Question

Assume that a normal distribution has a mean of 21 and a standard deviation of 2. What is the percentage of values that lie above 23?

Use the empirical rule.

Answer 1

`16%` of values are expected to be above `23`

Explanation:

The question asks us to apply the empirical rule for normal distributions which states that `68%, 95%,` and `99.7%` of values lie within `1, 2,` and `3` standard deviations of the mean, respectively. First, we need to know which of these ranges we are in, i.e. how many standard deviations from the mean. (68-95-99.7 rule)

We are looking for the percentage of the population above `23` where the mean is `mu=21` and the standard deviation is `sigma=2` which means that the point we were given was the mean plus one standard deviation, i.e. `mu + 1 sigma`.

The empirical rule tells us that `68%` of our population lies within `+-1sigma` from the mean. That tells us that `32%` lies outside that range, but on both sides - both above and below. Since the normal distribution is symmetric (the same on both sides) we know that `16%` is below `mu - 1 sigma` and that `16%` is above `mu + 1 sigma`.

Therefore, `16%` of values are expected to be above `23`.

Check:

We can use the following diagram of a normal curve to check our answer:

All we need to do is add up the percentages of the population above the `1 sigma` point which are:

`13.6% + 2.1% + 0.1% = 15.8% ~= 16%`

Therefore our answer makes sense.