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

STUDENT PROJECTS MAKESTATISTICS COME ALIVE FORMATHEMATICS MAJORSAgnes M. Rash PhD aa Department of Mathematics and ComputerScience , Saint Joseph's University1 ,Philadelphia, PA, 19131, USA E-mail:Published online: 13 Aug 2007.

To cite this article: Agnes M. Rash PhD (1998) STUDENT PROJECTS MAKESTATISTICS COME ALIVE FOR MATHEMATICS MAJORS, PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 8:3, 193-202, DOI:10.1080/10511979808965896

To link to this article: http://dx.doi.org/10.1080/10511979808965896

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Rash Student Projects Make Statistics Come Alive

STUDENT PROJECTS MAKESTATISTICS COME ALIVE FOR

MATHEMATICS MAJORS

Agnes M. Rash

ADDRESS: Depa rt ment of Mathematics and Computer Science, SaintJoseph's University", P hiladelphia PA 19131 USA. arash@sju.edu

ABSTRACT: This article suggests an approach in courses for mathemat­ics majors and minors similar to the approach becoming popular inapplied courses - using student projects. A natural place to introducemodern applications is in upper div ision, calculus-based probabilityand statistics courses for mathematics majors and minors. To providestudents with an opportunity to design and study a real problem,students can be required to complete a statistical analysis of a prob­lem in a semester-long project. The strategy and implementation isdescribed in the article.

KEYWORDS: Projects, statistics, probability, alternative assessment,real-world applications, data analysis, des ign, communicating mathe­matics .

INTRODUCTION

In entry-level mathematics courses for students majoring in social science orbusiness areas, instructors frequently attempt to make t he mathematics im­mediately relevant to students' course of study. Many of us do experiments,collect data, and use modern textbooks wit h numerous a pplications. Thequestion is: What are we do ing for ou r mathematics majors and minors? Anatural place to int roduce modern applications is in upper division, calculus­based probability and statistics courses for mat hema tics majors and minors.

I St . Joseph's University is a p rivate, church affil iated university with approximately3500 full-time students.

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Yet, these course tends to be theoretical in nature. Often we cover the top­ics necessary to be successful on an actuarial examination, and necessaryfor successful entry into statistical study. As important as these topics are,we need to provide mathematics majors (and minors) with the practicalaspects as well.

To provide students with an opportunity to design and study a realproblem, students in my upper division probability and statistics coursesare required to complete a statistical analysis of a problem in a semester­long project. I have taught a year-long course for over ten years usingprojects, and I am always surprised by the interests of the students andtheir access to data and information. Students' interests vary, and the useof projects allows for individual differences in learning style, content, andpresentation. The purposes of the project are

• to provide the students with a challenging problem, similar to one inwhich they may find themselves engaged after college;

• to assess students ability to apply the concepts to a new situation;

• to broaden the range of applications discussed in a course;

• to make topics presented relevant to the world of the students; and

• to improve the ability of the students to communicate mathematicalconcepts to their peers and to the instructor.

TOPIC CHOICES

Traditionally, the topics in the first semester of the two-semester sequenceprimarily concern theoretical probability, discrete and continuous randomvariables, distribution functions, moment-generating functions, etc. Thesecond semester consists mainly of applied techniques: random sampling,tests based on normal distributions, ANOVA, ranking, regression, nonpara­metric methods, etc.2 During the first semester of my course, studentschoose a problem to be analyzed. Through individual consultations with me,we try to limit the scope of the problem, insure that it is well-defined, anddetermine the feasibility of the study. Some students begin data-gatheringduring the fall semester or winter break. During the second semester, thebulk of the analysis takes place. Students are introduced to software pack­ages, such as Minitab and Excel to assist in data analysis. However, theyare free to use any other computer software with which they are familiar.

2Instructors and textbook authors are now beginning to intermingle the two themes.

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For example, some students use Harvard Graphics, Lotus, DERIVE, SPSS,BMDP, Quattro Pro, or Mathcad.

Once a problem is agreed upon, the design of the research is discussed.If appropriate techniques are not discussed in class, I meet with the studentindividually to design the study. Some sampling techniques and other meth­ods of data-gathering are presented during the fall semester so that studentscan begin to gather data. If the student cannot find a problem that inter­ests her/him, faculty members or administrators may contribute problemsof interest to them for the students to analyze. For example, the Office ofInstitutional Research at the University has many projects on which theyneed assistance. Currently our administration is interested in determiningthe effect of adding more categories of grades given for courses. Three yearsago, the administration instituted letter grades suffixed with a minus sign,in addition to the existing grades suffixed with a plus sign. Possible gradesin a course are now A, A-, B+, B, B-, C+, C, Co, D+, D, and F. The ques­tion is: Does the introduction of the new letter grades increase or decreasethe grade point average of students?

