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 8 months. Using the 68-95-99.7 rule, what is the approximate percentage of cars that remain in service between 48 and 56 months?

56 is 1 std below the mean

15.75

To find the approximate percentage of cars that remain in service between 48 and 56 months, we can use the 68-95-99.7 rule, also known as the empirical rule or the three-sigma rule.

According to this rule:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.

In this case, the mean is 64 months and the standard deviation is 8 months.

Step 1: Calculate the lower and upper bounds for the range of months
Lower Bound = mean - 2 * standard deviation
Upper Bound = mean + 2 * standard deviation

Lower Bound = 64 - 2 * 8 = 48
Upper Bound = 64 + 2 * 8 = 80

Step 2: Calculate the percentage of data that falls within the range
To find the approximate percentage of cars that remain in service between 48 and 56 months, we need to find the percentage of data that falls within 1 standard deviation of the mean (48 to 72 months) and subtract the percentage of data that falls above 56 months.

Percentage of data between 48 and 56 = Percentage of data between 48 and 72 - Percentage of data above 56

We know that approximately 68% of the data falls within one standard deviation of the mean.

Percentage of data between 48 and 72 = 68% - Percentage of data above 56

Step 3: Calculate the percentage of data above 56 months
To find the percentage of data above 56 months, we can subtract the percentage of data between 56 and 72 months from 68%.

Percentage of data above 56 = 68% - Percentage of data between 56 and 72

Step 4: Calculate the final percentage
Finally, we can substitute the values into the formulas and calculate the approximate percentage of cars that remain in service between 48 and 56 months.

Percentage of data between 48 and 56 = 68% - Percentage of data above 56
Percentage of data above 56 = 68% - (Percentage of data between 56 and 72)

After substituting the values, we can calculate the approximate percentage.