b. The boat was later rated to carry only 14 passengers, and the load limit was changed to $2,436 \mathrm{lb}$. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 174 (so that their total weight is greater than the maximum capacity of $2,436 \mathrm{lb}$ ).
The probability is (Round to four decimal placessas needed.)

Step 1 ：The problem is asking for the probability that the boat is overloaded, given that the mean weight of the passengers is greater than 174 lbs. This means that the total weight of the passengers is greater than the maximum capacity of the boat, which is 2436 lbs.

Step 2 ：We know that the mean weight of the passengers is greater than 174 lbs. We also know that the boat can carry only 14 passengers. Therefore, the total weight of the passengers that the boat can carry is \(174 \text{ lbs} \times 14 = 2436 \text{ lbs}\).

Step 3 ：We need to find the probability that the total weight of the passengers is greater than 2436 lbs. This is equivalent to finding the probability that the mean weight of the passengers is greater than 174 lbs.

Step 4 ：We can use the z-score formula to find this probability. The z-score is a measure of how many standard deviations an element is from the mean. In this case, we want to find the z-score for a weight of 174 lbs.

Step 5 ：The z-score formula is: \(z = \frac{(X - \mu)}{\sigma}\) where: - X is the value we are interested in (in this case, 174 lbs) - \(\mu\) is the mean - \(\sigma\) is the standard deviation

Step 6 ：We don't know the mean and the standard deviation of the weights of the passengers. However, we can assume that the weights are normally distributed. Therefore, we can use the properties of the normal distribution to find the mean and the standard deviation.

Step 7 ：The mean of a normal distribution is the value that has the highest probability (the peak of the distribution). The standard deviation is a measure of the spread of the distribution.

Step 8 ：The probability that the boat is overloaded because the mean weight of the passengers is greater than 174 lbs is 0.5. This means that there is a 50% chance that the boat will be overloaded if the mean weight of the passengers is greater than 174 lbs. This makes sense because 174 lbs is the mean weight of the passengers that the boat can carry. Therefore, if the mean weight of the passengers is greater than 174 lbs, the boat will be overloaded half of the time.

Step 9 ：Final Answer: The probability that the boat is overloaded because the mean weight of the passengers is greater than 174 lbs is \(\boxed{0.5}\).