SECTION 5.3 EXERCISES

For Exercises 5.43, 5.44, and 5.45, see page 312; for Exercises 5.46 and 5.47, see page 313; for Exercises 5.48 and 5.49, see page 317; for Exercises 5.50 and 5.51, see page 319; for Exercise 5.52, see page 320; for Exercise 5.53, see page 325; for Exercise 5.54, see page 328; and for Exercises 5.55 and 5.56, see page 330.

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 5.57

5.57 What is wrong? Explain what is wrong in each of the following scenarios.

  1. (a) 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. (b) If you toss a fair coin four times and observe the pattern HTHT, then the next toss is more likely to be a head than a tail.

  3. (c) The quantity is one of the parameters for a binomial distribution.

  4. (d) The binomial distribution can be used to model the daily number of pedestrian/cyclist near-crash events on campus.

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Question 5.58

5.58 What is wrong? Explain what is wrong in each of the following scenarios.

  1. (a) In the binomial setting, X is a proportion.

  2. (b) The variance for a binomial count is .

  3. (c) The Normal approximation to the binomial distribution is always accurate when n is greater than 1000.

  4. (d) We can use the binomial distribution to approximate the sampling distribution of when we draw an SRS of size n = 50 students from a population of 500 students.

Question 5.59

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

  1. (a) A poll of 200 college students asks whether or not you usually feel irritable in the morning. X is the number who reply that they do usually feel irritable in the morning.

  2. (b) You toss a fair coin until a head appears. X is the count of the number of tosses that you make.

  3. (c) 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. X is the number of times that a person is reached.

  4. (d) You deal 10 cards from a shuffled deck of standard playing cards and count the number X of black cards.

Question 5.60

5.60 Should you use the binomial distribution? In each of the following situations, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case.

  1. (a) In a random sample of students in a fitness study, X is the mean daily exercise time of the sample.

  2. (b) A manufacturer of running shoes picks a random sample of 20 shoes from the production of shoes each day for a detailed inspection. X is the number of pairs of shoes with a defect.

  3. (c) A nutrition study chooses an SRS of college students. They are asked whether or not they usually eat at least five servings of fruits or vegetables per day. X is the number who say that they do.

  4. (d) X is the number of days during the school year when you skip a class.

Question 5.61

5.61 Stealing from a store. A survey of more than 20,000 U.S. high school students revealed that 20% of the students say that they stole something from a store in the past year.17 This is down 7% from the last survey, which was performed two years earlier. You decide to take a random sample of 10 high school students from your city and ask them this question.

  1. (a) If the high school students in your city match this 20% rate, what is the distribution of the number of students who say that they stole something from a store in the past year? What is the distribution of the number of students who do not say that they stole something from a store in the past year?

  2. (b) What is the probability that four or more of the 10 students in your sample say that they stole something from a store in the past year?

Question 5.62

5.62 Illegal downloading. New regulations in Canada require all Internet service providers (ISPs) to send a notice to subscribers who are downloading files illegally asking them to stop. This “notice and notice” system was already in place with Rogers Cable. That company says that prior to these new regulations, 67% of its subscribers who received a notice did not reoffend.18 Consider a random sample of 50 of these Rogers subscribers who received a first notice.

  1. (a) What is the distribution of the number X of subscribers who reoffend? Explain your answer.

  2. (b) What is the probability that at least 18 of the 50 subscribers in your sample reoffend?

Question 5.63

5.63 Stealing from a store, continued. Refer to Exercise 5.61.

  1. (a) What is the expected number of students in your sample who say that they stole something from a store in the past year? What is the expected number of students who do not say that they stole? You should see that these two means add to 10, the total number of students.

  2. (b) What is the standard deviation σ of the number of students in your sample who say that they stole something?

  3. (c) Suppose that you live in a city where only 10% of the high school students say that they stole something from a store in the past year. What is σ in this case? What is σ if p = 0.01? What happens to the standard deviation of a binomial distribution as the probability of a success gets close to 0?

Question 5.64

5.64 Illegal downloading, continued. Refer to Exercise 5.62. Given the new regulations, suppose that 75% of the Canadian ISP subscribers will not reoffend after receiving a notice.

