Natural Foods sells Takeoff, a breakfast cereal, in one$-$ pound boxes. According to Food and Drug Administration $($FDA$)$ regulations, a one$-$pound box must contain at least $15\mathrm{.}5$ ounces of cereal. However, Natural Food$$s box$-$filling process is not perfect: its precision, expressed in terms of the standard deviation of the weight of a one$-$pound box filled, is $0\mathrm{.}5$ ounces.a$.$ Where should Natural Foods center its process in order to ensure that $98\%$ of boxes filled meet FDA requirements? What proportion of boxes would be overfilled beyond $16$ ounces?

b$.$ While underweight boxes might prompt FDA action, overweight boxes certainly cost Natural Foods in terms of higher material costs. Therefore, quality control manager Morris Nerdstat wants to monitor the cereal$-$filling process in order to ensure that its mean does not deviate from the level estab$-$ lished in Part a$.$ He plans to weigh nine randomly selected boxes at regular time intervals and plot the average weight on a chart. At one point, he finds an average weight of $15\mathrm{.}9$ ounces. The company$$s legal staff is pleased that this performance is better than the FDA requirement of $15\mathrm{.}5$ ounces. What action, if any, should Nerdstat take?

c$.$ What targets $($in terms of the mean and the stan$-$ dard deviation$)$ would result in the process with six$-$sigma capability?