Step 1 :Given that the weights of people are normally distributed with a mean of 178.6 lb and a standard deviation of 37.6 lb, we are asked to find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 145 lb.
Step 2 :We can use the z-score formula to find this probability. The z-score formula is: \(z = \frac{X - \mu}{\sigma / \sqrt{n}}\), where X is the value we are interested in (145 lb in this case), \(\mu\) is the mean (178.6 lb), \(\sigma\) is the standard deviation (37.6 lb), and n is the number of observations (50 passengers).
Step 3 :Substituting the given values into the z-score formula, we get: \(z = \frac{145 - 178.6}{37.6 / \sqrt{50}} = -6.318826555284041\).
Step 4 :The probability corresponding to this z-score is approximately 1.0000. This means that it is almost certain that the boat will be overloaded if the mean weight of the passengers is greater than 145 lb.
Step 5 :Final Answer: The probability that the boat is overloaded because the 50 passengers have a mean weight greater than 145 lb is approximately \(\boxed{1.0000}\).