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 56 months and a standard deviation of 4 months. Using the empirical rule, what is the approximate percentage of cars that remain in service between 44 and 52 months?
To find the approximate percentage of cars that remain in service between 44 and 52 months, we can use the empirical rule, also known as the 68-95-99.7 rule.
According to the empirical rule, for a bell-shaped distribution or a normal distribution:
- 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 56 months, and the standard deviation is 4 months. Therefore, to find the percentage of cars that remain in service between 44 and 52 months, we need to calculate the number of standard deviations away from the mean these values are.
To find the number of standard deviations, we can use the formula:
(Number - Mean) / Standard Deviation
For 44 months:
(44 - 56) / 4 = -3
For 52 months:
(52 - 56) / 4 = -1
Now we know that the values of 44 months and 52 months are 3 standard deviations and 1 standard deviation away from the mean, respectively.
According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean. Since the values we have are within one standard deviation, the approximate percentage of cars that remain in service between 44 and 52 months is 68%.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the Z scores. Multiply by 100.