Question

Tests such as the Wechsler Intelligence scales and the Stanford-Binet produce scores that are commonly called IQ scores. These tests are considered reliable measures of general cognitive ability. They have a mean of 100 and a standard deviation of 15. The scores on IQ tests are normally distributed.

a. Draw the distribution of IQ scores. Label the mean and the IQ scores which correspond to 1, 2, and 3 standard deviations from the mean.

b. Calculate the z-scores for the following IQ scores. i. 55 ii. 110 iii. 103 iv. 100

c. Answer the following questions for 250 fifth-grade students. i. How many would you expect to have IQs of 110 or higher? ii. How many would you expect to have IQs lower than 110? iii. How many would you expect to have IQs lower than 100?

d. Mensa is an organization of people who have high IQs. To be eligible for membership, a person must have an IQ higher than 98 percent of the population. What IQ is required to qualify?

Answer 1

a)

b) i) For X = 55

Z-score = (X - )/

              = (55 - 100)/15 = -3

ii) For X = 110

Z-score = (X - )/
              = (110 - 100)/15 = 0.67

iii) X = 103

Z-score = (X - )/
              = (103 - 100)/15 = 0.2

iv) X = 100

Z-score = (X - )/
              = (100 - 100)/15 = 0

c) P(X > 110)

= P(Z > 0.67)

= 1 - P(Z < 0.67)

= 1 - 0.7486

= 0.2514

Expected value = 250 * 0.2514 = 62.85 = 63

ii) P(X < 110)

= P(Z < 0.67)

= 0.7486

Expected value = 250 * 0.7486 = 187.15 = 187

iii) P(X < 100)

    = P(Z < 0)

    = 0.5

Expected value = 250 * 0.5 = 125

d) P(X > x) = 0.02

or, P((X - )/ > (x - )/) = 0.02

or, P(Z > (x - 100)/15) = 0.02

or, P(Z < (x - 100)/15) = 0.98

or, (x - 100)/15 = 2.05

or, x = 2.05 * 15 + 100

or, x = 130.75