AAA reports that the average age of a car on U.S. roads in recent years has risen to 7.5 years. Suppose that the distribution of the age of automobiles assumes a mound shaped distribution. If 99.7% of the ages are between 1 year and 14 years, what is the value of the standard deviation of car age?
3.4
To find the value of the standard deviation of car age, we can use the empirical rule (also known as the 68-95-99.7 rule) which applies to mound shaped distributions, such as the normal distribution.
The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and approximately 99.7% falls within three standard deviations of the mean.
From the given information, we know that 99.7% of the car ages are between 1 year and 14 years. This range represents three standard deviations from the mean (since 14 - 1 = 13, and 13/3 = 4.33 standard deviations, which is close enough to 3).
Therefore, we can say that the mean plus three standard deviations is equal to 14 years (since 14 is the upper limit). Let's call the mean of the car ages "μ" and the standard deviation "σ". Using this information, we can set up an equation:
μ + 3σ = 14
We also know that the average car age (mean) is given as 7.5 years:
μ = 7.5
Substituting this value into the equation, we get:
7.5 + 3σ = 14
Now we can solve for the standard deviation (σ):
3σ = 14 - 7.5
3σ = 6.5
σ = 6.5 / 3
σ ≈ 2.17 years
Therefore, the value of the standard deviation of car age is approximately 2.17 years.