# Normally Distribututed Data

Please help with this multi-part question on statistics:

The YUMM cereal company distributes Colored Sugar Cereal. Each box is supposed to contain 450 grams of cereal. They also sell cereal in a "2-pack," where each "2-pack" contains 2 boxes of cereal. The 2-packs are supposed to have a total weight of 900 grams. YUMM can choose Î¼, the actual mean amount of cereal to put in each of the boxes, but their filling process has some inaccuracies. Regardless of the value of Î¼ that they select (typically between 450 grams and 500 grams), the amount of cereal placed in the box by their filling

process is Normally distributed with mean Î¼ and standard deviation 10 grams. (The mean Î¼ is the same for all of the boxes.) Since each box is poured by the same machine that has been calibrated to the chosen value of Î¼, the correlation between the weights of any two boxes is CORR=0.63.

(a) Suppose that YUMM selects Î¼ = 470 grams. What is the probability that any given box is under the 450 grams than the box is supposed to weigh?

b) Suppose that YUMM selects Î¼ = 460. What is the expected total weight of the 2 boxes in a given "2-pack"? What are the variance and the standard deviation of the total weight of the given "2-pack"? What is the distribution of the total weight?

(c) Suppose that Î¼ = 460 grams. What is the probability that the total weight of a given "2-pack" is less than 900 grams?

(d) At what value should YUMM set Î¼ so that the probability is 0.95 that the weight in any given single box is at least 450 grams?

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#### Solution Preview

Solution:

a. Let X be mean weight of first pack and Y be mean weight of second pack

Since X is normally distributed, X~N(u, 10). And Z=(X-470)/10.

So ...

#### Solution Summary

The solution gives detailed steps on finding the probability for the normally distributed data. All formulas and workings are shown.