An elevator has a placard stating that the maximum capacity is 2355 lb for 15 passengers. Therefore, 15 adult male passengers can have a mean weight of up to [tex]\frac{2355}{15} = 157[/tex] 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 157 lb. (Assume that weights of males are normally distributed with a mean of 161 lb and a standard deviation of 28 lb.)

Does this elevator appear to be safe?

The probability that the elevator is overloaded is __

The probability that 15 adult males randomly selected would overload the elevator is approximately 29%. Given that the average weight of adult males is 161 pounds, this probability suggests that the elevator's rated capacity is too low for its intended use, posing a safety risk.

Explanation:

The problem is basically asking for the probability that the average weight of 15 randomly selected adult males would exceed 157 pounds, given that the mean weight is 161 pounds and the standard deviation is 28 pounds. In the real world, we would need to take a sample of adult males and measure their weights. However, given that we know the population mean and standard deviation, we can make use of the Central Limit Theorem to approximate this probability.

The Central Limit Theorem tells us that the mean of a large sample drawn from a population is approximately normally distributed. Since we are dealing with averages, we need to find the standard deviation of the means, which is the standard deviation of the population divided by the square root of the sample size (28/√15 = 7.23).

Next, we find the z-score, which is the difference between the measurement in question (157 pounds) and the mean (161 pounds) divided by the standard deviation of the means ((157-161)/7.23 = -0.55). The z-score tells us how many standard deviations our measurement is from the mean. To find the probability that a measurement is greater than indicated by our z-score, we need to look it up in a standard normal distribution table or use a calculator or computer program that can do the computation. The result is approximately 0.29, or 29%. So there's a 29% chance that 15 adult males chosen at random would overload this elevator.

If this is a standard service elevator used in everyday situations, a 29% chance of overload does not appear to be safe. Risk management best practices would recommend a maximum tolerable risk of 1% or even less for such routine operations, suggesting that the elevator's rated capacity is not appropriate for its intended use.

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