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Seeing Through Statistics 2nd Edition Jessica M. Utts - Solutions
11. A university is contemplating switching from the quarter system to the semester system. The administration conducts a survey of a random sample of 400 students and finds that 240 of them prefer to remain on the quarter system.a. Construct a 95% confidence interval for the true proportion of all
10. Confirm that the standard deviation is largest when the proportion used to calculate it is .50. Do this by using other values above and below .50 and comparing the answers to what you would get using .50. Try three values above and three values below .50.
9. Find the results of a poll reported in a weekly newsmagazine such as Newsweek or Time, in a newspaper such as the New York Times, or on the Internet in which a margin of error is also reported. Explain what question was asked and what margin of error was reported; then present a 95% confidence
8. Refer to Example 6 in this chapter. It is claimed that a 95% confidence interval for the percentage of placebo-patch users who quit smoking by the eighth week covers 13% to 27%. There were 120 placebo-patch users, and 24 quit smoking by the eighth week. Verify that the confidence interval given
7. Parade Magazine reported that “nearly 3200 readers dialed a 900 number to respond to a survey in our Jan. 8 cover story on America’s young people and violence” (19 February 1995, p. 20). Of those responding, “63.3% say they have been victims or personally know a victim of violent
6. Explain whether the width of a confidence interval would increase, decrease, or remain the same as a result of each of the following changes:a. The sample size is doubled, from 400 to 800b. The population size is doubled, from 25 million to 50 millionc. The level of confidence is lowered from
5. What level of confidence would accompany each of the following intervals?a. Sample proportion ± 1.0 (S.D.)b. Sample proportion ± 1.645 (S.D.)c. Sample proportion ± 1.96 (S.D.)d. Sample proportion ± 2.576 (S.D.)
4. A telephone poll reported in Time magazine (6 February 1995, p. 24) asked 359 adult Americans the question, “Do you think Congress should maintain or repeal last year’s ban on several types of assault weapons?” Seventy-five percent responded “maintain.”a. Compute the standard deviation
3. On September 10, 1998, the “Starr Report,” alleging impeachable offenses by President Bill Clinton, was released to Congress. That evening, the Gallup Organization conducted a poll of 645 adults nationwide to assess initial reaction(reported at www.gallup.com). One of the questions asked
2. Refer to Exercise 1. Of the 193 placebo takers, 43 reported headaches.a. Compute a 95% confidence interval for the true population proportion that would get headaches after taking a placebo.b. Notice that a higher proportion of placebo takers than Seldane-D takers reported headaches. Use that
1. An advertisement for Seldane-D, a drug prescribed for seasonal allergic rhinitis, reported results of a double-blind study in which 374 patients took Seldane-D and 193 took a placebo (Time, 27 March 1995, p. 18). Headaches were reported as a side effect by 65 of those taking Seldane-D.a. What is
2. The purpose of this mini-project is to help you verify the Rule for Sample Means. Suppose you are interested in measuring the average amount of blood contained in the bodies of adult women, in ounces. Suppose, in truth, the population consists of the values listed below. (Each value would be
1. The goal of this mini-project is to help you verify the Rule for Sample Proportions firsthand. You will use the population represented in Figure 18.1 to do so.It contains 400 individuals, of whom 160 (40%) are —that is, carry the gene for a disease—and the remaining 240 (60%) are —that is,
18. The administration of a large university wants to use a random sample of students to measure student opinion of a new food service on campus. Administrators plan to use a continuous scale from 1 to 100, where 1 is complete dissatisfaction and 100 is complete satisfaction. They know from past
17. Suppose the population of grade-point averages (GPAs) for students at the end of their first year at a large university has a mean of 3.1 and a standard deviation of .5. Draw a picture of the frequency curve for the mean GPA of a random sample of 100 students.
