1. M&M’s weigh 0.86 grams on average with SD = 0.04 grams, so the coefficient of variation is 0.04>0.86 < 0.047. Suppose that we decide to label packages by count rather than weight. The system adds candy to a package until the weight of the package exceeds a threshold. How large would we have to set the threshold weight to be 99.5% sure that a package has 60 pieces?

2. Suppose the same system is used (packaging by count), but this time we only want to have 10 pieces in a package. Where is the target weight? (Again, we want to be 99.5% sure that the package has at least 10 pieces.) What assumption is particularly relevant with these small counts?

3. In which situation (10 pieces or 60 pieces) would the packaging system be more likely to put more than the labeled count of candies into a package?

A SAMPLING EXPERIMENT
This experiment works well in a class, with some friends, or—if you really like chocolate—all by yourself. It requires two packages of milk chocolate M&M’s® if you are doing it alone, or one for each person for groups of two or more. We use milk chocolate M&M’s, but you can use any variety. Make sure that both bags are the same variety. You should also visit the company Web site to see if it shows the underlying proportions of colors for your variety; the distribution of colors varies among different types. We used packages labeled 1.69 ounces; these have about 60 pieces.

Milk chocolate M&M’s currently come in six colors: brown, yellow, red, blue, orange, and green. The colors are not likely equal. Brown, once the only color, is less common. Blue, a recent addition to the palette, is more common and usually accounts for more than one-sixth of plain M&M’s.

Open a bag of M&M’s and count the number of candies in each color. Figure out the percentage of blue candies in your bag and write it down.

Now open the second bag and find the percentage of blue M&M’s in this second bag. Write down this percentage as well.

A Model for Counts
Mars, the manufacturer of M&M’s, is privately held and secretive about its operations, but we can guess that elves don’t count the M&M’s that go into every bag. It’s more likely that the bags are filled from a large “bowl” that mixes the colors. The packaging process basically scoops the contents of each bag from this bowl of differently colored pieces. How can we model the colors in a bag?

The binomial model (Chapter 11) is a reasonable candidate to describe the number of blue M&Ms in a bag. Every binomial model has two characteristics, or parameters, that identify the model. These parameters are n, the total number of counted items, and p, the probability of a “success.” Let’s define a success in this case as finding a blue M&M. The total count n is the number of candies in the bag; n for your bags should be about 58.

All manufacturing processes have variation, and sometimes it is a surprise to consumers. You might think that every package of M&M’s has the same number of candies; after all, they are labeled to contain the same weight. Automated packaging systems, however, are not so consistent, and the number of pieces varies from bag to bag. We got the counts shown in the histogram in Figure 1 by counting the pieces in a case of 48 bags, each labeled to weigh 1.69 ounces.