Section 6.4

Question 3

Assume that females have pulse rates that are normally distributed with a mean of =72.0 beats per minute and a standard deviation of =12.5 beats per minute. Complete parts(a) through(c) below.

a. If 1 adult female is randomlyselected, find the probability that her pulse rate is less than 75 beats per minute.

The probability is _______.

(Round to four decimal places asneeded.)

b. If 16 adult females are randomlyselected, find the probability that they have pulse rates with a mean less than 75 beats per minute.

The probability is _______

(Round to four decimal places asneeded.)

c. Why can the normal distribution be used in part(b), even though the sample size does not exceed30?

A. Since the distribution is ofindividuals, not samplemeans, the distribution is a normal distribution for any sample size.

B. Since the distribution is of samplemeans, notindividuals, the distribution is a normal distribution for any sample size.

C. Since the original population has a normaldistribution, the distribution of sample means is a normal distribution for any sample size.

D. Since the mean pulse rate exceeds30, the distribution of sample means is a normal distribution for any sample size.

Question 4

Assume that females have pulse rates that are normally distributed with a mean of =72.0 beats per minute and a standard deviation of =12.5 beats per minute. Complete parts(a) through(c) below.

a. If 1 adult female is randomlyselected, find the probability that her pulse rate is between 66 beats per minute and 78 beats per minute.

The probability is _______.

(Round to four decimal places asneeded.)

b. If 16 adult females are randomlyselected, find the probability that they have pulse rates with a mean between 66 beats per minute and 78 beats per minute.

The probability is _______.

(Round to four decimal places asneeded.)

c. Why can the normal distribution be used in part(b), even though the sample size does not exceed30?

A. Since the original population has a normaldistribution, the distribution of sample means is a normal distribution for any sample size.

B. Since the distribution is ofindividuals, not samplemeans, the distribution is a normal distribution for any sample size.

C. Since the distribution is of samplemeans, notindividuals, the distribution is a normal distribution for any sample size.

D. Since the mean pulse rate exceeds30, the distribution of sample means is a normal distribution for any sample size.

Question 5

An elevator has a placard stating that the maximum capacity is 1272 lb8 passengers.So, 8 adult male passengers can have a mean weight of up to 1272/8=159pounds. If the elevator is loaded with 8 adult malepassengers, find the probability that it is overloaded because they have a mean weight greater than 159 lb.(Assume that weights of males are normally distributed with a mean of 163lb and a standard deviation of 27lb.) Does this elevator appear to besafe?

The probability the elevator is overloaded is ______.

(Round to four decimal places asneeded.)

Does this elevator appear to besafe?

A. Yes, 8 randomly selected adult male passengers will always be under the weight limit.

B. No, 8 randomly selected people will never be under the weight limit.

C. No, there is a good chance that 8 randomly selected adult male passengers will exceed the elevator capacity.

D. Yes, there is a good chance that 8 randomly selected people will not exceed the elevator capacity.

Question 6

A ski gondola carries skiers to the top of a mountain. Assume that weights of skiers are normally distributed with a mean of 202 lb and a standard deviation of 39 lb. The gondola has a stated capacity of 25 passengers, and the gondola is rated for a load limit of 3750 lb. Complete parts(a) through(d) below.

a. Given that the gondola is rated for a load limit of 3750 lb, what is the maximum mean weight of the passengers if the gondola is filled to the stated capacity of 25 passengers?

The maximum mean weight is ______ lb.

(Type an integer or a decimal. Do notround.)

b. If the gondola is filled with 25 randomly selectedskiers, what is the probability that their mean weight exceeds the value from part(a)?

The probability is _______.

(Round to four decimal places asneeded.)

c. If the weight assumptions were revised so that the new capacity became 20 passengers and the gondola is filled with 20 randomly selectedskiers, what is the probability that their mean weight exceeds 187.5 lb, which is the maximum mean weight that does not cause the total load to exceed 3750 lb?

The probability is _______.

(Round to four decimal places asneeded.)

d. Is the new capacity of 20 passengerssafe?

Since the probability of overloading is ________ (A. over 50 %, B.Under 5%) the new capacity ________( A. does not appear, B. appears) to be safe enough.

Question 7

Suppose that an airline uses a seat width of 16.6 in. Assume men have hip breadths that are normally distributed with a mean of 14.9 in. and a standard deviation of 1 in. Complete parts(a) through(c) below.

(a) Find the probability that if an individual man is randomlyselected, his hip breadth will be greater than 16.6 in.

The probability is _______.

(Round to four decimal places asneeded.)

(b) If a plane is filled with 111 randomly selectedmen, find the probability that these men have a mean hip breadth greater than 16.6 in.

The probability is ______.

(Round to four decimal places asneeded.)

(c) Which result should be considered for any changes in seatdesign: the result from part(a) or part(b)?

The result from ________ ( A. Part a, B. Part b ) should be considered because _________ ( A. the seats are occupied by individuals rather than means, B. only average individuals should be considered).

Question 9

A boat capsized and sank in a lake. Based on an assumption of a mean weight of 143 lb, the boat was rated to carry 50 passengers(so the load limit was 7,150 lb). After the boatsank, the assumed mean weight for similar boats was changed from 143 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 181.1 lb and a standard deviation of 40.5 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 143 lb.

The probability is ________.

(Round to four decimal places asneeded.)

b. The boat was later rated to carry only 14 passengers, and the load limit was changed to 2,436 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 2,436 lb).

The probability is _______.

(Round to four decimal places asneeded.)

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

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

This is the correct answer.

B. Because 181.4 is greater than 174, the new ratings do not appear to be safe when the boat is loaded with 14 passengers.

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

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

Question 10

An airliner carries 50 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 69.0 in and a standard deviation of 2.8 in. Complete parts(a) through(d).

a. If a male passenger is randomlyselected, find the probability that he can fit through the doorway without bending.

The probability is ______.

(Round to four decimal places asneeded.)

b. If half of the 100 passengers aremen, find the probability that the mean height of the 50 men is less than 72 in.

The probability is _______.

(Round to four decimal places asneeded.)

c. When considering the comfort and safety ofpassengers, which result is morerelevant: the probability from part(a) or the probability from part(b)? Why?

A. The probability from part(a) is more relevant because it shows the proportion of flights where the mean height of the male passengers will be less than the door height.

B.

The probability from part(a) is more relevant because it shows the proportion of male passengers that will not need to bend.

C. The probability from part(b) is more relevant because it shows the proportion of male passengers that will not need to bend.

D. The probability from part(b) is more relevant because it shows the proportion of flights where the mean height of the male passengers will be less than the door height.

d. When considering the comfort and safety ofpassengers, why are women ignored in thiscase?

A. There is no adequate reason to ignore women. A separate statistical analysis should be carried out for the case of women.

B. Since men are generally taller thanwomen, it is more difficult for them to bend when entering the aircraft.Therefore, it is more important that men not have to bend than it is important that women not have to bend.

C. Since men are generally taller thanwomen, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.

Question 11

Before everyflight, 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 35 passengers, and a flight has fuel and baggage that allows for a total passenger load of 5,775 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 5,775lb / 35=165 lb. What is the probability that the aircraft isoverloaded? Should the pilot take any action to correct for an overloadedaircraft? Assume that weights of men are normally distributed with a mean of 177 lb and a standard deviation of 36.6.

The probability is approximately ________.

(Round to four decimal places asneeded.)

Should the pilot take any action to correct for an overloadedaircraft?

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

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