16. In Case Study 18.1, we learned that about 56% of American adults actually voted in the presidential election of 1992, whereas about 61% of a random sample claimed that they had voted. The size of the sample was not specified, but suppose it were based on 1600 American adults, a common size for
15. Explain whether you think the Rule for Sample Means applies to each of the following situations. If it does apply, specify the population of interest and the measurement of interest. If it does not apply, explain why not.a. A researcher is interested in what the average cholesterol level would
14. Explain whether each of the following situations meets the conditions for which the Rule for Sample Proportions applies. If not, explain which condition is violated.a. Unknown to the government, 10% of all cars in a certain city do not meet appropriate emissions standards. The government wants
13. According to the Sacramento Bee (2 April 1998, p. F5), Americans get an average of 6 hours and 57 minutes of sleep per night. A survey of a class of 190 sta tistics students at a large university found that they averaged 7.1 hours of sleep the previous night, with a standard deviation of 1.95
12. Use the Rule for Sample Means to explain why it is desirable to take as large a sample as possible when trying to estimate a population value.
11. Suppose 20% of all television viewers in the country watch a particular program.a. For a random sample of 2500 households measured by a rating agency, describe the frequency curve for the possible sample proportions who watch the program.b. The program will be canceled if the ratings show less
10. According to USA Today (20 April 1998, Snapshot), a poll of 8709 adults taken in 1976 found that 9% believed in reincarnation, whereas a poll of 1000 adults taken in 1997 found that 25% held that belief.a. Assuming a proper random sample was used, verify that the sample proportion for the poll
9. Suppose that 35% of the students at a university favor the semester system, 60% favor the quarter system, and 5% have no preference. Is a random sample of 100 students large enough to provide convincing evidence that the quarter system is favored? Explain.
8. Suppose the population of IQ scores in the town or city where you live is bellshaped, with a mean of 105 and a standard deviation of 15. Describe the frequency curve for possible sample means that would result from random samples of 100 IQ scores.
7. Give an example of a situation of interest to you for which the Rule for Sample Proportions would apply. Explain why the conditions allowing the rule to be applied are satisfied for your example.
6. Refer to Exercise 5. Redraw the picture under the assumption that you will collect 100 measurements instead of only 9. Discuss how the picture differs from the one in Exercise 5.
5. Suppose you are interested in estimating the average number of miles per gallon of gasoline your car can get. You calculate the miles per gallon for each of the next nine times you fill the tank. Suppose, in truth, the values for your car are bell-shaped, with a mean of 25 miles per gallon and a
4. A recent Gallup poll found that of 800 randomly selected drivers surveyed, 70% thought they were better than average drivers. In truth, in the population, only 50% of all drivers can be “better than average.”a. Draw a picture of the possible sample proportions that would result from samples
3. According to the Sacramento Bee (2 April 1998, p. F5), “A 1997-98 survey of 1027 Americans conducted by the National Sleep Foundation found that 23%of adults say they have fallen asleep at the wheel in the last year.”a. Conditions 2 and 3 needed to apply the Rule for Sample Proportions are
2. Refer to Exercise 1. Suppose the truth is that .12 or 12% of the students are lefthanded, and you take a random sample of 200 students. Use the Rule for Sample Proportions to draw a picture similar to Figure 18.3, showing the possible sample proportions for this situation.
1. Suppose you want to estimate the proportion of students at your college who are left-handed. You decide to collect a random sample of 200 students and ask them which hand is dominant. Go through the conditions for which the rule for sample proportions applies (pp. 319–320) and explain why the
3. Conduct a survey in which you ask 20 people the two scenarios presented in Thought Question 5 at the beginning of this chapter and discussed in Section 17.5. Record the percentage who choose alternative A over B and the percentage who choose alternative C over D.a. Report your results. Are they
2. Ask four friends to tell you their most amazing coincidence story. Use the material in this chapter to assess how surprising each of the stories is to you. Pick one of the stories and try to approximate the probability of that specific event happening to your friend.
1. Find out the sensitivity and specificity of a common medical test. Calculate the probability of a true positive for someone who tests positive with the test, assuming the rate in the population is 1 per 100; then calculate the probability assuming the rate in the population is 1 per 1000.
23. Suppose you are trying to decide whether to park illegally while you attend class. If you get a ticket, the fine is $25. If you assess the probability of getting a ticket to be 1/100, what is the expected value for the fine you will have to pay? Under those circumstances, explain whether you
22. Refer to Case Study 17.2, in which the relationship between betting odds and probability of occurrence is explained.a. Suppose you are offered a bet on an outcome for which the odds are 2 to 1 and there is no handling fee. For you to have a break-even expected value of zero, what would the
21. It is time for the end-of-summer sales. One store is offering bathing suits at 50% of their usual cost, and another store is offering to sell you two for the price of one. Assuming the suits originally all cost the same amount, which store is offering a better deal? Explain.
