Texas A&M; University

Journal of Statistics Education v.2, n.1 (1994)

Copyright (c) 1994 by Stephen Eckert, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.

In elementary statistics courses, students often have difficulty understanding the principles of hypothesis testing. This paper discusses a simple yet effective demonstration using playing cards. The demonstration has been very useful in teaching basic concepts of hypothesis testing, including formulation of a null hypothesis, using data as evidence against the null hypothesis, and determining the strength of the evidence against the null hypothesis, i.e., the p-value.

1 I have taught an elementary, algebra-based statistics course for six semesters. During my teaching, I have tried to think of clever ways to communicate basic statistical ideas. I have always had difficulty in relating the basic concepts of hypothesis testing. I tried a "courtroom drama" skit using class members, but I never felt the idea caught on very well, or that students fully understood the implications. I have recently developed and tested a demonstration using cards which I found very successful in teaching the basic concepts of hypothesis testing.

2 The only item needed for this demonstration is a deck of 52 cards, all of which are the same color. (I have deck consisting of all red cards, for example.) I recommend purchasing two decks of cards and putting all the red cards in one deck and all the black cards in another deck. It is important that the back sides of these cards are identical. The resulting deck needs to look like a standard deck of cards.

3 Call for a volunteer from the class. In selecting the volunteer, pick someone who likes to participate in class and who is not easily embarrassed. Tell the class:

--> "I am going to have Cindy pull out a card from the deck. What is the probability that she pulls out a red card?"

4 Hopefully at this point, every person in the class will yell out 50% or 1/2. At this point, go to the chalkboard or overhead projector and write "p=0.5".

--> "If I have her select 10 different cards, and I replace the selected card and shuffle between picks, how many of the 10 cards do we expect to be red?"

5 Again, most students will answer with "5". I add:

--> "Is she guaranteed to get exactly 5?"

6 And at this point the students say "no" and roll their eyes, because I go over and over the idea of sampling variability, and am always asking questions to emphasize that the value of a statistic will change with repeated sampling.

7 Now the instructor should go to the chalkboard and write "RED" and "BLACK". After the student selects a card, the instructor will make a tally mark under RED if the card is red, or under BLACK if the card is black. (In this example, the instructor will of course make all his marks under the RED label.) Have the student pull out a card, look at it, call out "RED" to the class, and mark a tally under "RED" on the chalkboard. The student will then replace the card, and the instructor will give a quick shuffle. The student then selects another card. It is important to never show the faces of the cards while shuffling.

8 After about four or five red cards have been chosen, the students begin to get suspicious. A good thing to do at this point is say:

--> "Thanks for being the only student to ever ruin my demonstration."

9 This makes the others in the class laugh, embarrasses the student who is choosing the cards, and, most importantly, makes the students believe for a few more card selections that it is still possible that the deck is half red. 10 After the student has selected 10 cards, have her sit down.

11 After the student has taken her seat, point to the "p=0.5" on the chalkboard, ask:

--> "Do you believe that there is a 50% chance for drawing a red card with this deck?"

12 The students all say no. Then say:

--> "I agree. Now, is it possible that a person with a normal deck of half red and half black cards could pull out 10 red cards in a row?"

13 The students all chime in "yes".

--> "Is it very likely that a person would pull out 10 red cards if the deck were half red?"

14 Again, all the students say "no". (At this point, if the instructor has discussed binomial probabilities, students in the class can calculate the probability of all red cards, which in this demonstration serves to represent the p-value.)

--> "Now, we have two seemingly contradictory pieces of information about the deck of cards. We have a claim that p=0.5, and we have done an experiment in which 10 out of 10 cards chosen were red. The data which we collected seem 'inconsistent' with the hypothesis. That is, if the hypothesis were true, it would be very unlikely to have all 10 chosen cards be red. And yet, in our experiment, we selected 10 red cards. What should we conclude?"

15 The students should say that it does not seem likely that "p=0.5" is reasonable, given that all 10 chosen cards were red.

16 The instructor then begins a discussion about hypothesis testing. The important things to discuss, in my view, are:

- We set up a hypothesis, and assume that it is true.
- We gather data from some real-world experiment that is relevant to the hypothesis.
- We then make a determination about the hypothesis, based on the idea of "how likely is our data given the hypothesis?"

17 Another idea which can be discussed is that of "how much evidence is necessary?" The instructor can point out that after two cards had been drawn, no one was convinced that the deck was altered. But after 10 cards, everyone was sure that the deck was non-standard. I usually ask, by a show of hands, how many people began to be suspicious after 3 cards, after 4 cards, after 5 cards, etc. The instructor can discuss that different people require different amounts of convincing, and then lead into a discussion of p-values and fixed level tests.

18 I have given a demonstration which teaches some of the
principles in hypothesis testing. The card demonstration does not
*exactly* parallel a standard hypothesis test, but it does
do a good job in teaching the concepts behind a significance test
in a manner which is both entertaining and educational for the
students.

The author wishes to thank Dr. Elizabeth Eltinge, Texas A&M; University, for her helpful suggestions on previous drafts of the manuscript.

Stephen Eckert

Department of Statistics

Texas A&M; University

College Station, TX 77843-3143

eckert@stat.tamu.edu

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