An interest of Residence Life is the effectiveness of various strategies inreducing property damage in the dormitories. Others areas of the Univer­sity needing assistance in designing studies, collecting and analyzing dataand drawing conclusions are the offices of Admissions, Academic Comput­ing, Alumni Relations, Registrar and Academic Advising. After contactingthese offices, if suitable projects are available, I discuss them with the stu­dents. When a student expresses interest in working on one of the projectssuggested by a member of the university community, I call the office, sched­ule an appointment, and introduce the student. The student then goes tothe office and discusses the project directly with those involved.

PROGRESS THROUGHOUT THE SEMESTER

Throughout the semester, students present periodic progress reports duringmy office hours. When data is being collected, we discuss the strategy tobe used and the quality of the data obtained (whether the data is what weintended to collect or does not fit the design). Even if the proper method foranalyzing the data is not known until late in the second semester, studentscan be making progress on their studies. For example, if a random sampleis needed, students can determine the sampling method and collect thesample. If a questionnaire is needed, students can design the questionnaireand perform the pilot study. Hypotheses can be stated, delimiters identified,and so on . I make suggestions to the students to limit the scope of the

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project and to suggest methods of gathering and analyzing the data.Since the instructor must be available for consultation on a regular ba­

sis, implementation of this teaching strategy requires a commitment of theinstructor's time to the students. Each project includes the following stages:

• Define the problem.

• Limit the scope of the problem if necessary.

• Design the study.

• Collect the data.

• Analyze the data using the appropriate statistical tools.

• Draw conclusions and make recommendations for further study.

• Present results to the class.

• Discuss what slhe would change if slhe were to do the study again.

This list is given to the students when projects are discussed in class. I usethese stages when discussing problems with students during my office hours.My notes on a student's project indicate what progress slhe is making andat what stage slhe is in the process.

Class presentations are scheduled throughout the spring semester, as stu­dents conclude the projects. For some projects, the statistical techniquesare known relatively early. These students are urged to make their pre­sentations as soon as possible. Since revisions can be made to the reportbased on the questions and comments of their peers, students often decideto report their findings as soon as possible, thus allowing as much time aspossible for revisions. Final reports are due for all projects during the lastweek of class.

On the last day of class in the second semester, we discuss the finalstage as a group. Occasionally the study is too large, and a student does notanticipate the volume of data that is available. When problems are proposedby other faculty members or administrators, they may be ill formulated andrequire extensive refinement.. Students share these difficulties and how theyresolved the problems for the purpose of the study.

The interests that students have which are conducive to statistical studyare often surprising. Some have parents or friends in fields from which aproject can be obtained, while others have part-time jobs in which dataanalysis is helpful. There are usually some who rely on my assistance tofind a project.

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EXAMP LE S O F STUDENT P ROJE C T S

Below are a few exa m ples of s t ude nt projects.

1. Analysis of S u ccess in Cardiopulmonary Resuscitation- Metropolitan Emergency Ser vice.

In this st udy, the student was interested in the effectiveness of emergencyparamedic care in the city of Philadelphia. The first four minutes followinga heart attack are most crucial in saving the life of a heart attack victim.In this study, the availability of CPR through em ergency units, the speedof response, and death rates from cardiovascular problems in Philadelphiawas studied and compared with the national figures . The study invo lvedusing a z-t est for comparison of two means, a nd an F-test to comparethe variances. The study showed that the survival rate for Philadelphia issignificantly better than the national average. The explanation for this wasproposed.

2. Analysis of the Fruit F ly R a ja Zach a r i.

The Raja Zachari is a rare species of fruit fly about which little is known .The st udent, a biology minor, was participating in research on t he RajaZachari frui t fly. The expe rime nt was intended to explore several rela­tionships: (1) the average life span of a Raja Zachari in captivity, (2 ) thecourtship behavior of this fruit fly, (3) the number of offspring a Raja willproduce, (4) the relationship between length of courtship time and t he num­ber of offspring, and (5) the expected number of eggs laid in a given 24 hourperiod . The results were compared with other species of fruit fly using at-test.

3. A n a lysis of Wea t h er Patterns for t he PhiladelphiaM et r opolit a n Area .

The student had been collecting data on the wea t her for many years. Hewas interested in fitting curves to the data and using this to predict fut ureweather patters. Using a goodness of fit test, the student was able to showthat the mean daily temperatures follow a beta distribution. In fact, withhis data on temperature at various hours of the day he found values ofalpha and beta that fit the curve for days in each season. He was also ableto construct a cu rve of best fit for each season. As one would expect, the

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values of alpha and beta for the seasonal data were ext remely close to thevalues for t he daily mean te m pe ra tures throughout the day.

