Question

IQs are normally distributed with a mean of $100$ and standard deviation of $16$. How do you use the $68-95-99.7$ rule to find the percentage of people with scores below $84$?

Answer 1

Hint:Start with the necessary definition about the empirical rule. After the condition given in the question find out which slot of the standard deviation does our requirement lie. Once you have found this, use the formula, find the percentage of people with score less than 84 using necessary tricks.

Complete step by step answer:
First we will try to understand what a 68-95-99.7 rule is. This rule is commonly known as empirical rule. This rule states that the observed data will be within the three standard deviations for a normal distribution. We will understand the concept better after solving the problem.
We have been given that mean=100
And the standard deviation given is 16
$ \Rightarrow \sigma = 16$

As stated above empirical rule is only valid for normal population. Also empirical rule states that the population that lies within the first standard deviation is approximately 68%.
According to the formula, $\left( {\mu - \sigma } \right) = 100 - 15 = 85$
and $\left( {\mu + \sigma } \right) = 100 + 15 = 115$
According to empirical rules that implies that 68% people have an IQ between 85 and 115.
But we have to find the IQ of people with scores below 84.
This obviously means that if 68% of people have IQ between 85 and 115 then (1-0.68)=32% of people will have IQ outside the range of 85 and 115.
That implies that $\dfrac{{0.32}}{2} = 0.16 = 16\% $ of people will have an IQ below 84.

Hence the percentage of people with a score below 84 is 16%.

Note: Empirical rule helps you deal with problems related to estimating probabilities in case of normal distribution. As done above, creating the ranges of the standard deviation so that what percentages fall into the 1, 2 and 3 standard deviation can be found. This rule can only be used for normal distribution problems.