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 136 lb, the boat was rated to carry 50 passengers (so the load limit was 6,800 lb). After the boat sank, the assumed mean weight for similar boats was changed from 136 lb to 171 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 179.5 lb and a standard deviation of 36.4 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 136 lb. The probability is 1 (Round to four decimal places as needed.) b. The boat was later rated to carry only 13 passengers, and the load limit was changed to 2,223 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 171 (so that their total weight is greater than the maximum capacity of 2,223 lb). The probability is Round to four decimal places as needed.)

Explanation / Answer

a.

P(X>136) = P( (X- 179.5)/(36.4/500.5) > (136 - 179.5)/(36.4/500.5) ) = P(Z > -8.45) = 1

b.

P(X>171) = P( (X- 179.5)/(36.4/130.5) > (171 - 179.5)/(36.4/130.5) ) = P(Z > -0.842) = 0.8001

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