  1. (a) If you choose at random 15 subscribers who received a notice, what is the mean of the count X who will not reoffend? What is the mean of the proportion in your sample who will not reoffend?

  2. (b) 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?

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Question 5.65

image 5.65 More on illegal downloading. Consider the settings of Exercises 5.62 and 5.64.

  1. (a) Using the 67% rate of Rogers subscribers prior to the new regulations, what is the smallest number m out of n = 15 Canadian ISP subscribers who receive a notice such that is no larger than 0.05? You might consider m or more subscribers as evidence that the rate in your sample is larger than 67%.

  2. (b) Now using the 75% rate of Canadian ISP subscribers after the new regulations and your answer to part (a), what is ? This represents the chance of obtaining enough evidence given that the rate is 75%.

  3. (c) If you were to increase the sample size from n = 15 to n = 100 and repeat parts (a) and (b), would you expect the probability in part (b) to increase or decrease? Explain your answer.

Question 5.66

5.66 Attitudes toward drinking and studies of behavior. Some of the methods in this section are approximations rather than exact probability results. We have given rules of thumb for safe use of these approximations.

  1. (a) You are interested in attitudes toward drinking among the 75 members of a fraternity. You choose 30 members at random to interview. One question is “Have you had five or more drinks at one time during the last week?” Suppose that, in fact, 30% of the 75 members would say Yes. Explain why you cannot safely use the B(30, 0.3) distribution for the count X in your sample who say Yes.

  2. (b) The National AIDS Behavioral Surveys found that 0.2% (that’s 0.002 as a decimal fraction) of adult heterosexuals had both received a blood transfusion and had a sexual partner from a group at high risk of AIDS. Suppose that this national proportion holds for your region. Explain why you cannot safely use the Normal approximation for the sample proportion who fall in this group when you interview an SRS of 1000 adults.

Question 5.67

5.67 Random digits. Each entry in a table of random digits like Table B has probability 0.1 of being any given digit, and digits are independent of each other.

  1. (a) What is the probability that a group of six digits from the table will contain at least one digit greater than 5?

  2. (b) What is the mean number of digits greater than 5 in lines 40 digits long?

Question 5.68

image 5.68 Use the Probability applet. The Probability applet simulates tosses of a coin. You can choose the number of tosses n and the probability p of a head. You can therefore use the applet to simulate binomial random variables.

The count of misclassified sales records in Example 5.21 has the binomial distribution with n = 15 and p = 0.08. Set these values for the number of tosses and probability of heads in the applet. Table C shows that the probability of getting a sample with exactly 0 misclassified records is 0.2863. This is the long-run proportion of samples with no bad records. Click “Toss” and “Reset” repeatedly to simulate 25 samples of 15 tosses. Record the number of bad records (the count of heads) in each of the 25 samples.

  1. (a) What proportion of the 25 samples had exactly 0 bad records? Do you think this sample proportion is close to the probability?

  2. (b) Remember that this probability of 0.2863 tells us only what happens in the long run. Here we’re considering only 25 samples. If X is the number of samples out of 25 with exactly 0 misclassified records, what is the distribution of X?

  3. (c) Explain how to use the distribution in part (b) to describe the sampling distribution of in part (a).

Question 5.69

5.69 Cyberbullying. An online survey, in partnership with Habbo, was conducted to study cyberbullying among 13- to 25-year-olds in the United Kingdom. It was reported that 62% of the young people had received nasty private messages on a smartphone social network app.19 You randomly sample four young people from the United Kingdom and ask them if they’ve received nasty messages. Let X be the number who say Yes.

  1. (a) What are n and p in the binomial distribution of X?

  2. (b) Find the probability of each possible value of X, and draw a probability histogram for this distribution.

  3. (c) Find the mean number of positive responders and mark the location of this value on your histogram.

