Question

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 141 lb, the boat was rated to carry 50 passengers (so the load limit was 7 comma 050 lb). After the boat sank, the assumed mean weight for similar boats was changed from 141 lb to 173 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 lb and a standard deviation of 37.9 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 141 lb. The probability is 1. (Round to four decimal places as needed.) b. The boat was later rated to carry only 16 passengers, and the load limit was changed to 2 comma 768 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 comma 768 lb). The probability is ?

Answer #1

Solution :

Given that mean μ = 178.6 , standard deviation σ = 37.9

a. with n = 50

=> P(x > 141) = P((x - μ)/(σ/sqrt(n)) > (141 - 178.6)/(37.9/sqrt(50)))

= P(Z > -7.0151)

= 1

=> About 100% of chance that the boat is overloaded when the mean weight of the 50 passengers is greater than 141 lb.

b. with n = 16

=> P(x > 173) = P((x - μ)/(σ/sqrt(n)) > (173 - 178.6)/(37.9/sqrt(50)))

= P(Z > -1.0448)

= 0.8508

=> About 85.08% of chance that the boat is overloaded when the mean weight of the 16 passengers is greater than 173 lb.

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