Question: A manufacturer of automobile batteries claims that the average length of life for its entire population of Grade A battery is 60 months. However, the full-replacement guarantee on this brand is for just 36 months. Suppose the standard deviation of the life length of the population is known to be 10 months and the population distribution for life length is normal. The manufacturer plans to sell 300,000 batteries next year.

a. Approximately what percentage of the 300,000 batteries would we expect to fail during their fourth year of life? (This would be between month 36.000 and month 48.000.) (7 pts.)

b. Of the 300,000 forecasted to be sold next year (2008) how many batteries would we expect to be returned for a full replacement during the guarantee period? (7 pts.)

c. Competition is forcing this manufacturer to increase its guarantee from 3 years to 4.5 years. How many batteries would we expect to be returned for a full replacement during the guarantee period? (7 pts.)

d. Assume it costs this company $150 to replace a defective battery in the field that fails under warranty. Assume also this company’s sales are forecasted to be flat at 300,000 units per year in 2008, 2009, and 2010. If the original battery design is retained (mean life span of 36 months, standard deviation of 10 months) what will be the total forecasted warranty expense for ALL covered batteries in the field in 2010? (10 pts.)

e. The CEO of this battery manufacturer enjoys classic cars and lovingly maintains a fully-restored 1959 Chevrolet Impala as a hobby. When the car isn’t under cover in its climate controlled garage it is usually being trailered to shows, parades, and other special events. The Chevy is driven under its own power no more than 1500 miles a year and the car is started, driven a short distance, and allowed to idle 1 hour at least every 2 weeks. A single battery was taken directly from the production line and placed in the Impala in 1999. The battery is still performing well to this day. Statistically, what can we conclude about this particular battery? (5 pts.)