Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of

100

and a standard deviation of

13

Use the empirical rule to determine the following.

​(a) What percentage of people has an IQ score between

87

and

113​?

​(b) What percentage of people has an IQ score less than

74

or greater than

126​?

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

126​?

Answer 1

(a)

Z value for 87 is (87 - 100)/13 = -1

Z value for 113 is (113 - 100)/13 = +1

Thus, the values 87 and 113 are one standard deviations from the mean.

By  empirical rule, percentage of people has an IQ score between 87 and 113​ (one standard deviations from the mean) is 68%

(b)

Z value for 74 is (74 - 100)/13 = -2

Z value for 126 is (126 - 100)/13 = +2

Thus, the values 87 and 113 are two standard deviations from the mean.

By  empirical rule, percentage of people has an IQ score between 74 and 96​ (two standard deviations from the mean) is 95%

(c)

From part (a), percentage of people has an IQ score less than 74 + percentage of people has an IQ score greater than 126​

= 100 - 95 = 5

By symmetry of bell-shaped distribution, percentage of people has an IQ score less than 74 (less than 2 standard deviations from mean) is equal to the percentage of people has an IQ score greater than 126​ (greater than 2 standard deviations from mean)

Thus,  percentage of people has an IQ score greater than 126​ = 5 / 2 = 2.5%