If you could help me with a), b), c) and g) it would be tremendous. Thanks! The
ID: 3218015 • Letter: I
Question
If you could help me with a), b), c) and g) it would be tremendous.
Thanks!
The scores for the intelligence quotient (IQ) test are normally distributed with the mean of 100 and standard deviation of 15. a) What is the probability that at least two of five randomly selected people have IQ test score higher than 110? b) A person is randomly selected from all people whose IQ are higher than 110. What is the probability that this person has IQ higher than 120? c) One thousand people are randomly selected. Approximately what is the probability that fewer than 90 of them have IQ score higher than 120? d) Determine the minimum IQ scores for the top 5 % of the population. e) Find the probability that IQ score is between 88 and 112. f) Find the probability that IQ score is between 108 and 120. g) If we knew our subject's score is between 108 and 120, what will be the probability that this score fall between 88 and 112?Explanation / Answer
mean = 100 and std.dev. = 15
(A)
probability that a selected person will have IQ score higher than 110
P(X>110) = P(z>(110-100)/15) = P(z>0.6667) = 0.2525
Consider this probability as p = 0.2525
Probability that at least 2 of 5 have score greater than 110
P(X>=2) = 1 - P(X<2)
P(X<2) = P(X=0) + P(X=1)
P(X=0) = (1-p)^5 = 0.2334
P(X=1) = 5C1*p*(1-p)^4 = 0.3942
P(X>2) = 1 - 0.6276 = 0.3725
(B)
Probability that a randomly selected person has IQ score greater than 110, P(A) = 0.2525
Probability that a randomly selected person has IQ score greater than 120, P(B) =
P(X>120) = P(z>(120-100)/15) = P(z>1.333) = 0.0912
Probability that a student has score greater than 110 and 120 P(A and B) = 0.0912
Required probability P(B|A) = P(A and B)/P(B) = 0.0912/0.2525 = 0.3612
(C)
probability that a person selected has IQ score greater than 120, p = 0.0912
mean = np = 1000 * 0.0912 = 91.2
std. dev. = sqrt(np(1-p)) = 9.12
Probability that fewer than 90 of them have IQ score higher than 120
P(X<90) = P(z<(90-91.2)/9.12) = P(z<-0.1316) = 0.4477
(G)
Probabiltiy that score is between 108 and 120, P(A) = 0.2057
Probabiltiy that score is between 88 and 112, P(B) = 0.5763
probability of A and B, P(A and B) = 0.1206
Required probability P(B|A) = P(A and B)/P(A) = 0.1206/0.2057 = 0.5863