A boat capsized and sank in a lake. Based on an assumption of a mean weight of 142 lb , the boat was rated to carry 70 passengers (so the load limit was \( 9,940 \mathrm{lb} \) ). After the boat sank, the a for similar boats was changed from 142 lb to 172 lb . Complete parts a and b below. a. Assume that a similar boat is loaded with 70 passengers, and assume that the weights of people are normally distributed with a mean of 182.1 lb and a standard deviation of 38.6 lb . Find boat is overloaded because the 70 passengers have a mean weight greater than 142 lb . The probability is (Round to four decimal places as needed.)

In the world of boating safety, the capsizing of vessels can often lead to crucial changes in regulations. Following incidents, like the one you've described, the ratings for passenger limits and weights are reviewed to ensure enhanced safety. The shift in average weight considerations is vital, as it helps reflect not just average statistics, but also physical realities that need addressing for public safety. Now, to ensure you're set with the right tools for navigating similar problems, remember that normal distribution can be a bit tricky! A common mistake many make is ignoring the importance of checking the weight of every passenger individually, as well as rolling out the use of proper calculations for the mean and standard deviation. Always ensure that you're accounting for the full range of possibilities to avoid being caught off guard that could lead to dangerous outcomes.