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.