Estimate the proportion of brown M&M's and set a 90% Confidence Interval
Material:
Bags filled with candies (approximately the same numbers in each bag)
Clear plastic cups
Spoons to "sample" from the cups
Napkins, paper towels, or a clean sheet of paper
Large laminated graph chart to display results of our samples
(90% Boxplot charts( Exploring Surveys and Information from Samples, Landwehr, Swift and Watkins, Dale Seymour, 1987)
Calculators
Part I (Real M&M's)
1. Give each student or group a bag of candies, a cup, a spoon, and a napkin.
2. Count the number of M&M's in the bag and calculate the percent of each color.
3. Record the values of brown and total candies for the entire group.
4. Put all the candies in the cup and stir well.
5. Draw a "random" sample of size 20 and count the number of brown M&M's. Find a 90% confidence interval using the box plot for sample size n=20. Ask how many of these intervals capture to "true value" computed in step 3. Hopefully some (10%) did not contain the value.
6. Use the box plot charts to find a 90% confidence interval for the first sample collected, using increasing sample sizes (n=40, n=80, and n=100) formed by 2, 4, and 5 students.
7. Continue to stir and sample, recording the number of browns for each sample until a large number of samples have been collected, 100 is best. Agree that since the bags all came from the original population, we could collect data from the group to form these 100 values.
8. Now we want to display the sampling distribution of these 100 values on the large chart. Have the students read out the number of brown M&M's and mark an X for each value. An approximately normal distribution should emerge. To form a 90% confidence interval, remove 5 from the top and bottom, taking out an entire class, if necessary. Discuss the nature of the interval and its margin of error from the central estimate.
9. Ask the students to state the meaning of the 90% Confidence Interval from Step5 and Step 8.
10. Conclude by having each student write a statement of what a confidence interval tells us about the population.
Part II Calculator simulation (a random number table can also be used)
1. Determine how the digits 0-9 can be used to represent M&M's. Suggest 0,1,2 for brown (30%)
2. RandInt (0,9,20) will draw 20 digits.
3. Generate a list of 20 digits and count the number of 0, 1, 2 values (it helps to sort the list).
4. Use the 90% box to find the 90% confidence interval for the trial. Did all capture .3?
5. The class should continue to draw samples and collect the number of "browns" in each sample until 100 samples are gathered.
6. Chart the 100 sample results and remove the top and bottom 5 (as in Part I).
7. Each student should continue to draw 40, 80 and 100 digits and again use the box plots to find the 90% confidence intervals.
8. Conclude by having each student write down the meaning of each confidence interval.
Part III Binomial interpretation (Brown is a success)
1. Draw samples using the TI-83 command RandBin (20,.3).
2. Again use the box plots to find the 90% confidence interval.
3. Repeat the sampling to collect 100 trials. Chart the 100 (or store in a list and sort) and take out the top and bottom 5 values.
4. How many intervals did not capture the proportion of 30%.
NOTE: Simulation results can be compared with actual calculation of the confidence interval formulas.