SECTION 5.1 Exercises

For Exercises 5.1 and 5.2, see page 245; for 5.3 to 5.6, see page 246; for 5.7 to 5.9, see page 250; for 5.10 and 5.11, see page 253; for 5.12 and 5.13, see pages 254–255; for 5.14, see page 256; and for 5.15 to 5.17, see page 260.

Most binomial probability calculations required in these exercises can be done by using Table C or the Normal approximation. Your instructor may request that you use the binomial probability formula or software. In exercises requiring the Normal approximation, you should use the continuity correction if you studied that topic.

Question 5.18

5.18 What is wrong?

Explain what is wrong in each of the following scenarios.

  1. In the binomial setting, is a proportion.
  2. The variance for a binomial count is .
  3. The Normal approximation to the binomial distribution is always accurate when is greater than 1000.
  4. We can use the binomial distribution to approximate the distribution of when we draw an SRS of size students from a population of 500 students.

Question 5.19

5.19 What is wrong?

Explain what is wrong in each of the following scenarios.

  1. If you toss a fair coin four times and a head appears each time, then the next toss is more likely to be a tail than a head.
  2. If you toss a fair coin four times and observe the pattern HTHT, then the next toss is more likely to be ahead than a tail.
  3. The quantity is one of the parameters for a binomial distribution.
  4. The binomial distribution can be used to model the daily number of pedestrian/cyclist near-crash events on campus.

5.19

(a) Each flip is independent, and prior tosses have no impact on the outcome of a new toss. (b) Each flip is independent, and prior tosses have no impact on the outcome of a new toss. (c) is a parameter for the binomial, not . (d) There is no fixed number of trials .

Question 5.20

5.20 Should you use the binomial distribution?

In each of the following situations, is it reasonable to use a binomial distribution for the random variable ? Give reasons for your answer in each case. If a binomial distribution applies, give the values of and .

  1. In a random sample of 20 students in a fitness study, is the mean daily exercise time of the sample.
  2. A manufacturer of running shoes picks a random sample of 20 shoes from the production of shoes each day for a detailed inspection. is the number of pairs of shoes with a defect.
  3. A college tutoring center chooses an SRS of 50 students. The students are asked whether or not they have used the tutoring center for any sort of tutoring help. is the number who say that they have.
  4. is the number of days during the school year when you skip a class.

Question 5.21

5.21 Should you use the binomial distribution?

In each of the following situations, is it reasonable to use a binomial distribution for the random variable ? Give reasons for your answer in each case. If a binomial distribution applies, give the values of and .

  1. A poll of 200 college students asks whether or not they usually feel irritable in the morning. is the number who reply that they do usually feel irritable in the morning.
  2. You toss a fair coin until a head appears. is the count of the number of tosses that you make.
  3. Most calls made at random by sample surveys don’t succeed in talking with a person. Of calls to New York City, only one-twelfth succeed. A survey calls 500 randomly selected numbers in New York City. is the number of times that a person is reached.
  4. You deal 10 cards from a shuffled deck of standard playing cards and count the number of black cards.

5.21

(a) , where is the probability that a student says he or she usually feels irritable in the morning. (b) This is not binomial; there is not a fixed . (c) . (d) This is not binomial because separate cards are not independent.

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Question 5.22

5.22 Checking smartphone.

A 2014 Bank of America survey of U.S. adults who own smartphones found that 35% of the respondents check their phones at least once an hour for each hour during the waking hours.6 Such smartphone owners are classified as “constant checkers.” Suppose you were to draw a random sample of 10 smartphone owners.

  1. The number in your sample who are constant checkers has a binomial distribution. What are and ?
  2. Use the binomial formula to find the probability that exactly two of the 10 are constant checkers in your sample.
  3. Use the binomial formula to find the probability that two or fewer are constant checkers in your sample.
  4. What is the mean number of owners in such samples who are constant checkers? What is the standard deviation?

Question 5.23

5.23 Random stock prices.

As noted in Example 5.4(a) (page 246), the S&P 500 index has a probability 0.56 of increasing in any week. Moreover, the change in the index in any given week is not influenced by whether it rose or fell in earlier weeks. Let be the number of weeks among the next five weeks in which the index rises.

  1. has a binomial distribution. What are and ?
  2. What are the possible values that can take?
  3. Use the binomial formula to find the probability of each value of . Draw a probability histogram for the distribution of .
  4. What are the mean and standard deviation of this distribution?

5.23

(a) . (b) . (c) . . . . . . (d) . .

Question 5.24

5.24 Paying for music downloads.

A survey of Canadian teens aged 12 to 17 years reported that roughly 75% of them used a fee-based website to download music.7 You decide to interview a random sample of 15 U.S. teenagers. For now, assume that they behave similarly to the Canadian teenagers.

