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------------------------------------------------ An elevator has a placard stating that the maximum capacity is 1980 lb for 12 passengers. Therefore, 12 adult male passengers can have a mean weight of up to [tex]$\frac{1980}{12} = 165$[/tex] 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 165 lb. Assume that the weights of males are normally distributed with a mean of 174 lb and a standard deviation of 35 lb.

Does this elevator appear to be safe?

The probability the elevator is overloaded is _______ (Round to four decimal places as needed.)

The probability that the elevator is overloaded because the 12 adult male passengers have a mean weight greater than 165 lb is approximately 0.0684.

To solve this problem, we need to calculate the probability that the mean weight of 12 adult male passengers exceeds 165 lb. Since the weights are normally distributed with a mean of 174 lb and a standard deviation of 35 lb, we can use the Central Limit Theorem.

First, we calculate the standard deviation of the sample mean by dividing the standard deviation of the population by the square root of the sample size:

Standard deviation of sample mean = 35 lb / sqrt(12) ≈ 10.097 lb

Next, we need to standardize the mean weight of 165 lb using the formula:

Z = (X - μ) / (σ / sqrt(n))

where X is the mean weight, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Using this formula, we can calculate the Z-score:

Z = (165 - 174) / (10.097) ≈ -0.8903

Next, we find the probability of the Z-score being greater than -0.8903 using a standard normal distribution table or calculator. The probability comes out to be approximately 0.8091.

However, we are interested in the probability that the mean weight exceeds 165 lb, so we subtract the above probability from 1:

Probability (mean weight > 165 lb) = 1 - 0.8091 ≈ 0.1909

Therefore, the probability that the elevator is overloaded because the 12 adult male passengers have a mean weight greater than 165 lb is approximately 0.1909 or 19.09%.

Since this probability is higher than the acceptable threshold, the elevator does not appear to be safe for carrying 12 adult male passengers with a mean weight greater than 165 lb.

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