StatisticsCourses So That Nonstatisticians Experience Statistical Reasoning. TheAmerican Statistician . 50: 69-78.

2. Dell, Diane J . 2000. Memorable Quotations: Humorists, Wits andSatirists of the Past. Lincoln NE: iUniverse.com, Inc.

3. Hogg, R.V. 1991. Statistical Education: Improvements Are BadlyNeeded. The American Statistician. 45: 342-343.

4. Iman, Ronald L. 1994. A Data-Based Approach to Statistics . BelmontCA: Duxbury Press.

5. Marasinghe, M. G., W. Q. Meeker, D. Cooke, and T . Shin.1996 . Us­ing Graphics and Simulation to Teach Statistical Concepts. The AmericanStatistician. 50: 342-351.

6. Nolan, D. and T . P. Speed. 1999. Teaching Statistics Theory ThroughApplications. The American Statistician. 53: 370-375.

7. Singer, Judith D. and John B. Willett . 1990. Improving the Teachingof Applied Statistics: Putting the Data Back Into Data Analysis. TheAmerican Statistician . 44: 223-230.

8. Smith, Gary. 1998. Learning Statistics By Doing Statistics. Journalof Statistics Education . 6(3). (www.amstat. org/publications/j se).

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9. Tardiff, Robert M. and Barbara A. Wainwright. Statistical Thinking:A Course in Statistical Reasoning using a Laboratory Approach. Presentedat the Joint Meetings of the MAA and the AMS on January 15, 1999 inSan Antonio, TX

10. Wainwright, Barbara A. and Homer W. Austin. 1997. A Perspectiveon Teaching Elementary Statistics. PRIMUS. 8(2): 1997.

BIOGRAPHICAL SKETCHES

Barbara Wainwright is an Associate Professor of Mathematics and Statis­tics in the Department of Mathematics and Computer Science at SalisburyUniversity, and she has been at Salisbury since 1984.

Robert Tardiff is Professor of Mathematics and Statistics in the Departmentof Mathematics and Computer Science and Associate Dean of the HensonSchool of Science and Technology at Salisbury University, and he has beenat Salisbury since 1982.

Homer Austin is Professor of Mathematics and Statistics in the Departmentof Mathematics and Computer Science at Salisbury University, and he hasbeen at Salisbury since 1983.

Austin, Tardiff, and Wainwright all have common mathematical interestsin education and statistics.

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