Question

According to an article in American Scientist, IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the following probabilities:

a) The probability that a randomly selected person has an IQ score of at least 111.

b) The probability that a randomly selected person has an IQ of less than 96.

c) The probability that a randomly selected person has an IQ score between 119 and 145.

(Please show all work.)

Answer 1

Solution:-

Mean = 100, S.D = 15

a) The probability that a randomly selected person has an IQ score of at least 111 is 0.232.

x = 111

By applying normal distruibution:-

z = 0.733

P(z > 0.7333) = 0.232.

b) The probability that a randomly selected person has an IQ of less than 96 is 0.605.

x = 96

By applying normal distruibution:-

z = - 0.267

P(z < - 0.267) = 0.605

c) The probability that a randomly selected person has an IQ score between 119 and 145 is 0.102.

x₁ = 119

x₂ = 145

By applying normal distruibution:-

z₁ = 1.2667

z₂ = 3.0

P( 1.2667 < z < 3.00) = P(z > 1.2667) - P(z > 3.0)

P( 1.2667 < z < 3.00) = 0.103 - 0.001

P( 1.2667 < z < 3.00) = 0.102