# 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 μ, 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 μ 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 μ and standard deviation 10 grams. (The mean μ 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 μ, the correlation between the weights of any two boxes is CORR=0.63.

(a) Suppose that YUMM selects μ = 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 μ = 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 μ = 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 μ 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.