Question

5	Assume that the heights of U.S. residents are normally distributed with mean 65 inches and
	standard deviation of 5 inches.
a	What is the probability that a randomly selected resident is over 70 inches tall?
b	What is the probability that a randomly selected resident will be between 62 and 72 inches tall?
c	What is the probability that a randomly selected resident will be less than 58 inches tall?

Answer 1

Solution:

The heights of U.S. residents are normally distributed with mean 65 inches and standard deviation of 5 inches.

Mean =

Standard Deviation =

Part a) What is the probability that a randomly selected resident is over 70 inches tall?

P( X > 70) =...........?

Find z score:

Thus we get:

P(X > 70) =P( Z > 1.00)

P(X > 70) =1 - P( Z < 1.00)

Look in z table for z = 1.0 and 0.00 and find area.

From z table for z = 1.0 and 0.00, we get: P( Z < 1.00 ) = 0.8413

Thus

P(X > 70) =1 - P( Z < 1.00)

P(X > 70) =1 - 0.8413

P(X > 70) = 0.1587

Part b) What is the probability that a randomly selected resident will be between 62 and 72 inches tall?

P( 62 < X < 72) = ........?

Thus we get:

P( 62 < X < 72) = P( -0.60 < Z < 1.40)

P( 62 < X < 72) = P( Z < 1.40) - P( Z < -0.60)

Look in z table for z= 1.4 and 0.00 as well as for z = -0.6 and 0.00

and find corresponding area.

Thus from z table , we get:

P( Z < 1.40) = 0.9192

P( Z< -0.60) = 0.2743

Thus

P( 62 < X < 72) = P( Z < 1.40) - P( Z < -0.60)

P( 62 < X < 72) = 0.9192 - 0.2743

P( 62 < X < 72) = 0.6449

Part c) What is the probability that a randomly selected resident will be less than 58 inches tall?

P( X < 58) = ........?

P( X < 58) = P( Z < -1.40)

Look in z table for z   = -1.4 and 0.00 and find area.

Thus from z table we get:P( Z < -1.40) = 0.0808

Thus we get:

P( X < 58) = P( Z < -1.40)

P( X < 58) = 0.0808