Question

A ski gondola carries passengers to the top of a mountain. A plaque states that its maximum capacity is 12 people or 2019 lbs. Men’s weights are normally distributed with a mean of 179 lbs and a standard deviation of 30 lbs.

a)

What is the probability that 12 randomly-selected men will exceed the weight limit of the gondola?

b)

Suppose you are the engineer in charge of safety at the mountain. What is the maximum number of men you would allow to ride the gondola at a time in order to be confident they would not exceed the weight limit? Explain.

c)

Suppose the gondola carries the maximum number of men (as given in part b) on every trip. The gondola operates for eight hours a day, and each trip takes 15 minutes to complete. What is the expected amount of time before you lose your licence? Does this answer lead you to rethink your answer to part b)? If so, what would you do differently in part b) (you don’t have to redo part b)?

Answer 1

A ski gondola carries passengers to the top of a mountain. A plaque states that its maximum capacity is 12 people or 2019 lbs. Men’s weights are normally distributed with a mean of 179 lbs and a standard deviation of 30 lbs.

Let X be the random variable for a man's weight.

Then X ~ Normal(mean = 179, std dev = 30)

Let S be the random variable for the total weight of 12 men.

Then by the law for addition of normally distributed iid random variables, we have

S ~ Normal(mean= 12*179, std dev = Sqrt (12*(30*30)) )

S ~ Normal(mean= 2148, std dev = 103.92)

(a)

Probability that the 12 men would have weight exceeding weight limit is

P[S > 2019 ] , which is given by 1- P[ S <= 2019]

This is found using the norm.dist function in Excel and we get

P[S > 2019 ] = 0.8927

So there is 89.3% chance that the total weight would exceed the weight limit.

(b) Suppose you are the engineer in charge of safety at the mountain. What is the maximum number of men you would allow to ride the gondola at a time in order to be confident they would not exceed the weight limit? Explain.

Let n be the maximum number of men the engineer has allowed to ride

Then the total weight is given by Sn = X1 + X2 + X3 + ... + Xn

where each X on the right hand side is Normal(mean = 179, std dev = 30)

So Sn ~ Normal(mean = 179*n, std dev = Sqrt ( n*30*30) )

Let us assume that the engineer wants to have only a maximum 1% chance that the maximum number of men would exceed the weight limit.

So P[Sn >2019 ] = 0.01

Therefore P[Sn <= 2019 ] = 0.99

So P [ (Sn - 179*n)/Sqrt(n*30*30) <= (2019-179*n)/Sqrt(n*30*30) ] = 0.99

(Sn - 179*n)/Sqrt(n*30*30) is a standard normal distributed variable (denoted by "Phi" )

So Phi ((2019-179*n)/Sqrt(n*30*30) ) = 0.99

So (2019 - 179*n) / Sqrt(900*n) = 2.326348

So the engineer should allow maximum of 12 men to have a less than 1% chance of exceeding weight limit.

(d)

Suppose the gondola carries the maximum number of men (as given in part b) on every trip. The gondola operates for eight hours a day, and each trip takes 15 minutes to complete.

Each trip = 15 minutes, so in 1 day (8 hours) the gondola completes 32 trips.

For every trip, when the above limit is followed strictly, then there is only 1% chance of exceeding weight limit, and for the gondola to break down.

What is the expected amount of time before you lose your licence?

So consider a geometric distribution which is the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set { 1, 2, 3, ... }

For our problem, we consider "success" = event of exceeding weight limit,

so probability of success at every trial = p = 0.01.

Then probability that the weight is exceeding at trip number k is given by

The formula for expected number of trips is

So expected number of trips = 1/0.01 = 100 for exceeding the weight limit

So expected amount of time to lose my license = 100/32 = 3.125 days or less than 4 days

Does this answer lead you to rethink your answer to part b)? If so, what would you do differently in part b) (you don’t have to redo part b)?

Yes, it does lead me to rethink the answer to part (b). As there is only 1 "error event" of exceeding weight limit required for the gondola, whereas the number of tests for testing this event is considerably large, the cumulative chance of an error occurring greatly increases with increasing number of trials.

Therefore, I should consider dividing the error term by a factor = number of trips that the elevator should make in a lifetime.

This might not make a large difference to the number of allowed persons, as this only decreases the level of significance alpha.