Question

23. NOTE: Answers using z-scores rounded to 3 (or more) decimal places will work for this problem.

The population of weights for men attending a local health club is normally distributed with a mean of 175-lbs and a standard deviation of 26-lbs. An elevator in the health club is limited to 32 occupants, but it will be overloaded if the total weight is in excess of 5952-lbs.

a. Assume that there are 32 men in the elevator. What is the average weight beyond which the elevator would be considered overloaded?
average weight = lbs

b. What is the probability that one randomly selected male health club member will exceed this weight?
P(one man exceeds) =  
(Report answer accurate to 4 decimal places.)

c. If we assume that 32 male occupants in the elevator are the result of a random selection, find the probability that the elevator will be overloaded?
P(elevator overloaded) =  
(Report answer accurate to 4 decimal places.)

d. If the elevator is full (on average) 6 times a day, how many times will the elevator be overloaded in one (non-leap) year?
number of times overloaded =  
(Report answer rounded to the nearest whole number.)

Answer 1

Solution:

The population of weights for men attending a local health club is normally distributed with a mean of 175-lbs and a standard deviation of 26-lbs. An elevator in the health club is limited to 32 occupants, but it will be overloaded if the total weight is in excess of 5952-lbs.

a.

Assume that there are 32 men in the elevator.

We have to find the average weight beyond which the elevator would be considered overloaded.

The elevator will be overloaded if the total weight is in excess of 5952-lbs.

So, the average weight would be =

Answer: The beyond the average weight 186-lbs elevator would be considered overloaded.

b.

Let X be the weight of a randomly chosen man from the given population.

We are given that,

Hence,

So, the probability that one randomly selected male health club member will exceed the weight 186-lbs is,

Answer: The probability that one randomly selected male health club member will exceed the weight 186-lbs is 0.3361.

c.

We assume that 32 male occupants in the elevator are the result of a random selection, we have to find the probability that the elevator will be overloaded.

We know the elevator would be considered overloaded if the average weight is more than 186-lbs.

The distribution of the sample mean is given by,

In our data,

So,

Hence,

The probability that the lift is overloaded is given by,

Answer: The probability that the lift is overloaded is 0.0083.

d. The elevator is full (on average) 6 times a day. The number of times the elevator will be overloaded in one (non-leap) year is = 6*36 = 216.