A tire manufacturer warranties its tires to last at least 20,000 miles or "you get a new set of tires." In its experience, a set of these tires lasts on average 26,000 miles with SD 5,000 miles. Assume that the wear is normally distributed. The manufacturer profits $200 on each sold set, and replacing a set costs the manufacturer $400.
a) If the manufacturer sells 500 sets of tires, what is the probability that it earns a profit after paying for any replacements? Assume that the purchases are made around the country and that the drivers experience independent amounts of wear.
b) The tire manufacturer decides that there will be too much negative publicity if more than ½% of its tires fail before the 20,000 mile mark. With some additional effort in the manufacturing process they can produce tires with a mean of m and standard deviation 5000. How should they choose m so that only ½% of their tires fail before the 20,000 mile mark? (Assume that the failure mileage for new tires is normally distributed.)