1. According to Business Week, Maternity Chic, a purveyor of designer maternity wear, sells dresses and pants priced around $150 each for an average total sale of $1,200. The total sale has a normal distribution with a standard deviation of $350.
a. Calculate the probability that a randomly selected customer will have a total sale of more than $1,500.
b. Compute the probability that the total sale will be within 2 standard deviations of the mean total sales.
c. Determine the median total sale.
2. The Aberdeen Coca-Cola Bottling plant located in Aberdeen, North Carolina, is the bottler and distributor for Coca-Cola products in the Aberdeen area. The company's product line includes 12-ounce cans of Coke products. The cans are filled by an automated filling process that can be adjusted to any mean fill volume and that will fill cans according to a normal distribution. However, not all cans will contain the same volume due to variation in the filling process. Historical records show that regardless of what the mean is set at, the standard deviation in fill will be 0.035 ounces. Operations managers at the plant know that if they put too much Coke in a can, the company loses money. If too little is put in the can, customers are short-changed and the North Carolina Department of Weights and Measures may fine the company.
a. Suppose the industry standards for fill volume call for each 12-ounce can to contain between 11.98 and 12.02 ounces. Assuming that the Aberdeen manager sets the mean fill at 12 ounces, what is the probability that a can will contain a volume of Coke product that falls in the desired range?
b. Assume that the Aberdeen manager is focused on an upcoming audit by the North Carolina Department of Weights and Measures. She knows the process is to select one Coke can at random and that if it contains less than 11.97 ounces, the company will be reprimanded and potentially fined. Assuming that the manager wants at most a 5% chance of this happening, at what level should she set the mean fill level? Comment on the ramifications of this step, assuming that the company fills tens of thousands of cans each week.