Question

I need clear steps / formulas used with (F) ( I'm lost with current solutions )

To get full marks for the following questions you need to convert the question from words to a mathematical expression (i.e. use mathematical notation), defining your random variables where necessary, and using correct probability statements. Suppose that the IQ of adults is normally distributed with a mean of 100 and standard deviation of 15. (a) [2 marks] What IQ score distinguishes the highest 10%? (b) [3 marks] What is the probability that a randomly selected person has an IQ score between 91 and 118? (c) [2 marks] 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) [4 marks] 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) [2 marks] 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)

IQ Score of adults , X ~ N(100, 15)

The percentile equivalent to highest 10% = 100 - 10 = 90%

Z score for p = 0.90 is 1.28   

IQ score distinguishes the highest 10% = 100 + 1.28 * 15 =  119.2

(b)

Probability that a randomly selected person has an IQ score between 91 and 118

= P(91 < X < 118)

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

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

= P[Z < 1.2] - P[Z < -0.6]

= 0.8849 - 0.2743 (From standard normal tables)

= 0.6106

(c)

Probability that a randomly selected person has an IQ score above 125 = P(X > 125) =

P[Z > (125 - 100)/15] = P[Z > 1.6667] =  0.0478 = 4.78%

Thus, approximately 4.78% of people have an IQ score high enough (above 125) to be admitted to this particular club

(d)

The probability distribution X follows is Binomial distribution with parameters n = 25 and p = 0.0478

This is a binomial distribution because there are a fixed number of adults to be sampled, there are only two outcomes (IQ score above/below 125), and the probability of each outcome remains constant for each adult which is 0.0478.

(e)

Probability that at least 2 people in the random sample of 25 have IQ scores high enough to join the high-IQ club

= P(X 2) = 1 - P(X < 2)

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

= 1 - 0.2939033 - 0.3688452

= 0.3373

(f)

L ~ Uniform(a = 30, b = 60)

Probability that the applicant will finish the test in less than 45 minutes = P(L < 45)

= (45 - 30) / (60 - 30) {CDF of Uniform distribution is, P(L < k) = (k - a) / (b - a) }

= 0.5