Question

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 146 lb, the boat was rated to carry 50 passengers (so the load limit was 7 comma 300 lb). After the boat sank, the assumed mean weight for similar boats was changed from 146 lb to 172 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 180.8 lb and a standard deviation of 36.9 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 146 lb.

The probability is _____ (Round to four decimal places as needed.)

b. The boat was later rated to carry only 14 passengers, and the load limit was changed to 2,408 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 172 (so that their total weight is greater than the maximum capacity of 2,408 lb).

The probability is ____(Round to four decimal places as needed.)

Do the new ratings appear to be safe when the boat is loaded with 14 passengers? Choose the correct answer below.

A. Because 180.8 is greater than 172, the new ratings do not appear to be safe when the boat is loaded with 14 passengers.

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

C. Because there is a high probability of overloading, the new ratings do not appear to be safe when the boat is loaded with 14 passengers.

D. Because there is a high probability of overloading, the new ratings appear to be safe when the boat is loaded with 14 passengers.

Answer 1

a)

for normal distribution z score =(X-μ)/σ
here mean=       μ=	180.8
std deviation   =σ=	36.9000
sample size       =n=	50
std error=σx̅=σ/√n=	5.2184

probability =

P(X>146)

=

P(Z>-6.67)=

1-P(Z<-6.67)=

1-0=

1.0000

b)

sample size       =n=	14
std error=σx̅=σ/√n=	9.8619

probability =

P(X>172)

=

P(Z>-0.89)=

1-P(Z<-0.89)=

1-0.1867=

0.8133

A. Because 180.8 is greater than 172, the new ratings do not appear to be safe when the boat is loaded with 14 passengers.