# An In-Class Experiment to Estimate Binomial Probabilities

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#### Summary

This hands-on activity is appropriate for a lab or discussion section for an introductory statistics class, with 8 to 40 students. Each student performs a binomial experiment and computes a confidence interval for the true binomial probability. Teams of four students combine their results into one confidence interval, then the entire class combines results into one confidence interval. Results are displayed graphically on an overhead transparency, much like confidence intervals would be displayed in a meta-analysis. Results are discussed and generalized to larger issues about estimating binomial proportions/probabilities.

## Learning Goals

Students will learn how to identify and conduct a binomial experiment, how to compute a confidence interval for a binomial experiment, the role sample size plays in interval estimation, and how confidence intervals vary from one sample to the next.

## Context for Use

This is a hands-on activity for use in class (or lab/discussion) in an elementary statistics course. There should be between 8 and 40 students for the activity to work well. It can be done in as little time as half an hour, or can occupy a full 50-minute class, depending on how the instructor wants to conduct a discussion before and after the hands-on activity.

## Description and Teaching Materials

Each student will conduct a binomial experiment with n = 100, and compute a 95% confidence interval for p. Groups of 4 students then combine their results and get a 95% confidence interval for p with their combined n = 400. These intervals will be half as wide as the individual intervals. Finally, the entire class combines results to get a 95% confidence interval for p, with n = 100k, where k is the number of students in the class. This interval will be even more narrow, with its width depending on the number of students in the class. Students display their intervals on a "grid" with each interval as a horizontal line. (See below for an attachment with the grid.) Thus, students learn that:
1. Intervals based on n = 100 have different centers but are all about the same width. (Note that if you use something with a true p reasonably close to .5, the half-width of the intervals will be about 0.1.)
2. Intervals based on n = 400 have different centers, but are all about the same width and are half as wide as those with n = 100.
3. The interval based on the entire class is narrow and provides a more accurate estimate of p than the individual intervals.
4. The larger your sample size, the more accurately you can estimate a parameter.
Motivate the activity by explaining how prevalent binomial experiments are in the real world. Explain that each individual will conduct a binomial experiment to estimate the same unknown parameter. Discuss why the conditions for a binomial experiment are met, both for each individual and for the combined data.
There are three files to download. Two of them are provided in pdf and Word format. The pdf file is provided to make sure users can see the correct page layout. The Word files are provided so that users can modify the files. The first file contains detailed instructions for the teacher, including logistical issues and discussion questions. The second file is to be handed out to each team, with instructions for the students and spaces to record their results. The third file is the grid for recording the results, which should be printed onto an overhead transparency.

Instructions for the Instructor:
Instructor Instructions as pdf file (Acrobat (PDF) 115kB May17 07) Instructor Instructions as Word file (Microsoft Word 46kB May17 07)

Team Record Sheet to be handed out to students:
Student Team Record Sheet as pdf file (Acrobat (PDF) 74kB May17 07) Student Team Record Sheet as Word file (Microsoft Word 34kB May17 07)

Grid for overhead transparency to record results:
Grid for Overhead transparency (Microsoft Word 124kB May3 07)

The activity as described uses "animal eyes." These are plastic pieces used to glue onto a stuffed animal, which have a flat side and a curved side that looks like an eye. They can be purchased in a craft shop or online (type "animal eyes craft" into a search engine). The goal is to have a binomial situation for which the success probability is unknown. The animal eyes can be flipped and a "success" is that the eye lands looking up. You could use a different binomial experiment, such as estimating the proportion of one color in a multi-colored candy. The important criteria are that the experiment mimics a binomial setting, that each student can get an individual result with n = 100 trials, and that it makes sense to combine the results across students.

## Teaching Notes and Tips

Students have fun doing this activity. I provide paper bowls for holding the eyes, and students flip them into the bowl. I provide each pair of students with five eyes, so they can be efficient by flipping all five of them 20 times, to get n = 100. It's important to discuss the way they flip them, to make sure the trials are independent with constant p. Students work in pairs, with two pairs making up a team of four students. In each pair, one student flips the eyes while his/her partner keeps track of the number of flips and records the results. Then they switch roles. After each student has computed a confidence interval with n = 100 for themselves, the team comes together and computes a team confidence interval with n = 400. Additional teaching notes are provided in the attachment "Instructor Instructions" and directions to the students are provided in the attachment "Team Record Sheet."
The hardest part of this activity is reconvening the class when they have completed the experiment, and holding a discussion about the process and concepts. One possibility is to have students discuss the relevant questions in their teams, then report back to the class.

## Assessment

I provide a team record sheet to each team, so I know who was in class. I don't assess them beyond whether they participated or not. However, assessment could be built into the discussion of results. For instance, rather than having the whole class discuss issues like how confidence intervals widths change as number of trials change, students could be asked to write answers to these questions and turn them in.

## References and Resources

1. Activity-Based Statistics Richard L. Scheaffer, Mrudulla Gnanadesikan, Ann E. Watkins, and Jeffrey Witmer, Revised by Tim Erickson, Key Curriculum Press.
2. Workshop Statistics: Discovery with Data, 2nd Edition, Allan J. Rossman and Beth L. Chance, Key Curriculum Press
3. Mind On Statistics (2006), Jessica Utts and Robert Heckard, Duxbury/Brooks-Cole