A boat capsized and sank in a lake Based on an assumption of

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 140 lb, the boat was rated to carry 50 passengers (so the load limit was 7,000 lb). After the boat sank, the assumed mean weight for similar boats was changed from 140 lb to 172 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 177.4 lb and a standard deviation of 37.3 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 140 lb.

The probability is____

(Round to four decimal places as needed.)

b.?The boat was later rated to carry only 15 passengers, and the load limit was changed to 2,580 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 172 (so that their total weight is greater than the maximum capacity of 2,580 lb).

The probability is _____

(Round to four decimal places as needed.)

Do the new ratings appear to be safe when the boat is loaded with 15 passengers??Choose the correct answer below.

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

___B. Because there is a high probability of overloading, the new ratings appear to be safe when the boat is loaded with 15 passengers.

___C. Because 177.4 is greater than 172, the new ratings do not appear to be safe when the boat is loaded with 15 passengers.

___D. Because there is a high probability of overloading, the new ratings do not appear to be safe when the boat is loaded with 15 passengers

Solution

a) x bar =177.4 and mu =140

z value = 37.4/37.3 =1.003

P(Z>1.003) = 0.3413

----------------------------------

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