A boat capsized and sank in a lake. Based on an assumption of a mean weight of 1

Question

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 60 passengers (so the load limit was 8, 400 lb). After the boat sank, the assumed mean weight for similar boats was changed from 140 lb to 175 lb. Complete parts a and b below. a. Assume that a similar boat is loaded with 60 passengers, and assume that the weights of people are normally distributed with a mean of 178.9 lb and a standard deviation of 39.9 lb. Find the probability that the boat is overloaded because the 60 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 17 passengers, and the load limit was changed to 2, 975 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 175 (so that their total weight is greater than the maximum capacity of 2, 975 lb). The probability is. (Round to four decimal places as needed.)

Explanation / Answer

Solution:-

a)
Given, that the weights of people are normally distributed,
where, n = 60
mean, = 178.9 lb, and
Standard deviation, = 39.9 lb
Cummulative probability, P( xbar 140)
z = (xbar - ) / (/sqrt(n))
= (140 - 178.9) / (39.9/sqrt(60))
= -7.5518
Therefore, P( z > -7.5518) = 1
b)
Given, n = 17,
Mean, = 178.9 lb
Standard deviation, = 39.9 lb
Cummulative probability, P( xbar 175)
z = (xbar - ) / (/sqrt(n))
= (178.9 - 175 / (39.9/sqrt(17))
= 0.4030
Therefore, P( z > 0.4030) = 0.6869

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