Question

Suppose that the IQs of university​ A's students can be described by a normal model with mean 140 and standard deviation 8 points. Also suppose that IQs of students from university B can be described by a normal model with mean 120 and standard deviation 11. ​

a) Select a student at random from university A. Find the probability that the​ student's IQ is at least 130 points. The probability is nothing. ​(Round to three decimal places as​ needed.) ​

b) Select a student at random from each school. Find the probability that the university A​ student's IQ is at least 10 points higher than the university B​ student's IQ. The probability is nothing. ​(Round to three decimal places as​ needed.)

​c) Select 3 university B students at random. Find the probability that this​ group's average IQ is at least 125 points. The probability is nothing. ​(Round to three decimal places as​ needed.) ​

d) Also select 3 university A students at random.​ What's the probability that their average IQ is at least 10 points higher than the average for the 3 university B​ students? The probability is nothing. ​(Round to three decimal places as​ needed.)

Answer 1

We are given the distributions for Student A and student B here as:

a) The probability that the student A's IQ is at least 130 is computed here as:
P(A >= 130)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.8944 is the required probability here.

b) As both A and B are normal variables, therefore any linear combination of the two scores would also be a normal variable. The distribution of A - B is obtained here as:

The probability that the university A​ student's IQ is at least 10 points higher than the university B​ student's IQ is computed here as:

P(A - B >= 10)

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.7689 is the required probability here.

c) Probability that the average IQ score of the three B students selected is at least 125 points is computed here as:

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.2156 is the required probability here.

d) The distribution of the sample means for A and B here is given as:

The probability required here is:

Converting it to a standard normal variable, we have here:

Getting it from the standard normal tables, we have here:

Therefore 0.8986 is the required probability here.