A)

An elevator has a placard stating that the maximum capacity is 3500 lb for 25 passengers. So, 25 adult male passengers can have a mean weight of up to [tex]$3500/25 = 140$[/tex] pounds. Assume that the weights of males are normally distributed with a mean of 188 lb and a standard deviation of 36 lb.

a. Find the probability that 1 randomly selected adult male has a weight greater than 140 lb.

b. Find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140 lb.

c. What do you conclude about the safety of this elevator?

B)

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 143 lb, the boat was rated to carry 70 passengers (so the load limit was 10,010 lb). After the boat sank, the assumed mean weight for similar boats was changed from 143 lb to 175 lb. Complete parts a and b below.

a. Assume that a similar boat is loaded with 70 passengers, and assume that the weights of people are normally distributed with a mean of 182.3 lb and a standard deviation of 37.7 lb. Find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 143 lb.

The probability is (Round to four decimal places as needed.)

C)

A water taxi carries passengers from one harbor to another. Assume that the weights of passengers are normally distributed with a mean of 190 lb and a standard deviation of 38 lb. The water taxi has a stated capacity of 25 passengers and was rated for a load limit of 3500 lb. Complete parts (a) through (d) below.

a. Given that the water taxi was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers? The maximum mean weight is ____ lb. (Type an integer or a decimal. Do not round.)

We need to find the probability that a randomly selected adult male has a weight greater than 140lb (part a) and the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140lb (part b). These probabilities will help us assess the safety of the elevator.

In the second scenario, we are given the weight distribution of people on a boat, with a mean of 182.3lb and a standard deviation of 37.7lb. We need to find the probability that a similar boat carrying 70 passengers has a mean weight greater than 143lb (part a). This probability will help us determine if the boat is overloaded.

In the third scenario, we are given the weight distribution of passengers on a water taxi, with a mean of 190lb and a standard deviation of 38lb. We need to calculate the maximum mean weight of passengers if the water taxi is filled to its stated capacity of 25 passengers (part a).

In the first scenario, to find the probability that a randomly selected adult male has a weight greater than 140lb, we can use the normal distribution and calculate the area under the curve to the right of 140lb.

Similarly, to find the probability that a sample of 25 randomly selected adult males has a mean weight greater than 140lb, we can use the distribution of the sample mean and calculate the area under the curve to the right of 140lb. These probabilities will help determine if the elevator is safe for carrying 25 adult males.

In the second scenario, we need to find the probability that a similar boat carrying 70 passengers has a mean weight greater than 143lb. We can use the normal distribution and calculate the area under the curve to the right of 143lb. If this probability is sufficiently low, it indicates that the boat is not likely to be overloaded.

In the third scenario, we are given the capacity of the water taxi (25 passengers) and the load limit (3500lb). To calculate the maximum mean weight of passengers, we divide the load limit by the stated capacity. This will provide an estimate of the maximum weight that each passenger can have on average to ensure the load limit is not exceeded.

Overall, these calculations and probabilities help assess the safety and capacity limits of the elevator, boat, and water taxi based on the weight distributions and stated specifications.

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