A boat capsized and sank in a lake. Based on an assumption of a mean weight of $145 \mathrm{lb}$, the boat was rated to carry 50 passengers (so the load limit was 7,250 lb). After the boat sank, the assumed mean weight for similar boats was changed from $145 \mathrm{lb}$ to $174 \mathrm{lb}$. 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 $178.6 \mathrm{lb}$ and a standard deviation of $37.6 \mathrm{lb}$. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than $145 \mathrm{lb}$.
The probability is 1.0000 .
(Round to four decimal places as needed.)
b. The boat was later rated to carry only 14 passengers, and the load limit was changed to $2,436 \mathrm{lb}$. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 174 (so that their total weight is greater than the maximum capacity of $2,436 \mathrm{lb}$ ).
The probability is
(Round to four decimal places as needed.)
Answer
Final Answer: The probability that the boat is overloaded because the 50 passengers have a mean weight greater than 145 lb is approximately \(\boxed{1.0000}\).
Step 1 ：Given that the weights of people are normally distributed with a mean of 178.6 lb and a standard deviation of 37.6 lb, we are asked to find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 145 lb.
Step 2 ：We can use the z-score formula to find this probability. The z-score formula is: \(z = \frac{X - \mu}{\sigma / \sqrt{n}}\), where X is the value we are interested in (145 lb in this case), \(\mu\) is the mean (178.6 lb), \(\sigma\) is the standard deviation (37.6 lb), and n is the number of observations (50 passengers).
Step 3 ：Substituting the given values into the z-score formula, we get: \(z = \frac{145 - 178.6}{37.6 / \sqrt{50}} = -6.318826555284041\).
Step 4 ：The probability corresponding to this z-score is approximately 1.0000. This means that it is almost certain that the boat will be overloaded if the mean weight of the passengers is greater than 145 lb.
Step 5 ：Final Answer: The probability that the boat is overloaded because the 50 passengers have a mean weight greater than 145 lb is approximately \(\boxed{1.0000}\).