19. Explain why it would be much more surprising if someone were to flip a coin and get six heads in a row after telling you they were going to do so than it would be to simply watch them flip the coin six times and observe six heads in a row.20. We learned in this chapter that one idea researchers
18. If you wanted to pretend that you could do psychic readings, you could perform“cold readings” by inviting people you do not know to allow you to tell them about themselves. You would then make a series of statements like“I see that there is some distance between you and your mother that
17. Suppose a friend reports that she has just had a string of “bad luck” with her car. She had three major problems in as many months and now has replaced many of the worn parts with new ones. She concludes that it is her turn to be lucky and that she shouldn’t have any more problems for a
16. Suppose the sensitivity of a test is .90. Give either the false positive or the false negative rate for the test, and explain which you are providing. Could you provide the other one without additional information? Explain.
15. A statistics professor once made a big blunder by announcing to his class of about 50 students that he was fairly certain that someone in the room would share his birthday. We have already learned that there is a 97% chance that there will be 2 people in a room of 50 with a common birthday.
14. Explain why the story about George D. Bryson, reported in Example 1 in this chapter, is not all that surprising.
13. Using the data in Table 17.1, give numerical values and explain the meaning of the sensitivity and the specificity of the test.
12. Using the data in Table 17.1 about a hypothetical population of 100,000 women tested for breast cancer, find the probability of each of the following events:a. A woman whose test shows a malignant lump actually has a benign lump.b. A woman who actually has a benign lump has a test that shows a
11. You are at a casino with a friend, playing a game in which dice are involved.Your friend has just lost six times in a row. She is convinced that she will win on the next bet because she claims that, by the law of averages, it’s her turn to win. She explains to you that the probability of
10. Suppose a rare disease occurs in about 1 out of 1000 people who are like you.A test for the disease has sensitivity of 95% and specificity of 90%. Using the technique described in this chapter, compute the probability that you actually have the disease, given that your test results are positive.
9. Many people claim that they can often predict who is on the other end of the phone when it rings. Do you think that phenomenon has a normal explanation?Explain.
8. In financial situations, are businesses or individuals more likely to make use of expected value for making decisions? Explain.
7. The U.C. Berkeley Wellness Encyclopedia (1991) contains the following statement in its discussion of HIV testing: “In a high-risk population, virtually all people who test positive will truly be infected, but among people at low risk the false positives will outnumber the true positives. Thus,
6. Find a dollar bill or other item with a serial number. Write down the number.I predict that there is something unusual about it or some pattern to it. Explain what is unusual about it and how I was able to make that prediction.
5. Why is it not surprising that the night before a major airplane crash several people will have dreams about an airplane disaster? If you were one of those people, would you think that something amazing had occurred?
4. Suppose two sisters are reunited after not seeing each other since they were 3 years old. They are amazed to find out that they are both married to men named James and that they each have a daughter named Jennifer. Explain why this is not so amazing.
3. Explain why it is not at all unlikely that in a class of 50 students two of them will have the same last name.
2. Give an example of a sequence of events to which the gambler’s fallacy would not apply because the events are not independent.
1. Although it’s not quite true, suppose the probability of having a male child(M) is equal to the probability of having a female child (F). A couple has four children.a. Are they more likely to have FFFF or to have MFFM? Explain your answer.b. Explain which sequence in part a of this exercise a
4. Estimate the probability of some event in your life using a personal probability, such as the probability that a person who passes you on the street will be wearing a hat. Use an event for which you can keep a record of the relative frequency of occurrence over the next week. How well calibrated
3. Find a journal article that describes an experiment designed to test the kinds of biases described in this chapter. Summarize the article and discuss what conclusions can be made from the research. You can find such articles by searching appropriate bibliographic databases and trying key words
2. Find and explain an example of a marketing strategy that uses one of the techniques in this chapter to try to increase the chances that someone will purchase something. Do not use an exact example from the chapter, such as “buy one, get one free.”
