Question: (15 points) The Rice Cereal Company (RCC) makes Road Runner cereal. Each box of Road Runner is “guaranteed” to contain at least 500 grams of cereal. RCC also sells Road Runner cereal in 10-packs, where each 10-pack contains 10 boxes of cereal. RCC can set ?, the mean amount of cereal to put in each of the boxes, but its filling process has some random variation. Regardless of the amount ? that they set (typically between 500 grams and 550 grams), the amount of cereal placed in one box is independent of the amount in the other boxes, and it obeys a Normal distribution with mean ? and standard deviation ???20 grams.

(a) Suppose that RCC’s production manager has set the mean to be ? = 510 grams. What is the probability that a randomly selected cereal box does not meet the “guaranteed” weight of 500 grams?

(b) RCC wants 99% of the 10-packs it produces to contain at least 5,000 grams of cereal in total. What value of the mean ? should RCC set to achieve this goal?

(c) Suppose that RCC sets ? = 540 grams. A government agency buys a 10-pack of Road Runner cereal from a randomly selected store. If two or more boxes in the 10-pack are found to contain less than the guaranteed 500 grams of cereal, then the agency will impose a penalty on RCC. What is the probability that a penalty will be imposed?