Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 17. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 66 and 134​? ​(b) What percentage of people has an IQ score less than 49 or greater than 151​? ​(c) What percentage of people has an IQ score greater than 134​?

Answer 1

Solution:

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 17.

​(a) What percentage of people has an IQ score between 66 and 134​?

We have

Mean – 2*SD = 100 – 2*17 = 100 – 34 = 66

Mean + 2*SD = 100 + 2*17 = 100 + 34 = 134

So, according to empirical rule, about 95% of the area lies within two standard deviations from the mean.

So, about 95% of people have an IQ score between 66 and 134.

Answer: 95%

Part b

​(b) What percentage of people has an IQ score less than 49 or greater than 151​?

We have

Mean – 3*SD = 100 – 3*17 = 100 – 51 = 49

Mean + 3*SD = 100 + 3*17 = 100 + 51 = 151

So, according to empirical rule, about 99.7% or 0.997of the area lies within three standard deviations from the mean.

Probability of people has an IQ score less than 49 or greater than 151 = 1 – 0.997 = 0.003

Percentage of people has an IQ score less than 49 or greater than 151 = 0.3%

Answer: 0.3%

Part c

c) What percentage of people has an IQ score greater than 134​?

We have P(66<X<134) = 95% = 0.95

P(X<66) + P(X>134) = 1 – 0.95 = 0.05

P(X>134) = 0.05/2 = 0.025

Due to symmetry, P(X<66) and P(X>134) are same.

Required percentage = 2.5%