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A water taxi carries passengers from one harbor to another. Assume that the weights of passengers are normally distributed with a mean of 198 lb and a standard deviation of 42 lb. The water taxi has a stated capacity of 25 passengers and is rated for a load limit of 3750 lb. Complete parts (a) through (d) below.

a) Given that the water taxi was rated for a load limit of 3750 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?

b) If the water taxi is filled with 25 randomly selected passengers, what is the probability that their mean weight exceeds the value from part (a)?
The probability is?

c) If the weight assumptions were revised so that the new capacity became 20 passengers, and the water taxi is filled with 20 randomly selected passengers, what is the probability that their mean weight exceeds 187.5 lb, which is the maximum mean weight that does not cause the total load to exceed 3750 lb?
The probability is?

Answer :

Final answer:

a) The maximum mean weight of the passengers if the water taxi is filled to its capacity of 25 passengers is 150 lb. b) The probability that the mean weight of 25 randomly selected passengers exceeds the maximum mean weight is calculated using the z-score. c) If the weight assumptions were revised and the water taxi had a capacity of 20 passengers, the probability that the mean weight exceeds a certain value can be calculated using the same method as in part (b).

Explanation:

a) To find the maximum mean weight of the passengers if the water taxi is filled to the stated capacity of 25 passengers, we need to calculate the maximum weight in total. The maximum total weight is the load limit of 3750 lb divided by the number of passengers, which is 25: 3750 lb / 25 passengers = 150 lb. Therefore, the maximum mean weight of the passengers is 150 lb.

b) To find the probability that the mean weight of 25 randomly selected passengers exceeds the value from part (a), we need to calculate the z-score for the value from part (a) and then use the z-table to find the probability. Let's assume the population standard deviation is the same as the sample standard deviation, which is 42 lb. The z-score can be calculated as (150 lb - 198 lb) / (42 lb/√ (25)). Using the z-table, we can find the probability that the z-score is greater than the calculated value. This probability represents the probability that the mean weight exceeds the value from part (a).

c) If the weight assumptions were revised so that the new capacity became 20 passengers, we can use the same calculation as in part (b) to find the probability that the mean weight exceeds 187.5 lb. The only difference is that the sample size is now 20, instead of 25.

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