Question: Question 1: 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) [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?

b) [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?

c) [10 points] Suppose that μ = 460 grams. What is the probability that the total weight of a “2-pack” is less than 900 grams?