4. Escalation of the Cost of Higher Education-Predic­tions for the Future.

After reading "Bloate d Administration, Blighted Campuses," by Barbara R.Bergmann, in Academe Magazine, Dec. 1991, a student decid ed to compareour instituti on 1 with t he gen eral trend noted in the article. After findinga regression equation to represent the data given in t he study, t he st udentcompared t he data from this insti tution with t he report ed resul ts in t hearticle using a Chi-squared (goodness of fit ) test. He t he n ob tai ned da tafrom comparable institutions in our geographic region from their annualfinancial reports and rep eated th e analysis.

5. Relationship Between Length of Time a Song Is onthe "Top 10" List of Popular Music and Its MonetarySuccess.

The student who performed this st udy was very interested in popular music.The first di fficulty was delimiting t he scope of the pro ject . We decided toconsid er da t a for t he las t 10 years . He was able to ob tain informa ti onon t he len gth of ti me a particula r song was on the "Top 10" chart, andinformation on t he number of copies of t he record sold. However, a songcan be sold as a "single" or in an album. We limit ed t he data to sale ofsingle records, with the rat ionale that many songs appear in a n album t hatwould not be popular enough to buy as a "single." Once into the st udy,t he student realized t hat t he number of t imes t he song is played by a discjockey influ ences it s po pularity. For a fu r th er study, t he student suggesteddetermining how a disc jockey makes t he decision on how ofte n to play anumber, and t he rela ti onship between the frequen cy of play and t he successof the song. Rank correlation was used to analyze the data.

6. Precision of an Analytical Balance-Effect of BuildingVibrations on Instrumentation in the Science Building.

A chemistry major was concerned that a cons ta nt vibra t ion of t he scie ncebuilding was cau sing inaccu rate readings on deli cate instruments. To studythis phenomenon we agreed to limi t the study to one instrument: a st udyof the precision of a Mettler a naly t ical balan ce. Using a z-test, t he st ude nt

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compared the experimental results with the specifications of the Mettlerbalance. Data was collected digitally to compare the balance readings anddetermine the relative error in measurement. This was then compared tothe specifications for the instrument using a z-test of the means and anF -test of the variances.

7. Men and Women - 100 Meter Race Comparison.

This was a comparative study of men's and women's Olympic gold medaltimes for the one hundred meter run as they have been recorded since 1896.A theory presented in U. S. News (summer, 1991) was that women areincreasing their speed at a faster rate than men are and would soon run asfast as men run. Regression equations for the men's time and also for thewomen's times over the years, were used to compare the two data sets. As aresult of the analysis, the student predicted that in the year 2087 both menand women would run the 100-meter race in 7.71 seconds. There are manyassumptions underlying this study, which the student discussed at the endof the paper. To mention only one of the problems with the data, women'stimes do not begin until 1928, since women were not permitted to competebefore this date. A thorough discussion of the adequacy of the model wasgiven.

8. Indicators of Construction Trends and the Use ofThese Trends as Economic Indicators.

The problem was to determine which variables in the new home constructionmarket are factors in determining construction trends and general economictrends. The student used the Statistical Abstracts of the United States andCurrent Statistics (printed by Standard and Poor Corp., 1992) to obtaindata. From these sources, variables were identified. Regression analysisand correlation were used for this study. At the conclusion of this study,the student thought there were other variables in the construction businessthat might be useful as indicators of economic trends and made suggestionsfor further research.

9. An Analysis of Crime.

The investigator used data from the Statistical Abstracts of the UnitedStates. He considered the Crime Index Total, Violent Crime and PropertyCrime, over a six year period, by region of the country, by individual states,and by the age of the offender and its relationship to the type of crime. In

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all there were eleven factors considered . At th e conclusion of t he study, thest ude nt found many factors could be associated with crime . Am ong t heseare population density, stability of t he population, modes of t ranspor tationand highway syste m, economic conditions, climate, a t ti tu des of cit izenrytowards crime , and crime reporting pr acti ces of citi zenry. A continge ncyta ble was used to det ermin e which fact ors rela ted to t he incidences of crime

10. A Comparison of Standardized Test Scores for Boysand Girls in Eighth Grade.

Test scores were taken from a standardized test of eight h grade studentsin the areas of Reading , Language, Math ematics, Science, Social Studies,Sp elling and Total score . In ad di tion, scores were obtained on t he CognitiveSkills Index (CSI) , which represents the student's overall ability. The in­vestigator was interested in th e following qu estions: Is there a signifi cantlyhigher mean score in each area tested, for mal es or females? Is a stude nt'sscore in t he three areas of reading, lan guage and math em atics an acc uratereflection of t he stude nt 's total score? Which sub ject, if t he re is one, mostclosely rela t ed to the CSI ? Can the total score predict a st ude nt 's CSI ? Ist he vari ance in scores differ ent in a particular sub ject area for males andfemales? For manageability, this project was limited to a single school.Further st udy is needed for a complete analysis. This study used t-tests,F -tests , and ANOVA to an alyze t he da ta.

