Question

A set of data has a normal distribution with a mean of 180 and a standard deviation of 20. What percent of the data is in the interval 140 - 220?

Answer 1

About `95%`

Explanation:

We are given the information that this distribution is normal with a mean `mu` of 180 and a standard deviation `sigma` of 20. This describes a distribution `N(180,20^2)`.

To answer the question, we will convert this problem into a standard normal distribution `N_s(0,1^2)` question by determining the z-scores for the interval 140-220. This can be done either by "eyeballing it" (since `sigma` is 20, and each endpoint of the interval is a multiple of `sigma` away from the mean `mu`), or we can use the z-score formula:

`z = (x-mu)/sigma`

Thus:

`z_140 = (140 - 180)/20 = -40/20 = -2`

`z_220 = (220 - 180)/20 = 40/20 = +2`

This tells us the interval we're being asked about is analogous to determining what percent of the standard normal distribution `N_s` lies between z-scores of -2 and 2.

In statistics there is a handy "rule of thumb" sometimes called the Empirical Rule which says the approximately 95% of the data in a normal distribution lies in the interval `[-2sigma,2sigma]`, which is exactly what we're being asked. (The actual answer is more like 95.45%.)