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 64 months and a standard deviation of 5 months. Using the empirical rule, what is the approximate percentage of cars that remain in service between 69 and 74 months?

Answer 1

By the Empirical Rule, the approximate percentage of cars that remain in service between 69 and 74 months is 13.5%.

Explanation:

The Empirical Rule states that, for a normally distributed(bell-shaped) random variable:

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

95% of the measures are within 2 standard deviation of the mean.

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

In this problem, we have that:

Mean = 64

Standard deviation = 5.

Using the empirical rule, what is the approximate percentage of cars that remain in service between 69 and 74 months?

69 = 64 + 1*5

So 69 is one standard deviation above the mean.

74 = 64 + 2*5

So 74 is two standard deviations above the mean.

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

95% of the measures are within 2 standard deviation of the mean.

This means that:

95 - 68 = 27% between 1 and 2 standard deviations of the mean.

The normal distribution is symmetric, which means that 27/2 = 13.5% of those are below the mean(within 1 and 2 standard deviations below the mean) and 13.5% of those are above the mean(between 69 and 74).

So

By the Empirical Rule, the approximate percentage of cars that remain in service between 69 and 74 months is 13.5%.