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 42 months and a standard deviation of 7 months. Using the empirical rule (as presented in the book), what is the approximate percentage of cars that remain in service between 21 and 35 months?
21 = µ - 3σ
35 = µ - σ
so check your Z table for this interval
To find the approximate percentage of cars that remain in service between 21 and 35 months, we can use the empirical rule, also known as the 68-95-99.7 rule.
According to the empirical rule, approximately 68% of values fall within one standard deviation of the mean, approximately 95% fall within two standard deviations of the mean, and approximately 99.7% fall within three standard deviations of the mean, assuming a bell-shaped distribution.
Given that the mean is 42 months and the standard deviation is 7 months, we can calculate the range of values that fall within one standard deviation of the mean:
Lower Limit = mean - 1 * standard deviation = 42 - 7 = 35 months
Upper Limit = mean + 1 * standard deviation = 42 + 7 = 49 months
Therefore, approximately 68% of the cars will have a service time between 35 and 49 months.
To find the percentage of cars that remain in service between 21 and 35 months, we need to calculate the percentage of values that fall within one standard deviation below the mean:
Percentage = (Upper Limit - Value) / (Upper Limit - Lower Limit) * 68%
= (35 - 21) / (49 - 35) * 68%
= 14 / 14 * 68%
= 68%
Therefore, approximately 68% of the cars will remain in service between 21 and 35 months.
To answer this question using the empirical rule, also known as the 68-95-99.7 rule, we need to understand the properties of the normal distribution and its relationship with standard deviations.
The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation (mean ± standard deviation),
- Approximately 95% of the data falls within two standard deviations (mean ± 2 * standard deviation),
- Approximately 99.7% of the data falls within three standard deviations (mean ± 3 * standard deviation).
In this case, the mean is 42 months and the standard deviation is 7 months.
To find the approximate percentage of cars that remain in service between 21 and 35 months, we need to determine how many standard deviations away from the mean these values are.
First, let's calculate the z-scores for 21 and 35 months:
z-score = (x - mean) / standard deviation
For 21 months:
z-score = (21 - 42) / 7
= -3
For 35 months:
z-score = (35 - 42) / 7
= -1
Now, let's use the empirical rule to find the approximate percentage of cars that remain in service between 21 and 35 months.
Between 21 and 35 months is one standard deviation below the mean (-1) to two standard deviations below the mean (-3).
According to the empirical rule, approximately 95% of the data falls within two standard deviations.
So, the approximate percentage of cars that remain in service between 21 and 35 months is approximately 95%.
Note: The empirical rule provides an approximation and assumes that the data follows a perfect bell-shaped normal distribution.