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 63 months and a standard deviation of 10 months. Using the 68-95-99.7 rule, what is the approximate percentage of cars that remain in service between 73 and 83 months?
13.5
To find the approximate percentage of cars that remain in service between 73 and 83 months, we can use the 68-95-99.7 rule, also known as the empirical rule, which states that:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% of the data falls within two standard deviations of the mean.
- About 99.7% of the data falls within three standard deviations of the mean.
In this case, the mean is 63 months and the standard deviation is 10 months. Therefore, we can calculate the range within two standard deviations of the mean.
The lower range is: mean - 2 * standard deviation = 63 - 2 * 10 = 43 months.
The upper range is: mean + 2 * standard deviation = 63 + 2 * 10 = 83 months.
So, approximately 95% of the cars will remain in service between 43 and 83 months.
Now, to find the approximate percentage of cars that remain in service between 73 and 83 months, we can subtract the percentage of cars that remain in service between 43 and 73 months from the total 95%.
To calculate the percentage of cars that remain in service between 43 and 73 months, we need to calculate the z-scores for the respective values using the formula:
z = (x - mean) / standard deviation
For 43 months:
z_lower = (43 - 63) / 10 = -2
For 73 months:
z_upper = (73 - 63) / 10 = 1
Using a z-table or calculator, we can find the percentage associated with these z-scores.
The percentage for z = -2 is approximately 2.28%.
The percentage for z = 1 is approximately 84.13%.
Therefore, the approximate percentage of cars that remain in service between 73 and 83 months is:
95% - (84.13% - 2.28%) = 95% - 81.85% = 13.15%.
Approximately 13.15% of the cars will remain in service between 73 and 83 months.
To find the approximate percentage of cars that remain in service between 73 and 83 months, we can use the 68-95-99.7 rule, also known as the empirical rule for normal distributions.
According to the rule:
- Approximately 68% of the values fall within one standard deviation of the mean.
- Approximately 95% of the values fall within two standard deviations of the mean.
- Approximately 99.7% of the values fall within three standard deviations of the mean.
Given that the mean is 63 months and the standard deviation is 10 months, we can calculate the range within one standard deviation of the mean:
Lower limit = mean - standard deviation = 63 - 10 = 53 months
Upper limit = mean + standard deviation = 63 + 10 = 73 months
Approximately 68% of the cars will fall within the range of 53 to 73 months.
Now, let's calculate the range within two standard deviations of the mean:
Lower limit = mean - (2 * standard deviation) = 63 - (2 * 10) = 43 months
Upper limit = mean + (2 * standard deviation) = 63 + (2 * 10) = 83 months
Approximately 95% of the cars will fall within the range of 43 to 83 months.
Finally, let's calculate the range within three standard deviations of the mean:
Lower limit = mean - (3 * standard deviation) = 63 - (3 * 10) = 33 months
Upper limit = mean + (3 * standard deviation) = 63 + (3 * 10) = 93 months
Approximately 99.7% of the cars will fall within the range of 33 to 93 months.
Therefore, the approximate percentage of cars that remain in service between 73 and 83 months is approximately 95%.
cars in service after 73 months ... (100% - 68%) / 2
cars in service after 83 months ... (100% - 95%) / 2
take the difference to find the desired percentage