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Answer :
Final answer:
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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