«1 Introduction The key to effective public speaking, I have been told, is to begin with a funny story. Thus, I will begin this article with a story. ...»
Small Student Projects in an Introductory Statistics Course
Robert L. Wardrop∗
Department of Statistics
University of Wisconsin-Madison
July 3, 1999
The key to effective public speaking, I have been told, is to begin with a funny story. Thus, I will begin this
article with a story.
There was a one-room school house in a remote rural area. Periodically a regional supervisor
would visit the school to evaluate the performance of its teacher. On one visit the supervisor viewed a unit on the addition of fractions. The teacher wrote on the board, 2/3 + 4/5 = 6/8. After school that day the supervisor spent a great deal of time ensuring that the teacher understood the correct way to add fractions.
On the next visit to the school the supervisor was dismayed to ﬁnd the teacher again was using the incorrect method of adding fractions. Confronted later in the day, the teacher replied, “I tried your method, but mine is easier to teach.” The method’s beneﬁt to the teacher is, of course, overwhelmed by its disservice to the students. I will share brieﬂy some ideas I have on what makes a statistics course good or bad. In particular, I will argue that on certain occasions a teacher should opt for a more difﬁcult way to teach statistics.
My ideas can be summarized quite well by the paradigm: be willing, not willful. Below is an example of a willful statistics exercise.
Sally has a random sample of size n = 10 from a normal population with unknown mean and standard deviation. Use Sally’s data to construct a 95 percent conﬁdence interval for the mean of the population.
This exercise is willful because its author tells the student exactly what to assume in order to complete it.
This exercise is easy for the instructor because there is a known unique correct answer.
A willing exercise would provide a description of the scientiﬁc problem that motivates Sally’s work. It would carefully describe how Sally obtained her data. Finally, a willing exercise would ask the student to explain what can be learned from Sally’s data. The willful exercise amounts to teaching the student to turn a crank or push a button. The willing exercise forces the student to think; to draw upon ideas presented in the class along with other knowledge the student might have and combine these in a creative way. The willing exercise is difﬁcult for the instructor because he or she does not possess the unique correct answer. I learned ∗ Robert Wardrop is Professor, Department of Statistics, University of Wisconsin-Madison, 1210 West Dayton Street, Madison, WI 53706.
long ago that if I assign willing exercises, I must check my ego at the door—frequently one of my students ﬁnds a clever and creative answer that I missed.
Certainly, some willful exercises have a place in the best of courses. My complaint is with courses whose only goal is to prepare students to satisfactorily complete willful exercises. There is a great temptation to omit any willing exercises from a course. Willing exercises take time. As long as there are powerful people who judge a statistics course by “How many topics are covered,” teachers will face pressure to omit willing exercises, leaving more time to cover more topics.
I consider the subject of statistics to be a collection of concepts, principles, and methods that help scientists learn about the world. (I deﬁne a scientist as any person who wants to understand better his or her natural, political and social worlds.) Mathematics is a critical component of statistics, and, I believe, all other things being equal, the person who is the better mathematician will be the better statistician. If my view has any validity, should not our introductory statistics courses make some effort to help students become better scientists? For example, the ﬁrst day of class I announce that the fundamental goal of my
To enable students to discover that statistics can be an important tool in daily life.
I want my students to learn that statistics is useful because it can help them learn about things that they ﬁnd interesting.
My fundamental goal has an impact on various aspects of my course: the choice of text, the material presented, the methods of presenting the material, and the methods of evaluating the student’s learning. In this paper I will limit attention to one aspect of my course: my use of small student projects. See , , , and  for related work. As will become immediately obvious in Section 3, student projects can give rise only to willing exercises.
2 Limitations Beginning in the next section, I will present several examples of student projects from my classes. As will become obvious, my students deserve all the credit for the creativity and cleverness their work demonstrates.
My main contribution is that I assign this work to them. I believe that an ideal introductory class would require each student to do several projects. Limits on my time, however, require that I assign only two projects in my course.
Based on my teaching experiences, I fervently believe that a student learns more by analyzing data that he or she has collected, than by analyzing data collected by another. This discovery did not surprise me.
What did surprise me is that most of my students prefer studying other students’ projects to “professional” research that I present. I do not know why this is true; I conjecture that there is less variation in interests within students than there is between the students and me. This is the ﬁrst of two reasons I use numerous student projects as examples in lecture—my students ﬁnd them to be interesting. The second reason is that I learned in the early days of assigning projects that few students are creative without models of good projects, but almost every student is creative when provided models of good projects. (This partly explains why this paper contains numerous projects; if you choose to assign projects you might want to make the project topics below available to your students.) Figure 1, taken from , displays four possible study designs and notes their allowable inference(s), if any. Units can be selected at random or not at random, and can be allocated to groups by randomization or not. If units are selected at random, then inferences to populations are valid. If units are allocated to groups by randomization, then causal inferences are valid. All introductory Statistics texts present inferences to populations, but relatively few discuss causal inferences. This is unfortunate. Arguably a key component
Figure 1: Statistical inferences permitted by study designs (from The Statistical Sleuth by Ramsey and Schafer). For the designs in the ﬁrst row, inferences to populations can be drawn; for the designs in the ﬁrst column, causal inferences can be drawn.
of statistical literacy is the ability to distinguish between causal and noncausal links between variables. It is difﬁcult to distinguish a difference if one is never taught that such differences exist.