1. Design and conduct an experiment to try to elicit misjudgments based on one of the phenomena described in this chapter. Explain exactly what you did and your results.
18. Guess at the probability that if you ask five people when their birthdays are, you will find someone born in the same month as you. For simplicity, assume that the probability that a randomly selected person will have the same birth month you have is 1/12. Now use the material from Chapter 15
17. Suppose you have a friend who is willing to ask her friends a few questions and then, based on their answers, is willing to assess the probability that those friends will get an A in each of their classes. She always assesses the probability to be either .10 or .90. She has made hundreds of
16. Explain which of the concepts in this chapter might contribute to the decision to buy a lottery ticket.
15. Give one example of how each of the following concepts has had or might have an unwanted effect on a decision or action in your daily life:a. Conservatismb. Optimismc. Forgotten base ratesd. Availability
14. Suppose you go to your doctor for a routine examination, without any complaints of problems. A blood test reveals that you have tested positive for a certain disease. Based on the ideas in this chapter, what should you ask your doctor in order to assess how worried you should be?
13. Explain how the concepts in this chapter account for each of the following scenarios:a. Most people rate death by shark attacks to be much more likely than death by falling airplane parts, yet the chances of dying from the latter are actually 30 times greater (Plous, 1993, p. 121).b. You are a
12. Barnett (1990) examined front page stories in the New York Times for 1 year, beginning with October 1, 1988, and found 4 stories related to automobile deaths but 51 related to deaths from flying on a commercial jet. These correspond to 0.08 story per thousand U.S. deaths by automobile and 138.2
11. There are approximately 5 billion people in the world. Plous (1993, p. 5) asked readers to estimate how wide a cube-shaped tank would have to be to hold all of the human blood in the world. The correct answer is about 870 feet, but most people give much higher answers. Explain which of the
10. Explain how an insurance salesperson might try to use each of the following concepts to sell you insurance:a. Anchoringb. Pseudocertaintyc. Availability
9. Determine which statement (A or B) has a higher probability of being true and explain your answer. Using the material in this chapter, also explain which statement you think a statistically naive person would think had a higher probability.A. A car traveling 120 miles per hour on a two-lane
8. Research by Slovic and colleagues (1982) found that people judged that accidents and diseases cause about the same number of deaths in the United States, whereas in truth diseases cause about 16 times as many deaths as accidents.Using the material from this chapter, explain why the researchers
7. In this chapter, we learned that one way to lower personal-probability assessments that are too high is to list reasons why you might be wrong. Explain how the availability heuristic might account for this phenomenon.
6. A telephone solicitor recently contacted the author to ask for money for a charity in which typical contributions are in the range of $25 to $50. The solicitor said, “We are asking for as much as you can give, up to $300.00” Do you think the amount people give would be different if the
5. Explain why you should be cautious when someone tries to convince you of something by presenting a detailed scenario. Give an example.
4. Suppose a defense attorney is trying to convince the jury that his client’s wallet, found at the scene of the crime, was actually planted there by his client’s gardener.Here are two possible ways he might present this to the jury:Statement A: The gardener dropped the wallet when no one was
3. There are many more termites in the world than there are mosquitoes, but most of the termites live in tropical forests. Using the ideas in this chapter, explain why most people would think there were more mosquitoes in the world than termites.
2. Suppose a television advertisement were to show viewers a product and then say, “You might expect to pay $25, $30, or even more than this. But we are offering it for only $16.99.” Explain which of the ideas in this chapter is being used to try to exploit viewers.
1. Explain how the pseudocertainty effect differs from the certainty effect.
4. Find two lottery or casino games that have fixed payoffs and for which the probabilities of each payoff are available. (Some lottery tickets list them on the back of the ticket. Some books about gambling give the payoffs and probabilities for various casino games.)a. Compute the expected value
3. Pick an event that will result in the same outcome for everyone, such as whether it will rain next Saturday. Ask ten people to assess the probability of that event, and note the variability in their responses. (Don’t let them hear each other’s answers, and make sure you don’t pick
2. Flip a coin 100 times. Stop each time you have done 10 flips (that is, stop after 10 flips, 20 flips, 30 flips, and so on) and compute the proportion of heads using all of the flips up to that point. Plot that proportion versus the number of flips.Comment on how the plot relates to the
1. Refer to Exercise 12. Present the question to ten people, and note the proportion who answer with alternative B. Explain to the participants why it cannot be the right answer, and report on their reactions.
