# Struggling with simple standard deviation problem

I don't know how to grasp what this question is asking, nor how to attempt to solve it...

The Polk Company reported that the average age of a car on US roads in a recent year was 7.5 years. Suppose the distribution of ages of cars on US roads is approximately bell-shaped. If 95% of the ages are between 1 year and 14 years, what is the standard deviation of car age?

I could calculate the variance but I don't know N. Not sure what the 95% part is there for either or what to do with it.

This isn't technically a homework question, but it's on a practice exam. I'd really like some help as to how to go about solving this, the wording is messing me up.

• Question that is a hint: How many standard deviations does a 95% CI of a standard bell curve (aka a normal distribution) cover? Dec 17 '13 at 21:38
• Two standard deviations. The empirical rule right? The only close answer I have to that is 2.167... Dec 17 '13 at 21:42

Given that you got the hints, I'll give you the full solution for comparison:

For a random variable $X\sim N(\mu,\sigma^2)$ that follows a gaussian distribution with mean $\mu$ and variance $\sigma^2$, you can show that:

$$\mathbb{P}(\mu-2\sigma \leq X \leq \mu+2\sigma)\approx0.95.$$

In your case, $\mu=7.5$ and you wish to estimate the variance. However, interpreting the above probability as the frequency of cars between ages $\mu-2\sigma$ and $\mu+2\sigma$, this would translate into:

$$\mathbb{P}(7.5-2\sigma \leq X \leq 7.5+2\sigma)\approx0.95.$$

Because 95% of the cars are between 1 and 14 years, you could interpret this information as a way of estimating $7.5-2\sigma$ and $7.5+2\sigma$ because, following the interpretation of probabilities as frequencies, this information is telling you that:

$$\mathbb{P}(1\leq X \leq 14)\approx 0.95.$$

This gives:

$$\sigma \approx (7.5-1)/2 = (14-7.5)/2 = 3.25.$$

• For the questioners sake, 2 standard deviations is actually about 0.9545. Using 1.96 standard deviations is more accurate for 0.95. This is tagged self-study, and if it is to be turned in or a similar problem appears on a test, your instructor might be a stickler about it.
– TLJ
Dec 17 '13 at 21:54
• I agree on the accurateness of 1.96; my physicist mind, however, was unable to recall the number and approximated it as 2 :-). Dec 17 '13 at 22:35