Question

1.) A boat capsized and sank in a lake. Based on an assumption of a mean weight of 148 ​lb, the boat was rated to carry 60 passengers​ (so the load limit was 8 comma 880 ​lb). After the boat​ sank, the assumed mean weight for similar boats was changed from 148 lb to 170 lb. Complete parts a and b below.

a.) Assume that a similar boat is loaded with 60 ​passengers, and assume that the weights of people are normally distributed with a mean of 178.3 lb and a standard deviation of 40.9 lb. Find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 148 lb.

b. The boat was later rated to carry only 15 ​passengers, and the load limit was changed to 2 comma 550 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 170 ​(so that their total weight is greater than the maximum capacity of 2 comma 550 ​lb). The probability is nothing. ​(Round to four decimal places as​ needed.) Do the new ratings appear to be safe when the boat is loaded with 15 ​passengers? Choose the correct answer below.

A. Because there is a high probability of​ overloading, the new ratings appear to be safe when the boat is loaded with 15 passengers.

B. Because there is a high probability of​ overloading, the new ratings do not appear to be safe when the boat is loaded with 15 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 178.3 is greater than 170​, the new ratings do not appear to be safe when the boat is loaded with 15 passengers.

2.) An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 120 lb and 171 lb. The new population of pilots has normally distributed weights with a mean of 130 lb and a standard deviation of 29.4 lb.

a. If a pilot is randomly​ selected, find the probability that his weight is between 120 lb and 171 lb.

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

b. If 33 different pilots are randomly​ selected, find the probability that their mean weight is between 120 lb and 171 lb.

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

c. When redesigning the ejection​ seat, which probability is more​ relevant?

A. Part​ (b) because the seat performance for a single pilot is more important.

B. Part​ (a) because the seat performance for a single pilot is more important.

C. Part​ (a) because the seat performance for a sample of pilots is more important.

D. Part​ (b) because the seat performance for a sample of pilots is more important.

Answer 1

1)

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	178.3
std deviation   =σ=	40.900
sample size       =n=	60
std error=σx̅=σ/√n=	5.28017

probability =P(X>148)=P(Z>(148-178.3)/5.28)=P(Z>-5.74)=1-P(Z<-5.74)=1-0=1.0000

b)

probability =P(X>170)=P(Z>(170-178.3)/10.56)=P(Z>-0.79)=1-P(Z<-0.79)=1-0.2148=0.7852

D. Because 178.3 is greater than 170​, the new ratings do not appear to be safe when the boat is loaded with 15 passengers.

2)

a)

probability =P(120<X<171)=P((120-130)/7.591)<Z<(171-130)/7.591)=P(-1.32<Z<5.4)=1-0.0934=0.9066

b)

probability =P(120<X<171)=P((120-130)/5.118)<Z<(171-130)/5.118)=P(-1.95<Z<8.01)=1-0.0256=0.9744

B. Part​ (a) because the seat performance for a single pilot is more important.