Question

An elevator has a placard stating that the maximum capacity is 2040 10 -12 passengers. So, 12 adult male passengers can have a mean weight of up to 2040/12 = 170 pounds. If the elevator is loaded with 12 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 170 lb. (Assume that weights of males are normally distributed with a mean of 180 lb and a standard deviation of 30 lb.) Does this elevator appear to be safe?

The probability the elevator is overloaded is (Round to four decimal places as needed.)
Does this elevator appear to be safe?

A. Yes, 12 randomly selected adult male passengers will always be under the weight limit.
B. No, there is a good chance that 12 randomly selected adult male passengers will exceed the elevator capacity
C. Yes, there is a good chance that 12 randomly selected people will not exceed the elevator capacity
D. No, 12 randomly selected people will never be under the weight limit.

Answer 1

To calculate if the elevator is overloaded with 12 adult male passengers exceeding the mean weight of 170 lb, we use the normal distribution and Z-score. The Z-score of -2.3094 is calculated, and then the probability that the elevator is overloaded can be derived from the standard normal distribution table, leading to the conclusion about the safety of the elevator.

Step-by-step explanation:

To determine the probability of an elevator being overloaded when it is loaded with 12 adult male passengers, where it is given that the weights of males are normally distributed with a mean of 180 lb and a standard deviation of 30 lb, we use the concept of normal distribution and the Z-score formula for a sample mean.

Firstly, the maximum allowable mean weight per passenger is 170 lb for the elevator not to be overloaded. We need to find the probability that the actual mean weight of 12 passengers exceeds this weight limit. For this, we will calculate the Z-score using the formula:

Z = (X - μ) / (σ / √ n)

Where `X` is the sample mean that we want to compare against the mean (μ), σ is the standard deviation, and n is the sample size. Applying the values:

Z = (170 - 180) / (30 / √ 12) = -2.3094

We then consult a standard normal distribution table or use statistical software to find the probability that is associated with this Z-score. The more negative the Z-score, the lower the weight is compared to the mean, so we need the probability of Z being greater than -2.3094, which will give us the area to the right of Z-score in the standard normal distribution curve.

Assuming we have found from the table that this probability is p, the probability that the elevator is overloaded is 1 - p. If this probability is small, we can say that there is a good chance that the elevator will not be overloaded with 12 randomly selected adult male passengers. Conversely, if the probability is high, there is a good chance the elevator will be overloaded, which would indicate that the elevator is not safe according to these parameters.

Based on this analysis, if the probability we obtained is very low, ours falls under choice C. However, if the probability is high, then it matches with choice B. Since we do not have the exact probability value without a standard normal distribution table or software at hand, it is not possible to determine the correct answer between B or C from the choices given without additional calculation.