Question 2 (14 marks)

According to internal testing done by a car battery company over many years, the mean lifetime of car batteries sold on new cars is normally distributed with a mean of 5 years and a standard deviation of 4 months.

a) What is the probability that a random new car has a battery lasting greater than 5 years and 6 months?

If this is true, then we would like to know what the mean of battery-lifetimes is likely to be for the university's fleet of 36 such cars.

b) If the claim by the car battery company is true, what is the mean of the sampling distribution of x for samples of size n = 36?

c) If the claim by the car battery company is true, what is the standard deviation of the sampling distribution of x for samples of size n = 36?

d) Sketch this sampling distribution.

e) What is the probability that such a sample of 36 cars would have an average car battery lifetime greater than 5 years and 6 months?

f) What is the first quartile of the sampling distribution of average lifetimes of car batteries (36 cars)?

g) If the population distribution was really skewed and not Normal, explain whether or not these probability calculations on the sample of 36 cars are still valid. State the concept or theory that is involved in this situation.

h) On the same sketch in part d) above, sketch the distribution that would result for the same population if the sample size was n = 100.