An elevator has a placard stating that the maximum capacity is 1510 lb for 10 passengers. So, 10 adult male passengers can have a mean weight of up to [tex]\frac{1510}{10} = 151[/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 151 lb. (Assume that weights of males are normally distributed with a mean of 160 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).

The probability that the elevator is overloaded if carrying 10 adult males, given a population mean weight of 160 lbs and standard deviation of 30 lbs, is 97.13%. This is based on a normal distribution of male weights and the Z-score calculation, suggesting overload is likely, which can pose a question on elevator's safety.

Explanation:

The question falls under the category of normal distribution in Mathematics. Given that the adult male population is normally distributed with a mean weight of 160 pounds and a standard deviation of 30 pounds, and the elevator has a maximum capacity of 1510 pounds, we need to find the probability of the elevator being overloaded - that is, the combined weight of 10 males exceeding 1510 pounds, or an average weight per person of more than 151 pounds.

Since weights are normally distributed, we can use the Z-score equation: Z = (X - μ) / σ, where X is the value for which we want to find the probability, μ (miu) is the population mean and σ (sigma) is the standard deviation.

Therefore, Z = (151 - 160) / (30/sqrt(10)) = -1.897. From Z table, the probability that the weight is less than 151 is 0.0287. Therefore, the probability that the weight is more than 151 is 1 - 0.0287 = 0.9713 or 97.13%, meaning the elevator is likely to be overloaded if it carries 10 adult men.

Whether the elevator is safe or not will depend on the regularity of such overloaded occurrences, the exact weight of each male (though the normal distribution can give us a likelihood, individual weights may vary) and the elevator's ability to handle occasional overloads.

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