Question

Suppose a carnival director in a certain city imposes a patron 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 4.4 feet and a standard deviation of 0.8 feet (patrons are mostly children). Note: 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, show your rounded z-score (round to 2 decimal places), and state your found value from Table A4.

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

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? [5 marks]

Answer 1

a)

Here, μ = 4.4, σ = 0.8 and x = 4. We need to compute P(X >= 4). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (4 - 4.4)/0.8 = -0.5

Therefore,
P(X >= 4) = P(z <= (4 - 4.4)/0.8)
= P(z >= -0.5)
= 1 - 0.3085 = 0.6915

b)

Here, μ = 4.4, σ = 0.4619 and x = 4.5. We need to compute P(X >= 4.5). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (4.5 - 4.4)/0.4619 = 0.22

Therefore,
P(X >= 4.5) = P(z <= (4.5 - 4.4)/0.4619)
= P(z >= 0.22)
= 1 - 0.5871 = 0.4129