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 53 months and a standard deviation of 11 months. Using the Empirical Rule rule, what is the approximate percentage of cars that remain in service between 31 and 42 months? Do not enter the percent symbol. ans =

To determine the percentage of cars that remain in service between 31 and 42 months using the Empirical Rule, we first need to establish how many standard deviations these values are from the mean. The mean is 53 months, and the standard deviation is 11 months. - 31 months is \(53 - 2 \times 11 = 31\), which is 2 standard deviations below the mean. - 42 months is \(53 - 1 \times 11 = 42\), which is 1 standard deviation below the mean. According to the Empirical Rule, approximately 68% of the data lies within 1 standard deviation of the mean, while approximately 95% lies within 2 standard deviations. Therefore, the percentage of cars that remain in service between 31 and 42 months can be calculated by combining these percentages: - From 31 months to the mean (42 months), we consider the 68% in the range of 42 to 64 months (where 64 months is \(53 + 1 \times 11\)). - Since the distribution is symmetrical, half of the 68% (34%) is below the mean from 42 to 53 months. Adding together the entire 2-standard deviation range (95%) and the additional percentage from the average (34%): - We can estimate that the percentage of cars remaining between 31 and 42 months is approximately: 34% + 47.5% (the lower half of the bell curve). Thus, the final calculation results in: ans = 47.5