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 138 lb to 173 lb. Complete parts a and 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 179.3 lb and a standard deviation of 38.7 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 139 lb.

To find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 139lb, we can use the concept of the sampling distribution of the sample mean.

Explanation:

To find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 139lb, we can use the concept of the sampling distribution of the sample mean. The mean weight of the passengers is given as 179.3lb with a standard deviation of 38.7lb. We can use the Central Limit Theorem to approximate the sampling distribution as a normal distribution. The formula for the standard deviation of the sampling distribution is sigma/sqrt(n), where sigma is the standard deviation of the population and n is the sample size. In this case, the standard deviation of the sampling distribution is 38.7/sqrt(50).

To find the z-score for a mean weight greater than 139lb, we use the formula z = (x - mu) / (sigma/sqrt(n)), where x is the threshold value, mu is the mean of the sampling distribution, and sigma is the standard deviation of the sampling distribution. In this case, x = 139, mu = 179.3, sigma = 38.7/sqrt(50), and n = 50. Plugging these values into the formula, we get z = (139 - 179.3) / (38.7/sqrt(50)).

Finally, we can use a z-score table or a calculator to find the probability of a z-score greater than the calculated value. This probability represents the probability that the boat is overloaded.

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