Question

Let the random variable X represent a student’s score on an IQ test. Suppose that student IQ scores are Normally distributed with a mean of 100 and a standard deviation of 15.

Part A.Any student who scores at least 130 on an IQ test would be considered gifted. If a student has scored 130 on an IQ test, what percentile does this score represent? Part B.In a class of 50 students, how many would you expect to score at least 115 on an IQ test?

Part C.Suppose that 10 students are randomly selected from a large school. What is the chance that at least 3 of them will have an IQ score greater than 125 (use binomial PMF)

Answer 1

µ = 100

sd = 15

a)

                               

                                = P(Z < 2)

                                = 0.9772

                                = 97.72 %tile

b)

                               

                                = P(Z > 1)

                                = 1 - P(Z < 1)

                                = 1 - 0.8413

                                = 0.1587

n = 50

Expected value = 50 * 0.1587 = 7.935 = 8

c)

                               

                                = P(Z > 1.67)

                                = 1 - P(Z < 1.67)

                                = 1 - 0.9525

                                = 0.0475

This is a binomial distribution.

p = 0.0475

n = 10

P(X = x) = 10Cx * 0.0475^x * (1 - 0.0475)^10-x

P(X > 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))

              = 1 - (10C0 * 0.0475⁰ * 0.9525¹⁰ + 10C1 * 0.0475¹ * 0.9525⁹ + 10C2 * 0.0475² * 0.9525⁸ )

              = 1 - 0.9829

              = 0.0171