Question

A company has a policy of retiring company cars; this policy looks at number of miles driven, purpose of trips, style of car and other features. The distribution of the number of months in service for the fleet of cars is bell-shaped and has a mean of 45 months and a standard deviation of 3 months. Using the Empirical Rule rule, what is the approximate percentage of cars that remain in service between 36 and 39 months

Answer 1

The approximate percentage of cars that remain in service between 36 and 39 months is of 2.35%.

Explanation:

The Empirical Rule states that, for a normally distributed random variable:

Approximately 68% of the measures are within 1 standard deviation of the mean.

Approximately 95% of the measures are within 2 standard deviations of the mean.

Approximately 99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean of 45 months, standard deviation of 3 months.

What is the approximate percentage of cars that remain in service between 36 and 39 months?

36 = 45 - 3(3)

39 = 45 - 2(3)

So within 2 and 3 standard deviations below the mean.

99.7 - 95 = 4.7% of the measures are between 2 and 3 standard deviations of the mean, however, this is two-tailed, considering both above and below the mean.

In this case, both 36 and 39 are below the mean, and due to the symmetry of the normal distribution, this percentage is divided by half, so 4.7/2 = 2.35.

The approximate percentage of cars that remain in service between 36 and 39 months is of 2.35%.