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Question The company retires cars based on miles, purpose, and style. The car fleet's service duration follows a normal distribution with μ=62\mu=62 months and σ=5\sigma=5 months. What is the approximate percentage of cars in service between 47 and 57 months?

Studdy Solution

STEP 1

Assumptions
1. The distribution of the number of months in service for the fleet of cars is bell-shaped, which implies a normal distribution.
2. The mean (average) number of months in service is 62 months.
3. The standard deviation is 5 months.
4. The empirical rule applies, which states that for 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.
5. We are interested in the percentage of cars that remain in service between 47 and 57 months.

STEP 2

First, we need to determine how many standard deviations away from the mean the values 47 and 57 months are.
Number of standard deviations=Value−MeanStandard deviationNumber\, of\, standard\, deviations = \frac{Value - Mean}{Standard\, deviation}

STEP 3

Calculate the number of standard deviations for 47 months.
Number of standard deviations for 47 months=47−625Number\, of\, standard\, deviations\, for\, 47\, months = \frac{47 - 62}{5}

STEP 4

Perform the calculation for 47 months.
Number of standard deviations for 47 months=−155=−3Number\, of\, standard\, deviations\, for\, 47\, months = \frac{-15}{5} = -3

STEP 5

Calculate the number of standard deviations for 57 months.
Number of standard deviations for 57 months=57−625Number\, of\, standard\, deviations\, for\, 57\, months = \frac{57 - 62}{5}

STEP 6

Perform the calculation for 57 months.
Number of standard deviations for 57 months=−55=−1Number\, of\, standard\, deviations\, for\, 57\, months = \frac{-5}{5} = -1

STEP 7

Now we need to use the empirical rule to find the percentage of cars that remain in service between -3 and -1 standard deviations from the mean.
According to the empirical rule: - About 68% of data falls within one standard deviation of the mean (between -1 and 1 standard deviations). - This means that approximately 34% of data falls between the mean and -1 standard deviation, and another 34% falls between the mean and 1 standard deviation.

STEP 8

To find the percentage of cars between -3 and -1 standard deviations, we need to subtract the percentage of cars within one standard deviation from the percentage within three standard deviations.
According to the empirical rule: - About 99.7% of data falls within three standard deviations of the mean (between -3 and 3 standard deviations). - This means that approximately 99.7% / 2 = 49.85% of data falls between the mean and -3 standard deviations.

STEP 9

Subtract the percentage of cars within one standard deviation from the percentage within three standard deviations to find the percentage of cars between -3 and -1 standard deviations.
Percentage between −3 and −1 standard deviations=49.85%−34%Percentage\, between\, -3\, and\, -1\, standard\, deviations = 49.85\% - 34\%

STEP 10

Perform the subtraction to find the approximate percentage of cars that remain in service between 47 and 57 months.
Percentage between −3 and −1 standard deviations=49.85%−34%=15.85%Percentage\, between\, -3\, and\, -1\, standard\, deviations = 49.85\% - 34\% = 15.85\%
The approximate percentage of cars that remain in service between 47 and 57 months is 15.85.
ans = 15.8515.85

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