An elevator has a placard stating that the maximum capacity is 1600 lb for 10 passengers. Thus, 10 adult male passengers can have a mean weight of up to [tex]\frac{1600}{10} = 160[/tex] pounds.

If the elevator is loaded with 10 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 160 lb. Assume that the weights of males are normally distributed with a mean of 165 lb and a standard deviation of 25 lb.

Does this elevator appear to be safe?

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

To calculate the probability that the elevator is overloaded, we need to find the standard deviation of the mean weight of the 10 adult male passengers and use the z-score formula to find the cumulative probability. The probability of the elevator being overloaded is approximately 26.43%, indicating that it may not be completely safe to carry 10 adult male passengers with a mean weight greater than 160 lb.

Explanation:

To find the probability that the elevator is overloaded due to the mean weight of 10 adult male passengers exceeding 160 lb, first, we need to calculate the standard deviation of the mean weight. Since the weights of males are normally distributed, the standard deviation of the mean weight is calculated as the standard deviation of individual weights divided by the square root of the number of passengers. In this case, the standard deviation of the mean weight is 25 lb / √10 ≈ 7.905 lb.

Next, we need to calculate the area under the normal distribution curve to the right of 160 lb using the z-score formula: z = (x - mean) / standard deviation. With the mean weight of 165 lb and a standard deviation of 7.905 lb, the z-score for the mean weight of 160 lb is (160 - 165) / 7.905 ≈ -0.632. We can then use a standard normal distribution table or calculator to find the cumulative probability associated with a z-score of -0.632, which is approximately 0.2643. This means that the probability of the mean weight exceeding 160 lb is approximately 0.2643, or 26.43%.

Since the probability of the elevator being overloaded is 26.43%, it appears that the elevator is not completely safe, as there is a relatively high chance of it being overloaded when carrying 10 adult male passengers with a mean weight greater than 160 lb.

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