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 55 months and a standard deviation of 8 months. Using the Empirical Rule rule, what is the approximate percentage of cars that remain in service between 63 and 71 months?
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 two Z scores. Multiply by 100.
To determine the approximate percentage of cars that remain in service between 63 and 71 months, we can use the Empirical Rule (also known as the 68-95-99.7 rule). This rule states that for a bell-shaped distribution (i.e., a normal distribution), approximately:
- 68% of the data falls within one standard deviation of the mean,
- 95% falls within two standard deviations of the mean,
- and 99.7% falls within three standard deviations of the mean.
In this case, the mean is 55 months and the standard deviation is 8 months. We want to find the percentage of cars that fall between 63 and 71 months, which is within one standard deviation of the mean.
Step 1: Calculate the range within one standard deviation of the mean.
To find this range, we will add and subtract the standard deviation from the mean:
55 - 8 = 47 months
55 + 8 = 63 months
Step 2: Calculate the percentage of cars that fall within this range.
Since the Empirical Rule states that approximately 68% of the data falls within one standard deviation of the mean, we can conclude that approximately 68% of the cars remain in service between 47 and 63 months.
Step 3: Calculate the percentage of cars that fall between 63 and 71 months.
Since 63 months is at the upper end of the range within one standard deviation of the mean, and we want to find the percentage of cars between 63 and 71 months, we can estimate that approximately half of the 68% (34%) of cars fall within this range.
Therefore, the approximate percentage of cars that remain in service between 63 and 71 months is 34%.