Activity #2: Distributions

Below is the histogram of the total weight of a sample of 110 Fun Size Bags of M&Ms that you collected in Activity #1. The summary statistics for the distribution are Minimum = 18g, 1st Quantile = 20g, Median = 21g, 3rd Quantile = 22g, Maximum = 25g, Mean = 21.0091g and Standard Deviation = 1.2743g. Use this information to answer the questions below.

  1. Describe the distribution of the total weight of the Fun Size Bags of M&Ms.
  2. What do the values of the mean and standard deviation and the five number summary indicate about the symmetry/non-symmetry of the distribution?
  3. The bags are not labeled for individual sale, and so, do not include a weight. Based on the distribution of the total weight of the sample of 110 bags, how would you label a similar bag of M&Ms? Explain your reasoning.
  1. What does the shape of the histogram indicate about the probable shape of the density curve for the distribution of the total weight of Fun Size Bags of M&Ms?
  2. Describe the normal quantile plot for these data. What does this plot indicate about whether the distribution of the total weight of the Fun Size bags of M&Ms can be modeled with a normal distribution?
  3. Using your answers to questions 4 and 5 above, do you believe the total weight of the M&M bags can be modeled using a normal distribution? Explain your answer.

Fun Size Bags of M&Ms are sold as a part of a larger package. According to the Fair Packaging and Labeling Act, the law does not require each Fun Size Bag to carry a label weight; only the larger package must contain a weight of the contents in the package. If we wanted to sell Fun Size Bags of M&Ms separately, we would need to determine a label weight to place on each individual bag.

  1. Pick one summary value from the distribution and explain why you would use this summary value as the labeled weight of the Fun Size bags.
  2. M&Ms are also sold in larger individual packages. These packages are sold individually and so according to the law, must carry a labeled weight for the contents in the package. Below is the distribution of the total weight of a sample of 100 regular size bags of M&Ms. The summary statistics for these data are Minimum = 46g, 1st Quantile = 49g, Median = 50g, 3rd Quantile = 50g, Maximum = 53g, Mean () = 49.68g, Standard Deviation (s) = 1.2299g.

What labeled weight do you think is listed on regular size bags of M&Ms? Explain your answer.

  1. The actual labeled weight of the regular size bags of M&Ms is 47.9 grams. Given this information, pick one summary value from the distribution of the total weight of the sample of 110 Fun Size Bags to use as the labeled weight of the Fun Size bags. Explain why you now chose this value.
  2. The law that regulates the requirements for packaging and labeling of food and other products is called the Fair Packaging and Labeling Act. To determine a labeled weight for a package, the following portion of the law is relevant.

Variations from the stated weight or mass, measure, or numerical count shall be permitted when caused by unavoidable deviations in weighing, measuring, or counting the contents of individual packages which occur in good packaging practice: Provided, that such variations shall not be permitted to such extent that the average of the quantities in the packages comprising a shipment or other delivery of the commodity is below the quantity stated, and no unreasonable shortage in any package will be permitted even though overages in other packages in the same shipment or delivery compensate for such shortage. Variations from stated quantity of contents shall not be unreasonably large.

Given this information, make a final determination of the labeled weight of Fun Size Bags of M&Ms. Explain the reasoning behind your answer.

  1. The Fair Packaging and Labeling Act only deals with the weight of the contents (net weight), not the weight of the contents plus the packaging (total or gross weight). The distribution you were given is the total weight of the Fun Size Bags. How would you adjust your final labeled weight of the Fun Size bags to account for this difference?