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.)

**a. Maximum Mean Weight Calculation** The water taxi has a load limit of \(3750\) lb and a capacity of \(25\) passengers. Thus, the maximum allowable mean weight per passenger is calculated by \[ \text{Maximum Mean} = \frac{3750}{25} = 150 \text{ lb.} \] **b. Probability that the Mean Weight Exceeds 150 lb** The weights of the passengers are normally distributed with a mean \(\mu = 195\) lb and a standard deviation \(\sigma = 37\) lb. For a sample of \(n = 25\) passengers, the sampling distribution of the sample mean has: - Mean: \(\mu_{\overline{X}} = 195\) lb - Standard deviation (standard error): \(\sigma_{\overline{X}} = \frac{37}{\sqrt{25}} = \frac{37}{5} = 7.4\) lb We need the probability that the sample mean exceeds \(150\) lb. We convert \(150\) lb to a \(z\)-value: \[ z = \frac{150 - 195}{7.4} = \frac{-45}{7.4} \approx -6.0811. \] Since the \(z\)-score is extremely low, we find the probability: \[ P\left(\overline{X} > 150\right) = P\left(Z > -6.0811\right). \] For a standard normal distribution, \(P(Z > -6.0811)\) is practically \(1\) because the probability of \(Z\) being less than \(-6.0811\) is nearly \(0\). Therefore, rounding to four decimal places, we have \[ P\left(\overline{X} > 150\right) \approx 1.0000. \]