Question

Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 20. Use the empirical rule to determine the following

(a) What percentage of people has an IQ score between 40 and 160?

(b) What percentage of people has an IQ score less than 60 or greater than 140?

(c) What percentage of people has an IQ score greater than 120?

Answer 1

The empirical rule, also known as the 68-95-99.7 rule, states that:

- About 68% of data falls within one standard deviation of the mean.

- About 95% of data falls within two standard deviations of the mean.

- About 99.7% of data falls within three standard deviations of the mean.

Using this rule, we can answer the questions as follows:

(a) An IQ score between 40 and 160 is within two standard deviations of the mean, since it is between 100 - 2(20) = 60 and 100 + 2(20) = 140. Therefore, about 95% of people have an IQ score between 40 and 160.

(b) An IQ score less than 60 is more than two standard deviations below the mean, while an IQ score greater than 140 is more than two standard deviations above the mean. From the empirical rule, we know that about 2.5% of people have an IQ score less than 60, and about 2.5% have an IQ score greater than 140. Therefore, the percentage of people who have an IQ score less than 60 or greater than 140 is 2.5% + 2.5% = 5%.

(c) An IQ score greater than 120 is within one standard deviation above the mean, since it is between 100 and 100 + 20 = 120. From the empirical rule, we know that about 68% of people have an IQ score within one standard deviation above the mean. Therefore, the percentage of people who have an IQ score greater than 120 is approximately 68%.