Question

A. Construct a z distribution to find the area above z = 1.75

B. Construct a z distribution to find the area below z = -1.75

C. Construct a normal distribution with a mean of 50 and standard deviation of 10 to find the area above x = 67.5

D. Explain why your answers to A, B, and C are all the same.

Answer 1

Solution:

Part A) Construct a z distribution to find the area above z = 1.75

That is : P( Z > 1.75 ) = .........?

Thus we get :

P( Z > 1.75 ) = 1 - P ( Z < 1.75)

Look in z table for z = 1.7 and 0.05 and find area.

P( Z < 1.75) = 0.9599

Thus

P( Z > 1.75 ) = 1 - P( Z < 1.75 )

P( Z > 1.75 ) = 1 - 0.9599

P( Z > 1.75 ) = 0.0401

Part B. Construct a z distribution to find the area below z = -1.75

That is : P( Z < -1.75) = ..........?

Look in z table for z = -1.7 and 0.05 and find area.

P( Z < -1.75) = 0.0401

Part C. Construct a normal distribution with a mean of 50 and standard deviation of 10 to find the area above x = 67.5

That is we have to find : P( X > 67.5 ) = ............?

Thus find z score for x = 67.5

Thus we get :

P( X > 67.5 ) = P( Z > 1.75 )

P( X > 67.5 ) = 1 - P( Z < 1.75 )

Look in z table for z = 1.7 and 0.05 and find area.

P( Z < 1.75) = 0.9599

Thus

P( X > 67.5 ) = 1 - P( Z < 1.75 )

P( X > 67.5 ) = 1 - 0.9599

P( X > 67.5 ) = 0.0401

Part D. Explain why your answers to A, B, and C are all the same.

Answers to parts A, B, and C are all the same, since Standard Normal distribution Z is a Symmetric distribution with mean = 0 and standard deviation = 1.

Thus P( Z < -a) = P( Z > + a)

That means Area under the curve below Z = -a is same as Area under the curve above Z = +a.

So According to this: Area under the curve below Z = -1.75 is same as Area under the curve above Z = 1.75 which is 0.0401.