Question

Suppose a carnival director in a certain city imposes a height limit on an amusement park ride called Terror Mountain, due to safety concerns. Patrons must be at least 4 feet tall to ride Terror Mountain. Suppose patrons’ heights in this city follow a Normal distribution with a mean of 4.5 feet and a standard deviation of 0.8 feet (patrons are mostly children). Make sure to show all of your work in this question. Show the distribution that your random variable follows; state the probability you are asked to calculate; show any tricks you use; show how you standardize, and state your found value from Table A4.

a) [5 marks] What is the probability that a randomly selected patron would be tall enough to ride Terror Mountain?

b) [5 marks] A group of 3 friends want to ride Terror Mountain. What is the probability that their mean height is greater than 4.5 feet?

c) [7 marks] Another group of 5 friends wants to ride Terror Mountain. What is the probability that their mean height is between 4 and 4.25 feet, inclusive?

Answer 1

Solution:

Given: Patrons’ heights in this city follow a Normal distribution with a mean of 4.5 feet and a standard deviation of 0.8 feet.

Patrons must be at least 4 feet tall to ride Terror Mountain.

Part a) What is the probability that a randomly selected patron would be tall enough to ride Terror Mountain?

that is find:

Find z score for x = 4

Thus get:

Look in z table for z = -0.6 and 0.03 and find corresponding area.

P( Z < -0.63 ) = 0.2643

thus

Part b) A group of 3 friends want to ride Terror Mountain. What is the probability that their mean height is greater than 4.5 feet?

n = 3

find:

thus

Look in z table for z = 0.0 and 0.00 and find corresponding area.

P( Z < 0.00) = 0.5000

thus

Part c) Another group of 5 friends wants to ride Terror Mountain. What is the probability that their mean height is between 4 and 4.25 feet, inclusive?

n = 5

Find:

Find z scores:

thus

Look in z table for z = -1.4 and 0.00 as well as for  z = -0.7 and 0.00  and find corresponding area.

P( Z< -1.40) = 0.0808

and

P( Z < -0.70) = 0.2420

thus