11. Student Opinion Regarding Faculty and Curriculumin the Mathematics and Computer Science Department.

The st udent had two suspicions: (1) Student evaluat ions of mathem at ics andcomputer science majors ra te faculty in th e department higher th an t hosetaking mathematics only to sa tisfy the general education requirem ent , and(2) mathematics and computer science majors have a different view of t hevalue of t he curriculum t han non-majors . A questionnaire was designed . Apilot study was run and t he qu estions modified for cla rity. The questionn airewas given to classes of st udents, so a random sample was not available. Thedata was analyzed using at-t est.

12. An Analysis of the Assignment of Trolleys to Routesin the City of Philadelphia.

On e student , who commuted to college, kep t a dai ly recor d of the identifica­t ion number on th e t rolley on each route that she t raveled to get to and from

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St. Joseph 's University. Her suspicion was that trolleys were not assignedrandomly to routes. A nonparametric test of randomness was performed toanalyze the data.

As you can imagine, many other studies have been performed by studentsover the years. A few of these are: (1) The Effect of Lower Speed Limits(55 m. p. h.) on Death Rates , (2) Analysis of Television Viewing Habits ofCollege Students, (3) A Comparison of the Grades of Students in DifferentDormitories, (4) An Analysis of Resident Study Habits, and (5) AnalyzingPizza Shop Sales.

CONCLUSION

In conclusion, in a course of this type, mathematics majors and minors seea wide range of problems and applications in a variety of fields . Finding aproblem of interest to th e student is the most important asp ect of th e project.Students are anxious to work on an interesting problem. They usually havea hypothesis that they think will be supported by the research, althoughthey are sometimes surprised by the results. In many cases, the findings areshared with others and students gain a wealth of examples of applicationsof statistics. Many acquire descriptive techniques, and use the computer toillustrate the data graphically, which often clarifies a situation. In addition,the students gain experience in organizing and presenting a report. Theygain confidence in their ability to complete a lengthy project. Many of theprojects found on campus have been particularly useful to the Universityand to some individual professors who were assisted by students in thecourse. The exp erience is one which makes the students tackle full-scaleproblems and provides an experience similar to one they might encounterin the business or academic world. For some students, the results were ofgreat interest to their employer (or future employer).

From the description of the role of the instructor, you will note thatmany hours are invested in consultation. The drawbacks for the instructorare:

1. Finding suitable problems for students can require several consulta­tions and calls to colleagues.

2. Delimiting a broadly stated problem may require making the problemless interesting and the results may be less useful than anticipated.

3. Meetings with students concerning the progress being made on pro­jects are time-consuming.

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None t he less, the outcomes are well wor th the effort . In gen eral, theproject submit ted is useful to t he proposer and often t he problem s have con­seque nces beyond the course . Students be come highly mo ti vated to st udyappropriate analytical t ools. They exhibit inter est in t he projects and areproud of t he ir work. Stu den ts indicat e on course eval uation forms t hat t hepro jects are wor thwhile, incr ease understanding, and require an extens ivecom mit me nt of time and ene rgy. Studen ts have an opportunity to showthe depth of t he ir underst anding wh en time is not a cons t rai nt, as it is forin- clas s tests.

A few words of advice :

Do not be afraid to supe rvise a project about which you knowver y little. You may learn with the students and be a role modelfor your students.

To be successful, students will make frequ ent contac t with th e professor.Be accessible to your st ude nts - an op en door policy for students is helpful.Through t his course, stude nts have found career paths th at t hey did no tkn ow existed. \Vorking indep enden tly on research pr oj ect s enhances st u­den t s' confide nce, and pr ovid es an oppo rtunity to develop an expe r t ise t hatt hey did not have before t hey began.

BIOGRAPHICAL SKETCH

Professor R ash received her PhD from th e University of P ennsylvania. Sh eis curre nt ly Chair of t he Mathematics and Computer Science Depart mentat St. J oseph 's Univers ity and also Direct or of t he Honors P rogra m . Hercurrent research inte rest is usin g t echn ology to enhance learning. Sh e isworking on a multimedia pr oj ect for bu sin ess calculus under a National Sci­ence Foundati on gran t . Sh e is a speaker and a session chair at MathematicsAssociation of Am erica and National Council of Teachers of Mathemati csmeetings and a pr oponent for Wom en in Mathematics who interests youngwom en in ma themati cs. Sh e is also a textbook aut ho r.

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