Chapter 6.4 Homework

1.   Assume that females have pulse rates that are normally distributed with a mean of 76.0 beats per minute and a standard deviation of 12.5 beats per minute. Complete parts​ (a) through​ (c) below.

If 1 adult female is randomly​ selected, find the probability that her pulse rate is less than 83 beats per minute. 

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

If 25 adult females are randomly​ selected, find the probability that they have pulse rates with a mean less than 83 beats per minute.

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

Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

1. Since the distribution is of​ individuals, not sample​ means, the distribution is a normal distribution for any sample size.
2. Since the mean pulse rate exceeds​ 30, the distribution of sample means is a normal distribution for any sample size.
3. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.
4. Since the distribution is of sample​ means, not​ individuals, the distribution is a normal distribution for any sample size.

2.   The overhead reach distances of adult females are normally distributed with a mean of 202.5 cm and a standard deviation of 8 cm.

Find the probability that an individual distance is greater than 211.80 cm.

Find the probability that the mean for 15 randomly selected distances is greater than 200.70 cm.

Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

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

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

Choose the correct answer below.

1. The normal distribution can be used because the mean is large.
2. The normal distribution can be used because the original population has a normal distribution.
3. The normal distribution can be used because the probability is less than 0.5
4. The normal distribution can be used because the finite population correction factor is small.

3.  Assume that females have pulse rates that are normally distributed with a mean of 74.0 beats per minute and a standard deviation of 12.5 beats per minute. Complete parts​ (a) through​ (c) below.

If 1 adult female is randomly​ selected, find the probability that her pulse rate is between 70 beats per minute and 78 beats per minute.

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

If 25 adult females are randomly​ selected, find the probability that they have pulse rates with a mean between 70 beats per minute and 78 beats per minute.

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

Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

1. Since the distribution is of​ individuals, not sample​ means, the distribution is a normal distribution for any sample size.
2. Since the mean pulse rate exceeds​ 30, the distribution of sample means is a normal distribution for any sample size.
3. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.
4. Since the distribution is of sample​ means, not​ individuals, the distribution is a normal distribution for any sample size.

4.  An elevator has a placard stating that the maximum capacity is 1680 lb—10 passengers.​ So, 10 adult male passengers can have a mean weight of up to 1680/10=168 pounds. If the elevator is loaded with 10 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 168 lb.​ (Assume that weights of males are normally distributed with a mean of 171 lb and a standard deviation of 30 lb​.)

Does this elevator appear to be​ safe? The probability the elevator is overloaded is  _______​(Round to four decimal places as​ needed.)

Does this elevator appear to be​ safe?

1. ​No, 10 randomly selected people will never be under the weight limit.
2. ​No, there is a good chance that 10 randomly selected adult male passengers will exceed the elevator capacity.
3. ​Yes, there is a good chance that 10 randomly selected people will not exceed the elevator capacity.
4. ​Yes, 10 randomly selected adult male passengers will always be under the weight limit.

5.  A ski gondola carries skiers to the top of a mountain. Assume that weights of skiers are normally distributed with a mean of 186 lb and a standard deviation of 42 lb. The gondola has a stated capacity of 25 ​passengers, and the gondola is rated for a load limit of 3500 lb. Complete parts​ (a) through​ (d) below.

Given that the gondola is rated for a load limit of 3500 ​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 not​ round.)

If the gondola is filled with 25 randomly selected​ skiers, what is the probability that their mean weight exceeds the value from part​ (a)?

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

If the weight assumptions were revised so that the new capacity became 20 passengers and the gondola is filled with 20 randomly selected​ skiers, what is the probability that their mean weight exceeds 175 ​lb, which is the maximum mean weight that does not cause the total load to exceed 3500 ​lb?

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

Is the new capacity of 20 passengers​ safe?

Since the probability of overloading is ________________ the new capacity _______ to be safe enough.

6.  Suppose that an airline uses a seat width of 16.4 in. Assume men have hip breadths that are normally distributed with a mean of 14.9 in. and a standard deviation of 0.9 Complete parts​ (a) through​ (c) below.

​(a) Find the probability that if an individual man is randomly​ selected, his hip breadth will be greater than 16.4 in.

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

​(b) If a plane is filled with 123 randomly selected​ men, find the probability that these men have a mean hip breadth greater than 16.4 in.

