A water taxi carries passengers from harbor to another. Assume that weights of passengers are normally distributed with a mean of 195 lb and a standard deviation of 37 lb . The water taxi has a stated capacity of 25 passengers, and the water taxi was 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 150 lb . (Type an integer or a decimal. Do not round.) 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 (Round to four deafmal places as needed.)

To address the maximum mean weight of the passengers, given a load limit of 3750 lb for 25 passengers, you need to divide the limit by the number of passengers. Plugging in the numbers, we get: Maximum mean weight = 3750 lb / 25 = 150 lb. So, if the water taxi is filled to capacity, the passengers can only average 150 lb each! Now, regarding the probability of the mean weight exceeding 150 lb when 25 passengers are selected, we can use the Central Limit Theorem. The standard deviation of the mean weight of these passengers is calculated as the standard deviation divided by the square root of the sample size (n = 25): Standard error (SE) = standard deviation / √n = 37 lb / √25 = 7.4 lb. Next, we compute the z-score for the mean weight of 150 lb: Z = (X - μ) / SE = (150 - 195) / 7.4 ≈ -6.08. You would then look up this z-score in a standard normal distribution table or use a calculator to find: P(Z > -6.08). Since this z-score is quite extreme, the probability that their mean weight exceeds 150 lb is exceedingly close to 1. So, there you have it! The probability is virtually 1 (which in decimal form rounds to 1.0000). If you'd like to explore more about the Central Limit Theorem and its applications, I've got some fun tidbits!