Question

An elevator has a placard stating that the maximum capacity is 2400 lb - 15 passengers. So, 15 adult male passengers can have a mean weight of up to 2400 / 15 = 160 pounds. If the elevator is loaded with 15 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 160 lb. (Assume that weights of males are normally distributed with a mean of 170 lb and a standard deviation of 30 lb.) Does this elevator appear to be safe?

Select the most appropriate answer:

A. No, there is a good chance that 15 randomly selected people will exceed the elevator capacity.

B. Yes, 15 randomly selected people will always be under the weight limit.

C. Yes, there is a good chance that 15 randomly selected people will not exceed the elevator capacity.

D. No, 15 randomly selected people will never be under the weight limit.

Answer 1

To calculate the probability that the elevator is overloaded, we can use the sampling distribution of the sample mean. By calculating the z-score, we can find the probability that the mean weight of the 15 adult male passengers is greater than 160 lb. The probability is approximately 0.6306, indicating that there is a good chance that the elevator will not be overloaded.

Step-by-step explanation:

To find the probability that the mean weight of the 15 adult male passengers exceeds 160 pounds, we can use the concept of the sampling distribution of the sample mean. The sampling distribution of the sample mean follows a normal distribution when the sample size is sufficiently large (central limit theorem). In this case, the mean weight of the 15 adult male passengers is normally distributed with a mean of 170 lb and a standard deviation of 30 lb. We need to calculate the probability that the sample mean is greater than 160 lb.

We can use the formula for the z-score, which represents the number of standard deviations a value is away from the mean:

z = (x - mean) / standard deviation

In this case, x = 160 lb, mean = 170 lb, and standard deviation = 30 lb. Plugging in these values, we get:

z = (160 - 170) / 30 = -1/3

We need to find the probability that the z-score is greater than -1/3. Using a standard normal distribution table or a calculator, we can find this probability to be approximately 0.6306. Therefore, the probability that the elevator is overloaded is approximately 0.6306.

Based on this probability, we can conclude that there is a good chance (more than 50%) that 15 randomly selected adult male passengers will not exceed the elevator capacity. Therefore, option C is the most appropriate answer: Yes, there is a good chance that 15 randomly selected people will not exceed the elevator capacity.