Question

assume IQ scores are normally distributed with a mean of 100 and a standard deviation of 15 points. a) If you select a person at random what is the probability they will have an IQ score greater than 125? b) If you select a person at random what is the probability they will have an IQ score is between 70 and 108? c) If 70 people are randomly chosen, what is the probability that the mean of their IQ scores is greater than 105? g

Answer 1

a) 0.0478

b) 0.6803

c) 0.0026

Explanation:

If IQ scores are normally distributed with a mean of 100 and a standard deviation of 15 points, then, z-scores are computed as (x-100)/15. a) P(X > 125) = P((X-100)/15 > (125-100)/15) = P(Z > 1.6667) = 0.0478 b) P(70 < X < 108) = P((70-100)/15 < (X-100)/15 < (108-100)/15) = P(-2 < Z < 0.5334) = P(Z < 0.5334) - P(Z < -2) = 0.7031 - 0.0228 = 0.6803 c) If 70 people are randomly chosen, then, the mean is normally distributed with a mean of 100 and standard deviation of
`15/√(70)`, therefore
`P(\bar{X} > 105) = P((\bar{X}-100)/15/√(70) > (105-100)/15/√(70)) = P(Z > 2.7889) = 0.0026`