Question 5.70

5.70 The ideal number of children. “What do you think is the ideal number of children for a family to have?” A Gallup Poll asked this question of 1020 randomly chosen adults. Slightly less than half (48%) thought that a total of two children was ideal.20 Suppose that p = 0.48 is exactly true for the population of all adults. Gallup announced a margin of error of ±4 percentage points for this poll. What is the probability that the sample proportion for an SRS of size n = 1020 falls between 0.44 and 0.52? You see that it is likely, but not certain, that polls like this give results that are correct within their margin of error. We say more about margins of error in Chapter 6.

Question 5.71

5.71 Cyberbullying, continued. Refer to Exercise 5.69. Assume instead that that you sample n = 500 young people from the United Kingdom.

  1. (a) What is the probability that the sample proportion of those who received nasty messages is between 0.59 and 0.65 if the population proportion is p = 0.62?

  2. (b) What is the probability that the sample proportion is between 0.87 and 0.93 if the population proportion is p = 0.90?

  3. (c) Using the results from parts (a) and (b), how does the probability that falls within ±0.03 of the true p change as p gets closer to 1?

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Question 5.72

5.72 How do the results depend on the sample size? Return to the Gallup Poll setting of Exercise 5.70. We are supposing that the proportion of all adults who think that having two children is ideal is p = 0.48. What is the probability that a sample proportion falls between 0.44 and 0.52 (that is, within ±4 percentage points of the true p) if the sample is an SRS of size n = 300? Of size n = 5000? Combine these results with your work in Exercise 5.70 to make a general statement about the effect of larger samples in a sample survey.

Question 5.73

5.73 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%.21 Assume that 300,000 free throws will be attempted in the upcoming season.

  1. (a) What are the mean and standard deviation of if the population proportion is p = 0.69?

  2. (b) Using the 68–95–99.7 rule, we expect to fall between what two percents about 95% of the time?

  3. (c) 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.

Question 5.74

5.74 Online learning. 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.22 You decide to poll 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 373 prefer this type of course.

  1. (a) What is the sample proportion of incoming students at your school who prefer this type of blended instruction?

  2. (b) Assume the population proportion for all students nationwide is 85%. Assuming this is true for your insitution too, what is the standard deviation of ?

  3. (c) Using the 68–95–99.7 rule, you would expect to fall between what two percents about 95% of the time?

  4. (d) 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 5.75

image 5.75 Binge drinking. The Centers for Disease Control and Prevention finds that 28% of people aged 18 to 24 years binge drank. Those who binge drank averaged 9.3 drinks per episode and 4.2 episodes per month. The study took a sample of over 18,000 people aged 18 to 24 years, so the population proportion of people who binge drank is very close to p = 0.28.23 The administration of your college surveys an SRS of 200 students and finds that 56 binge drink.

  1. (a) What is the sample proportion of students at your college who binge drink?

  2. (b) If, in fact, the proportion of all students on your campus who binge drink is the same as the national 28%, what is the probability that the proportion in an SRS of 200 students is as large or larger than the result of the administration’s sample?

  3. (c) A writer for the student paper says that the percent of students who binge brink is higher on your campus than nationally. Write a short letter to the editor explaining why the survey does not support this conclusion.

Question 5.76

image 5.76 How large a sample is needed? The changing probabilities you found in Exercises 5.70 and 5.72 are due to the fact that the standard deviation of the sample proportion gets smaller as the sample size n increases. If the population proportion is p = 0.48, how large a sample is needed to reduce the standard deviation of to ? (The 68–95–99.7 rule then says that about 95% of all samples will have within 0.01 of the true p.)

Question 5.77

5.77 A test for ESP. In a test for ESP (extrasensory perception), the experimenter looks at cards that are hidden from the subject. Each card contains either a star, a circle, a wave, or a square. As the experimenter looks at each of 20 cards in turn, the subject names the shape on the card.

  1. (a) If a subject simply guesses the shape on each card, what is the probability of a successful guess on a single card? Because the cards are independent, the count of successes in 20 cards has a binomial distribution.

  2. (b) What is the probability that a subject correctly guesses at least 10 of the 20 shapes?

  3. (c) In many repetitions of this experiment with a subject who is guessing, how many cards will the subject guess correctly on the average? What is the standard deviation of the number of correct guesses?

