Question: [ 30 points]

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 .

1. [10 points] Suppose that YUMM selects μ = 470 grams. What is the probability that a single box is under the 450 grams that the box is supposed to weigh?
2. [10 points] At what value should YUMM set μ so that the probability is 0.95 that the weight in a single box is at least 450 grams?
3. [10 points] Suppose that μ = 460 grams. What is the probability that the total weight of a "2-pack" is less than 900 grams?

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