Bob-- Thanks for clarifying why I have been getting more and more uncomfortable about this M&M's thing. I had pretty much come to the conclusion that it was a terrible fit the the Chi-Square G.O.F. model, since you aren't sampling individual candies, but bags of them.
Do you (does anybody?) have any other ideas for a hands-on application of Chi-Square?
Bob Hayden's Message Text:
This only hides the problem. You will now get a lot more variability than sampling theory would suggest. Despite their claims to the contrary, Mars does not mix the candies thoroughly enough for them to be considered a random sample from a fixed population. I think they are great for demonstrating sampling variability (especially when they exaggerate it!) but not for any inference procedure based on random sampling, i.e., not for any inference procedure in an introductory course.
Here's one intuitive way to think about the problem. If the bags were random samples, the sampling distribution of the proportion of tan M&Ms would be close to normal. Suppose they mix them so badly that virtually every bag contian only one color. Then the sampling distribution of the proportion of tan ones has spikes at 0% and 100% and virtually nothing around the population proportion of tan M&Ms. It will not have the variance or shape that sampling theory predict (and use as the basis for inference).
To see how bad the problem is, open many bags of M&Ms and compare the empirical and theoretical variance of the sampling distribution of the proportion of each color. (I did it many years ago.)