13.8: Exercises

Generalizing from a Sample

1. Evaluate the following reasoning. In answering, specify the conclusion, say whether the conclusion follows, and explain why.

This survey of major corporate executives indicates that 60 percent of those sampled believe that some American businesses often engage in price fixing. Therefore, if you were to pick in the same way five of the surveyed major corporate executives, you could reasonably expect that three of them would believe that some American businesses often engage in price fixing.

2. If some members of the target population did not have an equal chance to be selected into the sample, then the sample must be non-representative of the population.

a. true
b. false

3. Rank the following three arguments in order of their strength, strongest first:

(1) Our local newspaper's film reviewer liked the film; so it's a good bet that everyone else will, too.

(2) Everyone else liked the film, so it's a good bet that our local newspaper's film reviewer will, too.

(3) Everyone liked the film, so it's a good bet that our local newspaper's film reviewer did, too.

a. 123
b. 32 1
c. 2 13
d. 3 12
e. 2 3 1

■ 4. Is a large random sample that is stratified on all the relevant characteristics in the population always representative of the population? Why?

■ 5. Why aren't all representative samples random? You may assume that any sample is less than the whole population being sampled.

6. For the following statistical report, (a) identify the sample, (b) identify the population, and (c) discuss bias and the representativeness of the sample, mentioning sample size, stratification, and so on.

The State Hornet, the State University student newspaper, conducted a survey by asking students a series of questions. The survey was conducted at noon in front of the University Union and involved 450 students out of a student body of 26,000. The interviewers were careful to get a sample with a racial, sexual, and age breakdown similar to that of the university as a whole. In the survey, 70 percent of the students interviewed said they opposed mixing sexes on the same floor of the dormitories. The newspaper presented the results of its survey in an article headlined "Majority of Student Body Opposes Mixing Sexes on Same Floor of Dorms."

Suppose that in response to this passage, Smith remarks, "There are several problems with this survey. For instance, the “70” is pseudoprecise, and just how do you tell from a distance what someone's age is?" (d) Discuss this response.

■ 7. After a gun control law was passed in the state of Washington, the murder rate in Washington dropped from 4.3 percent per thousand to 3.4 percent per thousand. If this drop is statistically significant, then

a. the drop is not due to random variation in the population of murders.
b. the difference between 4.3 percent and 3.4 percent is due to chance.
c. the difference between 4.3 percent and 3.4 percent is too small to be important statistically.
d. the difference between 4.3 percent and 3.4 percent either is due to chance or is too small to be important statistically, but not both, and you cannot tell which from the information given. ^[343]

8. For the following statistical report, (a) identify the sample and its size, (b) identify the population, and (c) discuss how the sampling could have been improved by stratifying on time (but don't mention other ways to improve it).

In an effort to determine U.S. truck driver attitudes about the new requirements, the Council for Population Studies asked U.S. truck drivers whether they thought the same smog requirements that automobile drivers must meet should apply to truck drivers as well. Of the several thousand who responded to the survey, most indicated that they believed trucks should be exempt from the automobile smog regulations. The voluntary survey was taken at random times of the twenty-four hour day at randomly selected truck stops throughout the United States.

■ 9. Hannah is getting sick and tired of following Ricardo’s advice. Every time he has recommended a film for her to see, she has been disappointed in the film. Once she even walked out before the film had ended. She decides that this time she is not going to go see “The Rise of Dracula” which Ricardo’s has just recommended.

In Hannah’s reasoning, what percentage of the items in the past have had the property in question that she is considering projecting into the future?

a. 25%
b. 20%
c. 100%
d. 0%
e. can’t tell

10. Examine the following dialogue, paying attention to the quality of the reasoning. Then answer the questions that follow.

Lesley: I think little Sam will soon be having dreams of giant needles.

Rico: What? Have you been reading the tabloids again?

Lesley: No, but his school says all elementary and pre-school kids should be vaccinated for measles.

Rico: Who is sick? Do you know anybody with measles?

Figure \(\PageIndex{1}\)

Lesley: No, but they might get sick. Evidently somebody high up thinks there's a chance. The school recommended the shots in a leaflet Sam brought home this afternoon.

Rico: What will this latest suggestion of theirs cost us?

Lesley: I don't know. That's a problem. We have to find a clinic, make the appointment, and all that. The leaflet recommended ten clinics in the county.

Rico: It may not be worth all the trouble. I don't know anybody in the last ten years who has ever gotten measles. Besides, can't you still get the disease even if you take the vaccine for protection? Do they say it's perfect? Can't the vaccine itself give you the disease? Shouldn't we consider all this?

Lesley: Well, the leaflet said something about a scientific report in some medical journal. Here it is. It says, "The new vaccine uses a live form of the measles virus that is expected to be the cause of most cases of measles in the U.S. over the next few years. However, the virus is weakened so it is very unlikely to cause a real case of the measles. In order to show that measles can be prevented in children, medical professors Carolyn Owen, Mary Pittman Lindemann, and Linda Bomstad gave injections last year to 1,244 children who had been admitted to Chicago hospitals for non-life-threatening problems. 622 received the vaccine; the rest of the children received an injection that looked identical but was actually a harmless placebo, just salt water. The nurses administering the injections were not told which children were getting which kind of injection. Seven months later, only one of those who received the vaccine had gotten measles, but 45 of the group whose injections contained no vaccine had been diagnosed as having the disease." How does that sound to you?

Rico: OK, the shot will help keep Sam safe, but I'd still like to know what it costs.

Lesley: Well, you go call a clinic and ask them.