  1. What is the distribution of the number who used a fee-based website to download music? Explain your answer.
  2. What is the probability that at least 12 of the 15 teenagers in your sample used a fee-based website to download music?

Question 5.25

5.25 Getting to work.

Many U.S. cities are investing and encouraging a shift of commuters toward the use of public transportation or other modes of non-auto commuting. Among the 10 largest U.S. cities, New York City and Philadelphia have the two highest percentages of non-auto commuters at 73% and 41%, respectively.8

  1. If you choose 10 NYC commuters at random, what is the probability that more than half (that is, six or more) are non-auto commuters?
  2. If you choose 100 NYC commuters at random, what is the probability that more than half (that is, 51 or more) are non-auto commuters?
  3. Repeat part (a) for Philadelphia.
  4. Repeat part (b) for Philadelphia.

5.25

(a) 0.8963. (b) > 0.9998 or almost 1. (c) 0.1834. (d) 0.0212.

Question 5.26

5.26 Paying for music downloads, continued.

Refer to Exercise 5.24. Suppose that only 60% of the U.S. teenagers used a fee-based website to download music.

  1. If you interview 15 U.S. teenagers at random, what is the mean of the count who used a fee-based website to download music? What is the mean of the proportion in your sample who used a fee-based website to download music?
  2. Repeat the calculations in part (a) for samples of size 150 and 1500. What happens to the mean count of successes as the sample size increases? What happens to the mean proportion of successes?

Question 5.27

5.27 More on paying for music downloads.

Consider the settings of Exercises 5.24 and 5.26.

  1. Using the 75% rate of the Canadian teenagers, what is the smallest number out of U.S. teenagers such that is no larger than 0.05? You might consider or fewer students as evidence that the rate in your sample is lower than the 75% rate of the Canadian teenagers.
  2. Now, using the 60% rate of the U.S. teenagers and your answer to part (a), what is ? This represents the chance of obtaining enough evidence with your sample to conclude that the U.S. rate is less than the Canadian rate.

5.27

(a) . (b) 0.2131.

Question 5.28

5.28 Internet video postings.

Suppose (as is roughly true) about 30% of all adult Internet users have posted videos online. A sample survey interviews an SRS of 1555 Internet users.

  1. What is the actual distribution of the number in the sample who have posted videos online?
  2. Use software to find the exact probability that 450 or fewer of the people in the sample have posted videos online.
  3. Use the Normal approximation to find the probability that 450 or fewer of the people in the sample have posted videos online. Compare this approximation with the exact probability found in part (b).

Question 5.29

5.29 Random digits.

Each entry in a table of random digits like Table B has probability 0.1 of being a 0, and digits are independent of each other.

  1. Suppose you want to determine the probability of getting at least one 0 in a group of five digits. Explain what is wrong with the logic of computing it as
  2. Find the probability that a group of five digits from the table will contain at least one 0.
  3. In Table B, there are 40 digits on any given line. What is the mean number of 0s in lines 40 digits long?

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5.29

(a) Answers will vary—the events are not disjoint, etc. (b) 0.4095. (c) .

Question 5.30

5.30 Online learning.

Recently, the U.S. Department of Education released a report on online learning stating that blended instruction, a combination of conventional face-to-face and online instruction, appears more effective in terms of student performance than conventional teaching.9 You decide to poll the incoming students at your institution to see if they prefer courses that blend face-to-face instruction with online components. In an SRS of 400 incoming students, you find that 311 prefer this type of course.

  1. What is the sample proportion who prefer this type of blended instruction?
  2. If the population proportion for all students nationwide is 85%, what is the standard deviation of ?
  3. Using the 68–95–99.7 rule, if you had drawn an SRS from the United States, you would expect to fall between what two percents about 95% of the time?
  4. Based on your result in part (a), do you think that the incoming students at your institution prefer this type of instruction more, less, or about the same as students nationally? Explain your answer.

Question 5.31

5.31 Shooting free throws.

Since the mid-1960s, the overall free throw percent at all college levels, for both men and women, has remained pretty consistent. For men, players have been successful on roughly 69% of these free throws, with the season percent never falling below 67% or above 70%.10 Assume that 300,000 free throws will be attempted in the upcoming season.

  1. What are the mean and standard deviation of if the population proportion is ?
  2. Using the 68–95–99.7 rule, we expect to fall between what two percents about 95% of the time?
  3. Given the width of the interval in part (b) and the range of season percents, do you think that it is reasonable to assume that the population proportion has been the same over the last 50 seasons? Explain your answer.

