Correcting for a Finite Population The Orange County Spa began with 300 members. Those members had weights with a distribution that is approximately normal with a mean of 177 lb and a standard deviation of 40 lb. The facility includes an elevator that can hold up to 16 passengers.

a. When considering the distribution of sample means from weights of samples of 16 passengers, should σ x ̄ be corrected by using the finite population correction factor? Why or why not? What is the value of σ x ̄ ?

b. If the elevator is designed to safely carry a load of up to 3000 lb, what is the maximum safe mean weight of passengers when the elevator is loaded with 16 passengers?

c. If the elevator is filled with 16 randomly selected club members, what is the probability that the total load exceeds the safe limit of 3000 lb? Is this probability low enough?

d. What is the maximum number of passengers that should be allowed if we want at least a 0.999 probability that the elevator will not be overloaded when it is filled with randomly selected club members?

Solution 24BB

- When considering the distribution of sample means from weights of 16 passengers, there should be corrected by using the finite population correction factor because the sample is without replacement and the sample size is 16 which are greater than 5% of the finite population of 300. The value for is 9.75 lb
- If the elevator is designed to safely carry a load of up to 3000 lb, the maximum safe mean weight of passengers when the elevator is loaded with 16 passengers is 187.5 lb
- If the elevator is filled with 16 randomly selected club members, the probability is that the total load exceeds the safe limit of 3000 lb is 0.1401. Yes, this is the low probability.
- The maximum number of passengers that should be allowed if we want at least 0.99 probabilities that the elevator will not be overloaded when it is filled with randomly selected club members is 200.