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Problem 66 Easy Difficulty

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.

Answer

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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Video Transcript

{'transcript': "Hi. This question would be laughed toe to find Chi square and justice is six for two different samples. So for the 1st 1 I used many tab on. I found that guys square is equal to seven point 16 degree. Freedom is nine because the real freedom is equal and minus one, and evil is 1758 on when we change the sample to 20 we found that. So the statistics is the team board 96 So and increases okay on the P value, it becomes now going forth to eight. So it decrease and that's actually take care of part C and the So what will have been for that? This is a six, as is gonna increase because, as you know, from the formula, it is and minus one here, this where with us in the square. So I'm assuming this part this fixed constant in some sense. So when you're in keys in Chi Square or the just statistics is gonna increase, and that's gonna have a consequence, which is we're gonna have ah better P values, which is, um, less big value here"}