5.31

(a) . (b) Between 0.6883 and 0.6917. (c) No, the actual percentages are much more variable than the interval, suggesting that the percent has changed from season to season.

Question 5.32

5.32 Finding .

In Example 5.5, we found when has a distribution. Suppose we wish to find using the Normal approximation.

  1. What is the value for if the Normal approximation is used without continuity correction?
  2. What is the value for if the Normal approximation is used now with continuity correction?

Question 5.33

5.33 Multiple-choice tests.

Here is a simple probability model for multiple-choice tests. Suppose that each student has probability of correctly answering a question chosen at random from a universe of possible questions. (A strong student has a higher than a weak student.) The correctness of an answer to a question is independent of the correctness of answers to other questions. Emily is a good student for whom .

  1. Use the Normal approximation to find the probability that Emily scores 85% or lower on a 100-question test.
  2. If the test contains 250 questions, what is the probability that Emily will score 85% or lower?
  3. How many questions must the test contain in order to reduce the standard deviation of Emily’s proportion of correct answers to half its value for a 100-item test?
  4. Diane is a weaker student for whom . Does the answer you gave in part (c) for the standard deviation of Emily’s score apply to Diane’s standard deviation also?

5.33

(a) 0.1788. (b) 0.0721. (c) . (d.) Yes.

Question 5.34

5.34 Are we shipping on time?

Your mail-order company advertises that it ships 90% of its orders within three working days. You select an SRS of 100 of the 5000 orders received in the past week for an audit. The audit reveals that 86 of these orders were shipped on time.

  1. If the company really ships 90% of its orders on time, what is the probability that 86 or fewer in an SRS of 100 orders are shipped on time?
  2. A critic says, “Aha! You claim 90%, but in your sample the on-time percent is only 86%. So the 90% claim is wrong.” Explain in simple language why your probability calculation in part (a) shows that the result of the sample does not refute the 90% claim.

Question 5.35

5.35 Checking for survey errors.

One way of checking the effect of undercoverage, nonresponse, and other sources of error in a sample survey is to compare the sample with known facts about the population. About 13% of American adults are black. The number of blacks in a random sample of 1500 adults should, therefore, vary with the binomial distribution.

  1. What are the mean and standard deviation of ?
  2. Use the Normal approximation to find the probability that the sample will contain 170 or fewer black adults. Be sure to check that you can safely use the approximation.

5.35

(a) . (b) 0.0274.

Question 5.36

5.36 Show that these facts are true.

Use the definition of binomial coefficients to show that each of the following facts is true. Then restate each fact in words in terms of the number of ways that successes can be distributed among observations.

  1. for any whole number .
  2. for any whole number .
  3. for any whole number .
  4. for any whole numbers and with .

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Question 5.37

5.37 Does your vote matter?

Consider a common situation in which a vote takes place among a group of people and the winning result is associated with having one vote greater than the losing result. For example, if a management board of 11 members votes Yes or No on a particular issue, then minimally a 6-to-5 vote is needed to decide the issue either way. Your vote would have mattered if the other members voted 5-to-5.

  1. You are on this committee of 11 members. Assume that there is a 50% chance that each of the other members will vote Yes, and assume that the members are voting independently of each other. What is the probability that your vote will matter?
  2. There is a closely contested election between two candidates for your town mayor in a town of 523 eligible voters. Assume that all eligible voters will vote with a 50%chance that a voter will vote for a particular candidate. What is the probability that your vote will matter?

5.37

(a) 0.2461. (b) 0.0320 using continuity correction (0.0350 from software).

Question 5.38

5.38 Tossing a die.

You are tossing a balanced die that has probability 1/6 of coming up 1 on each toss. Tosses are independent. We are interested in how long we must wait to get the first 1.

  1. The probability of a 1 on the first toss is 1/6. What is the probability that the first toss is not a 1 and the second toss is a 1?
  2. What is the probability that the first two tosses are not 1s and the third toss is a 1? This is the probability that the first 1 occurs on the third toss.
  3. Now you see the pattern. What is the probability that the first 1 occurs on the fourth toss? On the fifth toss?

Question 5.39

5.39 The geometric distribution.

Generalize your work in Exercise 5.38. You have independent trials, each resulting in a success or a failure. The probability of a success is on each trial. The binomial distribution describes the count of successes in a fixed number of trials. Now, the number of trials is not fixed; instead, continue until you get a success. The random variable is the number of the trial on which the first success occurs. What are the possible values of ? What is the probability for any of these values?

(Comment: The distribution of the number of trials to the first success is called a geometric distribution.)

5.39

has possible values 1, 2, 3, … , etc. , because we must have failures before the success on the kth trial.