Question

74. IQ is normally distributed with a mean of 100 and a standard deviation of 15. Suppose one individual is randomly chosen. Let X = IQ of an individual. X ~ _____(_____,_____) Find the probability that the person has an IQ greater than 120. Include a sketch of the graph, and write a probability statement. MENSA is an organization whose members have the top 2% of all IQs. Find the minimum IQ needed to qualify for the MENSA organization. Sketch the graph, and write the probability statement. The middle 50% of IQs fall between what two values? Sketch the graph and write the probability statement.

Answer 1

a) X ~ N(100, 15)

b) P(X > 120)

= P((X - )/ > (120 - )/)

= P(Z > (120 - 100)/15)

= P(Z > 1.33)

= 1 - P(Z < 1.33)

= 1 - 0.9082

= 0.0918

c) 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

d) P(X < x) = 0.25

or, P((X - )/ < (x - )/) = 0.25

or, P(Z < (x - 100)/15) = 0.25

or, (x - 100)/15 = -0.67

or, x = -0.67 * 15 + 100

or, x = 89.95

P(X > x) = 0.25

or, P((X - )/ > (x - )/) = 0.25

or, P(Z > (x - 100)/15) = 0.25

or, P(Z < (x - 100)/15) = 0.75

or, (x - 100)/15 = 0.67

or, x = 0.67 * 15 + 100

or, x = 110.05