Estimating Proportions: Just what is the proportion of M & M colors ?

In the lab, we will try to model the red M & M's by using beads. Hoping to determine a method for estimating the proportion accurately.

Part 1- Seeing the pattern

1. Repeat the sampling procedure used in the pre-lab four more times, for a total of five sample proportions for red beads. Each sample should be returned to the box before the next is selected, since all samples are supposed to come from the same population. After recording your  the give sample proportions, and give the results to the instructor.
2. Using the same bead color, repeat the procedure of step 1 for samples of size: 20, 40, 80, 100 Make sure you record give sample proportions for each sample size; give the results to your instructor.
3. Construct dot plots of the five sample proportions for each sample size on a single real number line. You may want to use different symbols for the different sample sizes. Do you see any pattern emerging?
4. Your instructor will now provide the data from all teams. Using a graphing calculator or computer, construct dot plots of the sample proportions for each sample size. (You should have five dot lots, one each for samples of size 10, 20, 40, 80, 100.) These distributions are approximations to the sampling distribution of these sample proportions.
5. Describe the patterns displayed by the individual sampling distributions. Then describe how the pattern changes as the sample size increases. In particular, where do the dot plots appear to center? What do you thin is the true proportion of red beads in the box?

Part 2 - Evaluating the distribution of the samples.

1. Calculate the Mean and Variance (this is the square of the standard deviation) for the sample proportions recorded in each dot plot.
2. Plot the sample variances against the sample size, with sample size on the horizontal axis. What pattern do you see? Does this look like the points should fall on a straight line?
3. It appears that the variances are related to the sample size, but the relationship is not linear. Plot the variances against (1/sample size), the reciprocal of the sample size. Observe the pattern that appears. "Fit" a straight line through this pattern either by eye or by using linear regression. Does the straight line appear to fit well?
4. What is the approximate slope of the line fit through the plot of variance versus reciprocal of sample size? Can you see any relationship between the slope and the true proportion of red beads in the box, as estimated step 5?
5. The above analysis should suggest that the slope of the line relating variance to (1/sample size) is approximately

where p represents the true proportion of red beads in the box

1. Write a formula for the variance of a sample proportion as a function of p and the sample size, n.
2. Write a formula for the standard deviation of a sample proportion as a function of p and n.

6. A two-standard-deviation interval is called the margin of sampling error (or the margin of error or the sampling error) by most pollsters.

1. In the repeated sampling, how often will the distance between a sample proportion and the true proportion be less than the margin of error?
2. Using the class dot plots constructed earlier in this lesson, count the number of times a sample proportion is less than two standard deviations from the true proportion for each plot. Do the results agree with your answer to part a?

1. Revisit the Scenario at the beginning of this lesson. For a Gallup poll, a typical sample size is 1,000. For the Clinton-bush election polls, the sample sizes were around 1,500.

1. Are the reported margins of error correct?
2. Interpret the results of these polls in light of the margin of error.

1. Does the concept of margin of error make sense if the data in a poll do not come from a random sample? Explain.
2. Write a brief report on what you learned about sampling distributions and margins of error for proportions.

1. Find examples of polls published in the media. If the margin of error is given, verify that is correct. If the margin of error is not given, calculate it. Discuss how the margin of error helps you to interpret the result of the poll.
2. The approximation to the standard deviation of a proportion was developed in step 10 for a single true value of p. How do we know it will work for other values of p? To see that it does, work through the steps outlined earlier for samples selected from a population with a different value of p. (The approximation does not work well for values of p very close to 0 or 1; so choose your new value of p between .1 and .9)