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------------------------------------------------ Men have an average weight of 172 pounds with a standard deviation of 29 pounds.

a. Find the probability that 20 randomly selected men will have a sum weight greater than 3600 lbs.

b. If 20 men have a sum weight greater than 3500 lbs, then their total weight exceeds the safety limits for water taxis. Based on (a), is this a safety concern? Explain.

The question involves using the Central Limit Theorem to calculate the probability that 20 randomly selected men have a total weight over a certain threshold. The probability of them weighing more than 3600 pounds is found to be 10.9%, indicating a potential safety concern for water taxis.

Explanation:

This is a problem of probability involving the Central Limit Theorem. The Central Limit Theorem tells us that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.

Here our random variable is the sum weight of 20 randomly selected men. The expectation of the sum of 20 men's weights is 20*172 = 3440 pounds and the standard deviation is sqrt(20)*29 = approx.130 pounds. We then convert 3600 to a z-score (z=(x-μ)/σ) getting z ≈ 1.23. Consulting a standard normal distribution table, we find that the area to the left of z=1.23 is about 0.891, so the probability that a group of 20 men have a sum weight greater than 3600 pounds is 1-0.891 = 0.109 or 10.9%. So, overloading water taxis could be a realistic concern, depending on the frequency of such groups.

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