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

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

The result from _______ should be considered because ___________

7.  The weights of a certain brand of candies are normally distributed with a mean weight of 0.8611 g and a standard deviation of 0.0517 g. A sample of these candies came from a package containing 460 ​candies, and the package label stated that the net weight is 392.9 g.​ (If every package has 460 ​candies, the mean weight of the candies must exceed 392.9460=0.8542 g for the net contents to weigh at least 392.9 ​g.)

If 1 candy is randomly​ selected, find the probability that it weighs more than 0.8542 g.

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

If 460 candies are randomly​ selected, find the probability that their mean weight is at least 0.8542g.The probability that a sample of 460 candies will have a mean of 0.8542 g or greater is _______  (Round to four decimal places as​ needed.)

Given these​ results, does it seem that the candy company is providing consumers with the amount claimed on the​ label? ________ because the probability of getting a sample mean of 0.8542 g or greater when 460 candies are selected _______ exceptionally small.

8.

An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 140 lb and 191 lb. The new population of pilots has normally distributed weights with a mean of 147 lb and a standard deviation of 27.1 lb.

If a pilot is randomly​ selected, find the probability that his weight is between 140 lb and 191 lb.

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

If 32 different pilots are randomly​ selected, find the probability that their mean weight is between 140 lb and 191 lb.

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

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

1. Part​ (b) because the seat performance for a sample of pilots is more important.
2. Part​ (a) because the seat performance for a sample of pilots is more important.
3. Part​ (a) because the seat performance for a single pilot is more important.
4. Part​ (b) because the seat performance for a single pilot is more important.

9.  A boat capsized and sank in a lake. Based on an assumption of a mean weight of 137 ​lb, the boat was rated to carry 50 passengers​ (so the load limit was 6,850 ​lb). After the boat​ sank, the assumed mean weight for similar boats was changed from 137 lb to 175 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 181.6 lb and a standard deviation of 36.4 lb. Find the probability that the boat is overloaded because the 50 passengers have a mean weight greater than 137 lb.

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

The boat was later rated to carry only 16 ​passengers, and the load limit was changed to 2,800 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 175 ​(so that their total weight is greater than the maximum capacity of 2,800 ​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 16 ​passengers? Choose the correct answer below.

1. 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. Because there is a high probability of​ overloading, the new ratings appear to be safe when the boat is loaded with 16 passengers.
3. Because the probability of overloading is lower with the new ratings than with the old​ ratings, the new ratings appear to be safe.
4. Because 181.6 is greater than 175​, the new ratings do not appear to be safe when the boat is loaded with16 passengers.

10.  An airliner carries 150 passengers and has doors with a height of 74 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).

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

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

If half of the 150 passengers are​ men, find the probability that the mean height of the 75 men is less than 74 in.

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

When considering the comfort and safety of​ passengers, which result is more​ relevant: the probability from part​ (a) or the probability from part​ (b)? Why?

1. 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.
2. 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.
3. The probability from part​ (a) is more relevant because it shows the proportion of male passengers that will not need to bend.
4. The probability from part​ (b) is more relevant because it shows the proportion of male passengers that will not need to bend.

When considering the comfort and safety of​ passengers, why are women ignored in this​ case?

1. Since men are generally taller than​ women, a design that accommodates a suitable proportion of men will necessarily accommodate a greater proportion of women.
2. Since men are generally taller than​ women, 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.
3. There is no adequate reason to ignore women. A separate statistical analysis should be carried out for the case of women.

11.  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

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 of179.5 lb and a standard deviation of 36.3.

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

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

1. No. Because the probability is​ high, the aircraft is safe to fly with its current load.
2. Yes. Because the probability is​ high, the pilot should take action by somehow reducing the weight of the aircraft.

12. Fill in the blank.

The​ _______ tells us that for a population with any​ distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.

13    Fill in the blank.

The standard deviation of the distribution of sample means is​ _______.

14.   Which of the following is NOT a conclusion of the Central Limit​ Theorem?

Choose the correct answer below.

1. The mean of all sample means is the population mean
2. The standard deviation of all sample means is the population standard deviation divided by the square root of the sample size.
3. The distribution of the sample means x ​will, as the sample size​ increases, approach a normal distribution.
4. The distribution of the sample data will approach a normal distribution as the sample size increases.