Question

Suppose that the distribution of IQ's of North Catalina State University's students can be approximated by a normal model with mean 130 and standard deviation 8 points. Also suppose that the distribution of IQ's of Chapel Mountain University's students can be approximated by a normal model with mean 120 and standard deviation 10 points.

Question 1. You select 3 students at random from each school. What is the probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students?

Answer 1

The probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students = 0.7517

Explanation:

let A =X-Y where X and Y are IQ of 3 North Catalina student and mean IQ of the 3 Chapel Mountain students

Hence expected value of A =130-120 =10

std error of mean difference =(82/3+102/3)1/2 =7.3937

Hence probability that the mean IQ of the 3 North Catalina students is at least 5 points higher than the mean IQ of the 3 Chapel Mountain students =P(A>5)=P(Z>(5-10)/7.3937)=P(Z>-0.68)=0.7517

Answer 2

Explanation:

here mean difference =130-120 =10

and standard deviation of mean difference =(8^2/3+10^2/3)=7.3937

therefore from normal distribution; probability that mean difference is higher than 5:

P(X>5)=P(Z>(5 - 10)/7.3937)

=P(Z>-0.68)

=0.7517

(please try 0.7506 if this comes wrong due to rounding of z score)