A boat capsized and sank in a lake. Based on an assumption of a mean weight of 149 lb, the boat was rated to carry 70 passengers (so the load limit was 10,430 lb). After the boat sank, the assumed mean weight for similar boats was changed from 149 lb to 173 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 177.2 lb and a standard deviation of 35.8 lb. Find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 149 lb.

The probability is

To find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 149 lb, we can calculate the z-score based on the normal distribution of weights and find the probability using a standard normal distribution table or calculator.

Explanation:

To find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 149 lb, we can use the concept of the sampling distribution of the sample mean.

We know that the weights of people are normally distributed with a mean of 177.2 lb and a standard deviation of 35.8 lb.

First, we need to calculate the standard error of the mean (SEM) using the formula SEM = standard deviation / square root of sample size. In this case, the sample size is 70.

SEM = 35.8 / sqrt (70) = 4.29 lb.

Next, we can calculate the z-score using the formula z = (x - mean) / SEM, where x is the value we want to find the probability for. In this case, x = 149 lb.

z = (149 - 177.2) / 4.29 = -6.57 (rounded to two decimal places).

Finally, we can use a standard normal distribution table or a calculator to find the probability of a z-score greater than -6.57. The probability is essentially 0, since a z-score of -6.57 is extremely unlikely in a standard normal distribution.