A boat capsized and sank in a lake. Based on an assumption of a mean weight of 144 lb, the boat was rated to carry 60 passengers (so the load limit was 8,640 lb). After the boat sank, the assumed mean weight for similar boats was changed from 144 lb to 173 lb.

Complete parts a and b below.

a. Assume that a similar boat is loaded with 60 passengers, and assume that the weights of people are normally distributed with a mean of 182.3 lb and a standard deviation of 37.6 lb. Find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 144 lb.

The probability is ________ (Round to four decimal places as needed.)

To find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 144 lb, we calculate the z-score for 144 lb and use a z-table to find the probability.

Explanation:

To find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 144 lb, we need to calculate the z-score for the mean weight of 144 lb using the given mean and standard deviation. The formula for the z-score is (x - mean) / standard deviation, where x is the value we want to find the probability for. In this case, x is 144 lb. After finding the z-score, we can use a z-table to find the probability. The z-score for 144 lb is (144 - 182.3) / 37.6 = -1.0197.

Using the z-table, we can find the probability that a random variable from a standard normal distribution is greater than -1.0197, which is the probability of having a mean weight greater than 144 lb. The probability is approximately 0.8431.