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 a student at random from each school. Determine the probability that the North Catalina State University student's IQ is at least 5 points higher than the Chapel Mountain University student's IQ.

Answer 1

The required probability is 0.6517

Explanation:

Consider the provided information.

North Catalina State University's students can be approximated by a normal model with mean 130 and standard deviation 8 points.

μ₁ = 130 and σ₁ = 8

Chapel Mountain University's students can be approximated by a normal model with mean 120 and standard deviation 10 points.

μ₂ = 120 and σ₂ = 10

As both schools have IQ scores which is normally distributed, distribution of this difference will also be normal with a mean of μ₁-μ₂ and standard deviation will be
`√(\sigma_1^2+\sigma_2^2)`

Therefore,

μ = 130-120=10

`\sigma = √(8^2+10^2)=12.806`

Now determine the probability of North Catalina State University student's IQ is at least 5 points higher than the Chapel Mountain University student's IQ:

`z=(\bar x-\mu)/(\sigma)`

`z=(5-10)/(12.806)\approx-0.39`

Now by using the z table we find the z- score of -0.39 is 0.6517.

Hence, the required probability is 0.6517