27. Find out your yearly car insurance cost. If you don’t have a car, find out the yearly cost for a friend or relative. Now assume you will either have an accident or not, and if you do, it will cost the insurance company $5000 more than the premium you pay. Calculate what yearly accident
26. We have seen many examples for which the term expected value seems to be a misnomer. Construct an example of a situation where the term expected value would not seem to be a misnomer for what it represents.
25. In 1991, 72% of children in the United States were living with both parents, 22% were living with mother only, and 3% were living with father only and 3%were not living with either parent (World Almanac and Book of Facts, 1993, p. 945). What is the expected value for the number of parents a
24. Suppose the probability that you get an A in any class you take is .3, and the probability that you get a B is .7. To construct a grade-point average, an A is worth 4.0 and a B is worth 3.0. What is the expected value for your grade-point average? Would you expect to have this grade-point
23. In the “3 Spot” version of the California Keno lottery game, the player picks three numbers from 1 to 40. Ten possible winning numbers are then randomly selected. It costs $1 to play. Table 15.7 shows the possible outcomes. Compute the expected value for this game. Interpret what it means.
22. Remember that the probability that a birth results in a boy is about .51. You offer a bet to an unsuspecting friend. Each day you will call the local hospital and find out how many boys and how many girls were born the previous day.For each girl, you will give your friend $1 and for each boy
21. Suppose you have to cross a train track on your commute. The probability that you will have to wait for a train is 1/5, or .20. If you don’t have to wait, the commute takes 15 minutes, but if you have to wait, it takes 20 minutes.a. What is the expected value of the time it takes you to
20. According to Krantz (1992, p. 161), the probability of being injured by lightning in any given year is 1/685,000. Assume that the probability remains the same from year to year and that avoiding a strike in one year doesn’t change your probability in the next year.a. What is the probability
19. Lyme disease is a disease carried by ticks, which can be transmitted to humans by tick bites. Suppose the probability of contracting the disease is 1/100 for each tick bite.a. What is the probability that you will not get the disease when bitten once?b. What is the probability that you will not
18. Suppose you routinely check coin-return slots in vending machines to see if they have any money in them. You have found that about 10% of the time you find money.a. What is the probability that you do not find money the next time you check?b. What is the probability that the next time you will
17. Read the definition of “independent events” given in Rule 3. Explain whether each of the following pairs of events is likely to be independent:a. A married couple goes to the voting booth. Event A is that the husband votes for the Republican candidate; Event B is that the wife votes for the
16. Use your own particular expertise to assign a personal probability to something, such as the probability that a certain sports team will win next week. Now assign a personal probability to another related event. Explain how you determined each probability, and explain how your assignments are
15. People are surprised to find that it is not all that uncommon for two people in a group of 20 to 30 people to have the same birthday. We will learn how to find that probability in a later chapter. For now, consider the probability of finding two people who have birthdays in the same month. Make
14. In Section 15.2, you learned two ways in which relative-frequency probabilities can be determined. Explain which method you think was used to determine each of the following probabilities:a. The probability that a particular flight from New York to San Francisco will be on time is .78.b. On any
13. Example 3 in this chapter states that “the probability that a piece of checked luggage will be temporarily lost on a flight with a U.S. airline is 1/176.” Interpret that statement, using the appropriate interpretation of probability.
12. A study by Kahneman and Tversky (1982, p. 496) asked people the following question: “Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in
11. The probability that a randomly selected American adult belongs to the American Automobile Association (AAA) is .10 (10%), and the probability that that person belongs to the American Association of Retired Persons (AARP) is .11(11%) (Krantz, 1992, p. 175). What assumption would we have to make
10. Suppose the probability that you get an interesting piece of mail on any given weekday is 1/10. Is the probability that you get at least one interesting piece of mail during the week (Monday to Friday) equal to 5/10? Why or why not?
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