A boat capsized and sank in a lake. Based on an assumption of a mean weight of 138 lb, the boat was rated to carry 70 passengers (so the load limit was 9,660 lb). After the boat sank, the assumed mean weight for similar boats was changed from 138 lb to 173 lb. Complete parts a through c 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 179.4 lb. and a standard deviation of 37.7 lb. Find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 138 lb.

The probability is .

(Round to four decimal places as needed)

b. The boat was later rated to carry only 17 passengers , and the load limit was changed to 2,941 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 173 ( so that their total weight is greater than the maximum capacity of 2,941 lb)

The probability is .

(Round to four decimal places as needed)

c. Do the new ratings appear to be safe when the boat is loaded with 17 passengers? Choose the correct answer below.

A. Because 179.4 is greater than 173, the new ratings do not appear to be safe when the boat is loaded with 17 passengers.

B. Because the probability of overloading is lower with the new ratings than with the old ratings, the new ratings appear to be safe.

C. Because there is a high probability of overloading, the new ratings do not appear safe when the boat is loaded with 17 passengers.

D. Because there is a high probability of overloading, the new ratings appear to be safe when the boat is loaded with 17 passengers.

a)

And using the complement rule we got:

b)

And using the complement rule we got:

c) B. Because the probability of overloading is lower with the new ratings than with the old ratings, the new ratings appear to be safe.

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:

Where and

For this case since the distirbution for X is normal then the distribution for the sample mean is also normal and given by:

If we apply this formula we got:

And using the complement rule we got:

Part b

And using the complement rule we got:

Part c

B. Because the probability of overloading is lower with the new ratings than with the old ratings, the new ratings appear to be safe.

step-by-step explanation:

through just a quick inspection, one can see that the graph of this cubing function starts in quadrant iii and ends in quadrant i.   when x = 0, y = 3, so the y-intercept is (0, 3).

if graphs are part of the problem statement, share them.   .

answer: 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

step-by-step explanation: