Question

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 130 ​lb, the boat was rated to carry 70 passengers​ (so the load limit was 9 comma 100 ​lb). After the boat​ sank, the assumed mean weight for similar boats was changed from 130 lb to 174 lb. Complete parts a and b below.

a. Assume that a similar boat is loaded with 70 ​passengers, and assume that the weights of people are normally distributed with a mean of 182.8 lb and a standard deviation of 39.9 lb. Find the probability that the boat is overloaded because the 70 passengers have a mean weight greater than 130 lb.

b. The boat was later rated to carry only 16 ​passengers, and the load limit was changed to 2 comma 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 comma 784 ​lb). The probability is Do the new ratings appear to be safe when the boat is loaded with 16 ​passengers? Choose the correct answer below.

A. Because 182.8 is greater than 174​, the new ratings do not appear to be safe when the boat is loaded with 16 passengers.

B. Because there is a high probability of​ overloading, the new ratings appear to be safe when the boat is loaded with 16 passengers.

C. Because the probability of overloading is lower with the new ratings than with the old​ ratings, the new ratings appear to be safe.

D. Because there is a high probability of​ overloading, the new ratings do not appear to be safe when the boat is loaded with 16 passengers.

Before every​ flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The aircraft can carry 42 ​passengers, and a flight has fuel and baggage that allows for a total passenger load of 6,888 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than 6,888 lb/42 = 164 lb. What is the probability that the aircraft is​ overloaded? Should the pilot take any action to correct for an overloaded​ aircraft? Assume that weights of men are normally distributed with a mean of 183 lb and a standard deviation of 36.8.

The probability is approximately

Should the pilot take any action to correct for an overloaded​ aircraft?

A. Yes. Because the probability is​ high, the pilot should take action by somehow reducing the weight of the aircraft.

B. No. Because the probability is​ high, the aircraft is safe to fly with its current load.

Answer 1

1)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	182.8
std deviation   =σ=	39.900
sample size       =n=	70
std error=σx̅=σ/√n=	4.76896

probability =P(X>130)=P(Z>(130-182.8)/4.769)=P(Z>-11.07)=1-P(Z<-11.07)=1-0=1.0000

b)

probability =P(X>174)=P(Z>(174-182.8)/9.975)=P(Z>-0.88)=1-P(Z<-0.88)=1-0.1894=0.8106

D. Because there is a high probability of​ overloading, the new ratings do not appear to be safe when the boat is loaded with 16 passengers.

2)

sample size       =n=	42
std error=σx̅=σ/√n=	5.67836

probability =P(X>164)=P(Z>(164-183)/5.678)=P(Z>-3.35)=1-P(Z<-3.35)=1-0.0004=0.9996

A. Yes. Because the probability is​ high, the pilot should take action by somehow reducing the weight of the aircraft.