The boat was later rated to carry only 16... A boat capsized and sank in a lake.

Question

The boat was later rated to carry only 16...

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 147 lb, the boat was rated to carry 50 passengers (so the load limit was 7, 350 lb). After the boat sank, the assumed mean weight for similar boats was changed from 147 lb to 174 lb. Complete parts a and b below. Assume that a similar boat is loaded with 50 passengers, and assume that the weights of people are normally distributed with a mean of 181.8 lb and a standard deviation of 39.4 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 147 lb. The probability is .9999. (Round to four decimal places as needed.) The boat was later rated to carry only 16 passengers, and the load limit was changed to 2, 784 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 174 (so that their total weight is greater than the maximum capacity of 2, 784 lb). The probability is (Round to four decimal places as needed.)

Explanation / Answer

total expected weight of 50 passengers =181.8*50=9090

and std deviation =(39.4)*(50)1/2=278.6

as load limit=147*50=7350

a) hence P(X>7350)=1-P(X<7350)=1-P(Z<(7350-9090)/278.6)=1-P(Z<-6.2455)=1-0.0000=1

b)load limit=2784

total weight of 16 people =16*181.8=2908.8

and std deviation =39.4*(16)1/2 =157.6

hence P(X>2784)=1-P(Z<(2784-2908.8)/157.6)=1-P(Z<-0.7919)=1-0.2142=0.7858

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