Question: Information gathered by the Quality Control Department of Major Automotive Parts Corporation revealed that the useful life X of the gear trains manufactured by the company is a normally distributed random variable with mean and standard deviation of 180,000 kilometers and 18,000 kilometers respectively.

(a) Find the probability that a randomly selected gear train will last less than 196,000 kilometers. Draw, clearly and neatly, the probability density function of X and indicate the area corresponding to the probability asked for.

(b) Compute the probability that the useful life of a randomly selected gear train will differ from the mean by less than 9,000 kilometers. Draw, clearly and neatly, the probability density function of X and indicate the area corresponding to the probability asked for.

(c) Let M represent the mean useful life of a sample of 36 gear trains. Write down the probability density function of X and M. Then draw on the same graph, clearly and neatly, the probability density function of X and M. Finally find the probability that M is grater than 185,000 kilometers.

(d) Assume now that Mister Friendly’s Garage buys five of these gear trains. (i) What is the probability that all five will have a useful life in excess of 180,000 kilometers? (ii) How many are expected to have a useful life in excess of 182,000 kilometers? (iii) What is the probability that all five will have a useful life in excess of 177,000 kilometers?