A boat capsized and sank in a lake. Based on an assumption of a mean weight of 139 lb, the boat was rated to carry 50 passengers (so the load limit was 6,950 lb). After the boat sank, the assumed mean weight for similar boats was changed from 139 lb to 174 lb.

Complete parts a and b below.

a. Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 182.4 lb and a standard deviation of 40.3 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 139 lb.

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

To find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 139 lb, we need to calculate the z-score for the given mean weight of 139 lb. The probability is 0.0000.

Explanation:

To find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 139 lb, we need to calculate the z-score for the given mean weight of 139 lb.

Z = (mean - population mean) / (standard deviation / sqrt(sample size))

Z = (139 - 182.4) / (40.3 / sqrt(50))

After finding the z-score, we can then find the probability using a z-table or a calculator. The probability is the area under the normal curve to the right of the z-score.

The probability that the boat is overloaded is 0.0000, rounded to four decimal places.

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