*Submitted by Dr. Ted Stanford, NMSU Mathematician.
The following box plots come from the 8th grade CMP unit, Samples and Populations. Some Natural brands of peanut butter and some Regular brands of peanut butter were each rated on a quality scale by a panel of trained tasters. Based on these box plots, if a Natural brand of peanut butter and a Regular brand of peanut butter are selected at random, what is the probability that Natural brand will have a higher quality rating than the Regular brand?
Answer: The probability must be greater than 9/16.
Explanation: 3/4 of the time you will get a Natural brand greater than the first quartile, which is a quality rating of about 57. Also 3/4 of the time you will get a Regular brand below the third quartile, which is about 54. Since the random choice of a Natural peanut butter is independent from the random choice of the Regular peanut butter, the probability that the Natural will be above 57 and the Regular will be below 54 is (3/4) x (3/4) = 9/16. Remark: There isn’t enough information in the box plot to give an exact answer to this question. You have to use some kind of estimation strategy or lower bound strategy. There are other possibilities besides the explanation above. For example, the 9/16 lower bound can be improved to 10/16 by observing that the lowest 1/4 of the Natural brands have a higher rating than the lower 1/4 of the Regular brands. If you want an exact answer to the problem, you can look up all the peanut butter data in Samples and Populations.