A manufacturer of booklets packages them in boxes of 100 booklets. It is known that, on the average, the booklets weigh 1 ounce, with a standard deviation of 0.024 oz. The manufacturer is interested in calculating the probability that a given box of 100 booklets weighs more than 100.5 ounces. This probability would help in detecting whether too many booklets are being put into a box.
(a) Explain how you would calculate the (approximate?) value of this probability, and calculate it. Mention any relevant results or assumptions needed.
(b) Suppose that these boxes of booklets are shipped in collections of 30 boxes. The manufacturer randomly selects one of these shipments and determines the weight (in ounces) of each of the 30 boxes in that shipment. If at least 4 of the boxes in the shipment exceed 100.5 ounces, the shipment will be rejected and returned for repackaging. What is the probability a randomly selected shipment will be rejected? State any assumptions made, define all terms used, and explain your reasoning clearly. Make sure to keep at least 4 decimal places in all proportions calculated.