Question

Suppose that the IQ of adults is normally distributed with a mean of 100 and standard deviation of 15.

(a) What IQ score distinguishes the highest 10%?

(b) What is the probability that a randomly selected person has an IQ score between 91 and 118?

(c) Suppose people with IQ scores above 125 are eligible to join a high-IQ club. Show that approximately 4.78% of people have an IQ score high enough to be admitted to this particular club.

(d) Let X be the number of people in a random sample of 25 who have an IQ score high enough to join the high-IQ club. What probability distribution does X follow? Justify your answer.

(e) Using the probability distribution from part (d), find the probability that at least 2 people in the random sample of 25 have IQ scores high enough to join the high-IQ club. (f) [3 marks] Let L be the amount of time (in minutes) it takes a randomly selected applicant to complete an IQ test. Suppose L follows a uniform distribution from 30 to 60. What is the probability that the applicant will finish the test in less than 45 minutes?

Answer 1

a) P(X > x) = 0.1

Or, P((X - )/ > (x - )/) = 0.1

Or, P(Z > (x - 100)/15) = 0.1

Or, P(Z < (x - 100)/15) = 0.9

Or, (x - 100)/15 = 1.28

Or, x = 1.28 * 15 + 100

Or, x = 119.2

b) P(91 < X < 118)

= P((91 - )/ < (X - )/ < (118 - )/)

= P((91 - 100)/15 < Z < (118 - 100)/15)

= P(-0.6 < Z < 1.2)

= P(Z < 1.2) - P(Z < -0.6)

= 0.8849 - 0.2743

= 0.6106

c) P(X > 125)

= P((X - )/ > (125 - )/)

= P(Z > (125 - 100)/15)

= P(Z > 1.667)

= 1 - P(Z < 1.667)

= 1 - 0.9522

= 0.0478 = 4.78%

d) The probability distribution of X is binomial.

Because, there is n = 25 identical trials.

The probability of success P = 0.0478 remains same for all trials.

Each trial is independent from the other trials.

e) n = 25

P = 0.0478

P(X = x) = nCx * px * (1 - p)n - x

P(X ​​​​​​> 2) = 1 - P(X < 2)

= 1 - (P(X = 0) + P(X = 1))

= 1 - (25C0 * (0.0478)^0 * (0.9522)^25 + 25C1 * (0.0478)^1 * (0.9522)^24)

= 1 - 0.6627 = 0.3373

f) P(X < 45) = (45 - 30)/(60 - 30)

= 15/30 = 0.5