  4. (d) A standard ESP deck actually contains 25 cards. There are five different shapes, each of which appears on five cards. The subject knows that the deck has this makeup. Is a binomial model still appropriate for the count of correct guesses in one pass through this deck? If so, what are n and p? If not, why not?

Question 5.78

5.78 Admitting students to college. A selective college would like to have an entering class of 1000 students. Because not all students who are offered admission accept, the college admits more than 1000 students. Past experience shows that about 83% of the students admitted will accept. The college decides to admit 1200 students. Assuming that students make their decisions independently, the number who accept has the B(1200, 0.83) distribution. If this number is less than 1000, the college will admit students from its waiting list.

  1. (a) What are the mean and the standard deviation of the number X of students who accept?

  2. (b) Use the Normal approximation to find the probability that at least 800 students accept.

  3. (c) The college does not want more than 1000 students. What is the probability that more than 1000 will accept?

  4. (d) If the college decides to decrease the number of admission offers to 1150, what is the probability that more than 1000 will accept?

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Question 5.79

image 5.79 Is the ESP result better than guessing? When the ESP study of Exercise 5.77 discovers a subject whose performance appears to be better than guessing, the study continues at greater length. The experimenter looks at many cards bearing one of five shapes (star, square, circle, wave, and cross) in an order determined by random numbers. The subject cannot see the experimenter as the experimenter looks at each card in turn, in order to avoid any possible nonverbal clues. The answers of a subject who does not have ESP should be independent observations, each with probability 1/5 of success. We record 900 attempts.

  1. (a) What are the mean and the standard deviation of the count of successes?

  2. (b) What are the mean and the standard deviation of the proportion of successes among the 900 attempts?

  3. (c) What is the probability that a subject without ESP will be successful in at least 24% of 900 attempts?

  4. (d) The researcher considers evidence of ESP to be a proportion of successes so large that there is only probability 0.01 that a subject could do this well or better by guessing. What proportion of successes must a subject have to meet this standard? (Example 1.45, on pages 65–66, shows how to do an inverse calculation for the Normal distribution that is similar to the type required here.)

Question 5.80

image 5.80 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 k successes can be distributed among n observations.

  1. (a) for any whole number n ≥ 1.

  2. (b) for any whole number n ≥ 1.

  3. (c) for any n and k with k ≤ n.

Question 5.81

5.81 English Premier League Goals. The total number of goals scored per soccer match in the English Premier League (EPL) often follows the Poisson distribution. In one recent season, the average number of goals scored per match (over 380 games played) was 2.768. Compute the following probabilities.

  1. (a) What is the probability that three or more goals are scored in a game?

  2. (b) What is the probability that a game will end in a 0−0 tie?

  3. (c) Explain why you cannot compute the probability that a game will end in a 1−1 tie but can provide an upper bound on this probability.

Question 5.82

5.82 Number of colony-forming units. In microbiology, colony-forming units (CFUs) are used to measure the number of microorganisms present in a sample. To determine the number of CFUs, the sample is prepared, spread uniformly on an agar plate, and then incubated at some suitable temperature. Suppose that the number of CFUs that appear after incubation follows a Poisson distribution with μ = 15.

  1. (a) If the area of the agar plate is 75 square centimeters (cm2), what is the probability of observing fewer than 4 CFUs in a 25 cm2 area of the plate?

  2. (b) If you were to count the total number of CFUs in five plates, what is the probability you would observe more than 90 CFUs? Use the Poisson distribution to obtain this probability.

  3. (c) Repeat the probability calculation in part (b), but now use the Normal approximation. How close is your answer to your answer in part (b)?

Question 5.83

5.83 Metal fatigue. Metal fatigue refers to the gradual weakening and eventual failure of metal that undergoes cyclic loads. The wings of an aircraft, for example, are subject to cyclic loads when in the air, and cracks can form. It is thought that these cracks start at large particles found in the metal. Suppose that the number of particles large enough to initiate a crack follows a Poisson distribution with mean μ = 0.5 per square centimeter (cm2).

  1. (a) What is the mean of the Poisson distribution if we consider a 100 cm2 area?

  2. (b) Using the Normal approximation, what is the probability that this section has more than 60 of these large particles?