Questions: 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 46 months and a standard deviation of 4 months. Using the 68-95-99.7 (Empirical) Rule, what is the approximate percentage of cars that remain in service between 50 and 58 months? The approximate percentage of cars that remain in service between 50 and 58 months is (enter the percent symbol.)

Solution Steps

To find the percentage of cars that remain in service between 50 and 58 months, we first calculate the Z-scores for the lower and upper bounds using the formula:

\[ z = \frac{X - \mu}{\sigma} \]

For the lower bound \(X = 50\):

\[ z_{start} = \frac{50 - 46}{4} = 1.0 \]

For the upper bound \(X = 58\):

\[ z_{end} = \frac{58 - 46}{4} = 3.0 \]

Next, we use the Z-scores to find the probability that the sample mean falls within the specified range. This is done using the cumulative distribution function \(P\):

\[ P = \Phi(z_{end}) - \Phi(z_{start}) \]

Using standard normal distribution tables or calculators, we find:

\[ \Phi(3.0) \approx 0.9987 \quad \text{and} \quad \Phi(1.0) \approx 0.8413 \]

Thus, the probability is:

\[ P \approx 0.9987 - 0.8413 = 0.1574 \]

To express this probability as a percentage, we multiply by 100:

\[ \text{Percentage} = 0.1574 \times 100 \approx 15.74\% \]

Final Answer