I want my students to be able to perform valid inference, but I do not want them to expend the time and effort necessary to obtain random samples. Hence, my students “work in” the lower left entry in Figure 1;
they can draw causal inferences, but not population inferences. I refer to such inference as randomizationbased inference. For details on randomization-based inference see , , and .
3 Eleven Projects with a Dichotomous Response For many of my students, their daily life includes working to obtain enough money to continue school, and these students are very interested in ways to earn more money. The ﬁrst four projects below were executed on-the-job. (Note: My students have given me permission to use their work and I will identify each project with the ﬁrst name(s) of its author(s).)
1. Lori worked as a waitress and wondered whether suggesting a speciﬁc appetizer upon greeting her customers would lead to an increase in the sales of appetizers. Her data, though not statistically significant, ran counter to her belief—mentioning a speciﬁc appetizer decreased the sales of all appetizers!
2. Nell worked at a coffee cart which offered two sizes, small and large. Of course, some customers speciﬁed the size when they ordered, but Nell was interested in those who simply ordered coffee.
She found that the question, “Would you like a large?” was statistically signiﬁcantly superior to the question, “Would you like a small or a large?” at eliciting the purchase of a large cup of coffee.
3. Andre’s work problem was similar to Nell’s. He sold ice cream which could be served in a plain or a wafﬂe cone. His store made a much larger proﬁt on the wafﬂe cones. For customers who did not specify the cone type in their order, Andre found that the question, “Would you like a plain cone or a homemade wafﬂe cone?” elicited signiﬁcantly more sales of wafﬂe cones than did the same question with the adjective “homemade” deleted.
4. Mary’s job duties included the purchase of used compact discs from customers. Mary wondered which of the following statements to the customer would be more effective.
• I will give you $X.
• How does $X sound?
(The value of X was appropriate for the number and quality of the compact discs offered for sale.) Mary found that the ﬁrst statement performed better (at getting an acceptance of the ﬁrst offer) than the second, but that the difference barely missed achieving statistical signiﬁcance.
The following project was motivated by the experiences of its author and many of his friends.
5. Tuan was interested in the problems an international student faces at the University of Wisconsin– Madison. He showed 25 graduate students part of an essay he said was written by “Jack McConnell, a student from Iowa.” He showed 25 other college graduates the same essay, but said it was written by “Hsiao-Ping Zhang, an international student from China.” When asked, “Do you detect any grammatical errors in this passage?” 64 percent who had read the Chinese student’s essay said yes, compared to only 20 percent who had read the Iowa student’s essay! This huge difference is highly statistically signiﬁcant.
6. Ruth visited a minimum-security federal prison camp to obtain her subjects, ﬁrst-time nonviolent offenders. The ﬁrst version of her question read, The prison is beginning a program in which inmates have the opportunity to volunteer for community service with developmentally disabled adults. Inmates who volunteer will receive a sentence reduction. Would you participate?
The second version was the same, except that there was no mention of sentence reduction. Ruth was surprised when her data revealed that the second version received a much higher proportion of yes responses than the ﬁrst version, but the difference did not quite achieve statistical signiﬁcance.
In the above studies, the experimental units are people. More often, my students choose units that are trials. The remaining examples are of this latter type. Frequently, my students choose to use statistics to investigate longstanding hobbies or pastimes.
7. Erin began her project with the following statement.
Ice-skating has been a part of my life for 12 years and one thing that stands out in my mind... were the numerous arguments I would stubbornly have with my coach.
Erin’s trial was an attempt at an axel (a ﬁgure skating jump with one and one-half completed turns).
Erin found that she was highly statistically signiﬁcantly better at completing an axel off her right foot than an axel off her left foot.
8. Kathy found that she was much better at successfully completing a cartwheel if she led with her right hand rather than her left.
9. When signing the alphabet, it is standard to use one hand as the primary hand and the other as the assister. Diana found that choosing her right hand as primary made her a better signer than when her left hand was primary, but the difference just failed to achieve statistical signiﬁcance.
10. Lisa S. found that she was a much better lacrosse shooter with her right hand than with her left.
11. Mike enjoyed riding his personal water craft very fast and making sharp turns. His pleasure was diminished, however, whenever he would be thrown from the craft and land 30 feet away from it.
Mike found that he was much better at staying on-board if he turned left than if he turned right.
In the next section I will present projects with a numerical response. Before turning to them, I want to share a bit of what my students and I have learned from projects like those above.
Over the past few years I have graded thousands of student projects. A small number of the projecttopics appears to have been chosen to complete the assignment with as little thought and effort as possible.
For example, a study of the effect of the hand used on the outcome of the toss of a coin would be of this type. I make it clear to my students that it is not necessary for me to consider the topic to be interesting; it is necessary, however, that the project report convincingly explains why the topic is of interest to the student.
The great majority of the projects are creative and interesting, and not qualitatively different from the ones described above. Students show great interest in studying ambidexterity, and various sports and games, especially darts, golf, basketball, archery and sharpshooting.
Analyzing their own data makes students appreciate the importance of the P-value. Obtaining a very small P-value seems to make them think, “Yes! I have achieved results that standard scientiﬁc practice says are real!” When a difference that is large enough to be important is not statistically signiﬁcant, most students realize in a very personal way the advantages of collecting more data. Unfortunately, some students misinterpret P-values larger than 0.05 as proving that the treatments are identical.