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The data below are the net weights (in grams) for a sample of 30 bags of $\mathrm{M} \& \mathrm{M}$ 's. The advertised net weight is 47.9 grams per bag.

$$\begin{array}{llllll}\hline 46.22 & 46.72 & 46.94 & 47.61 & 47.67 & 47.70 \\47.98 & 48.28 & 48.33 & 48.45 & 48.49 & 48.72 \\48.74 & 48.95 & 48.98 & 49.16 & 49.40 & 49.69 \\49.79 & 49.80 & 49.80 & 50.01 & 50.23 & 50.40 \\50.43 & 50.97 & 51.53 & 51.68 & 51.71 & 52.06 \\\hline\end{array}$$

The FDA requires that (nearly) every bag contain the advertised weight; otherwise, violations (less than 47.9 grams per bag) will bring about mandated fines. (M\&M's are manufactured and distributed by Mars Inc.

a. What percentage of the bags in the sample are in violation?

b. If the weight of all filled bags is normally distributed with a mean weight of 47.9 g, what percentage of the bags will be in violation?

c. Assuming the bag weights are normally distributed with a standard deviation of $1.5 \mathrm{g},$ what mean value would leave $5 \%$ of the weights below $47.9 \mathrm{g} ?$

d. Assuming the bag weights are normally distributed with a standard deviation of $1.0 \mathrm{g},$ what mean value would leave $5 \%$ of the weights below $47.9 \mathrm{g} ?$

e. Assuming the bag weights are normally distributed with a standard deviation of $1.5 \mathrm{g},$ what mean value would leave $1 \%$ of the weights below $47.9 \mathrm{g} ?$

f. Why is it important for Mars to keep the percentage of violations low?

g. It is important for Mars to keep the standard deviation as small as possible so that in turn the mean can be as small as possible to maintain net weight. Explain the relationship between the standard deviation and the mean. Explain why this is important to Mars.

a. $20 \%$

b. $50 \%$

c. 50.37 grams

d. 49.55 grams

e. 51.39 grams

f. percentage of violations low

g. standard deviation low

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University of Pittsburgh - Main Campus

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