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An elevator has a placard stating that the maximum capacity is 1760 lb for 10 passengers. Thus, 10 adult male passengers can have a mean weight of up to [tex]1760/10 = 176[/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 176 lb. Assume that weights of males are normally distributed with a mean of 179 lb and a standard deviation of 27 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. No, there is a good chance that 10 randomly selected adult male passengers will exceed the elevator capacity.
B. Yes, there is a good chance that 10 randomly selected people will not exceed the elevator capacity.
C. Yes, 10 randomly selected adult male passengers will always be under the weight limit.
D. No, 10 randomly selected people will never be under the weight limit.

Answer :

Final answer:

To find the probability that the elevator is overloaded, use the Central Limit Theorem to approximate the distribution of the mean weight of 10 adult male passengers. Calculate the z-score for a mean weight of 176 pounds and find the probability associated with this z-score using a z-table or calculator. The probability is approximately 0.2823, indicating that there is a good chance the elevator will be overloaded. Therefore, the elevator does not appear to be safe.

Explanation:

To find the probability that the elevator is overloaded, we need to calculate the probability that the mean weight of 10 adult male passengers is greater than 176 pounds. We can use the Central Limit Theorem to approximate the distribution of the mean weight. Given a mean of 179 pounds and a standard deviation of 27 pounds, we can calculate the z-score for a mean weight of 176 pounds by using the formula:

z = (x - μ) / (σ / sqrt(n))


Substituting in the values:

z = (176 - 179) / (27 / sqrt(10))

Calculating this, we find that the z-score is approximately -0.574. Using a z-table or calculator, we can find the probability associated with this z-score. The probability that the mean weight exceeds 176 pounds is the same as the probability that the z-score is less than -0.574. The lookup table or calculator will give us a probability of approximately 0.2823. Therefore, the probability that the elevator is overloaded is approximately 0.2823.

Based on this probability, we can determine that there is a good chance that 10 randomly selected adult male passengers will exceed the elevator capacity. Therefore, the elevator does not appear to be safe.

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