Rico: You're better at dealing with bureaucracies. You call.

a. What is the main issue in this conversation?
b. Rico implicitly makes an inductive generalization based on some statistics. What is the target population?
c. Describe the sample, but do not evaluate the sampling procedure itself.
d. Any problems with the sampling procedure? Comment on stratification of the sample.
e. What did this study say or show about how to cure measles in a child once the child has gotten the disease?
f. Is Rico being illogical anywhere in the conversation? If so, where and why?

■ 11. Could this be true? "I was trying to learn about the population, but my totally unbiased sampling method produced what I later learned was a non-representative sample."

12. About 95 percent of the sample of 94 resistors taken from the approximately 1,500 resistors in Tuesday's output at the factory are of good enough quality to be sold. From this information about the 94 resistors, which of the following statements about the 1,500 is most likely to be true?

a. All of Tuesday's total output of resistors work OK.
b. Exactly 95 percent of Tuesday's total output of resistors work OK.
c. Over 90 percent of Tuesday's total output of resistors work OK.
d. 94 to 96 percent of Tuesday's total output of resistors work OK.

13. If some members of the target population did not have an equal chance to be selected into the sample, then the sample must be nonrandom.

a. true
b. false

14. What is an important thing to do if you want to be able to trust the answer you get from taking a statistical survey, especially if you are paying people to participate in your survey?

a. Be sure to survey as many of your friends as possible.
b. Be sure to avoid surveying any of your friends.
c. Stratify your sample.
d. Avoid a representative sample in cases where the population is large.
e. Avoid a representative sample in cases where the population is small.

15. Suppose you were interested in whether the customers who buy heavy metal music from your store would like you to carry wall posters of the musicians. You can't ask all the customers, but you can ask a few by taking a poll. You happen to know that about 60 percent of your customers who buy heavy metal music are male. You know that about 50 percent of the people in the world are female. If you were going to stratify your sample on sex, how should you do the stratification?

■ 16. After examining the birth records of as many black persons as she could fine who were born between 1850 and 1950 in a Gulf Coast state, Dr. Gale Carswell discovered that 55 percent of those children were female. She then reported the remarkable result that there were significantly more female than male black children born in the Gulf Coast states during that period. In her study, the population was

a. as many black persons as she could find who were born between 1850 and 1950 in a Gulf Coast state.
b. black persons who were born between 1850 and 1950 in a Gulf Coast state.
c. people living in states along the Gulf Coast between 1850 and 1950.
d. 55 percent of the black persons born in a Gulf Coast state between 1850 and 1950.

17. Logical reasoners should not commit the fallacy of covering up counterevidence. In each of the following passages the reasoner is guilty of committing this fallacy, though you aren’t told why. What would you guess is the negative evidence that is being suppressed either intentionally or unintentionally?

a. Every day of my life the night has been followed by the sun's coming up. It is reasonable, therefore, to suppose that the sun will always come up in the future.
b. I’ve tried lungfish at three different restaurants over the last few years. Every time it has tasted awful to me. So, if I order the lungfish on this menu tonight, I won't like it.
c. The creation of the world happened long before anyone was around to witness it, so there can be no support for the theory of evolution from individual testimony. The only real evidence for evolution is in the bones embedded in rocks, but there are so many questions in this area of paleontology that even the paleontologists don’t agree. Besides, all the evidence is easily accounted for by the Noah’s flood that is mentioned in the Bible. Therefore, if you base your belief in evolution on geology or paleontology you are really being unscientific.

■ 18. If you obtained new theoretical knowledge that the population of objects you are about to study by statistical sampling is not very diverse, then you can make good use of this knowledge by

a. increasing your sample size.
b. decreasing your sample size.
c. assuming that similar effects are likely to have dissimilar causes.
d. avoiding a representative sample.

Other Types of Inductive Arguments

■ 1. Suppose someone offers the following argument: Amassing a fortune is like winning an election because it takes hard work, new ideas, and charisma. Well, behind every great fortune there is a great crime. So, you know what that means for elections. Explain the analogy by identifying the argument's conclusion and the A, B, and C that appear in the standard form of any argument by analogy.

■ 2. Create a short, serious argument by analogy for the following conclusion even if you don’t agree with it:

Abortion clinics deserve to be bombed.

3. Which one of the following three passages argues in a way that relies on an anecdote?

a. Uncle Antonio told me, "Don't bother checking," but I didn't listen to him. Somehow I just didn't believe Sandra when she said Sacajawea was some president's wife. I really wanted to find out more about Sacajawea, so I asked the librarian. She said to check the encyclopedia. It said that Sacajawea was an Indian woman who guided Lewis and Clark's expedition in 1804. She didn't marry any president. But think about that expedition. Knowing what you know now about U.S. history since 1804, do you think things would have turned out better if Sacajawea would have refused to be the guide for Lewis and Clark?
b. Mercy Otis Warren was a black activist who wrote political pamphlets during the American Revolution. I can still remember my grandmother saying to me, "When you grow up, you should read about that revolution. But don't read about it from your high school textbook. Read other books from big libraries." That's why I'm here. I want to know if you have any history books about Mercy Otis Warren. There is no listing for his name in the computerized catalog.
c. Paula Abdul and Wynton Marsalis are better singers than Lady Gaga. I went to the same concert that you are talking about, but I was closer to the stage than you were. Trust me; Lady Gaga didn't sing those songs; she just moved her lips to make it look that way. Once, when she tripped while dancing across the stage, she closed her mouth for a second, but the song kept right on going.

4. Identify the analogy that is used or mentioned in the following passage:

Hardly anybody likes to kill people. War is a messy, dirty, godforsaken business. Who wouldn't rather be home eating popcorn on the couch? But let's face it. You can't make an omelet without breaking eggs.

5. Create an original argument about some aspect of warfare. Your argument must be reasonable and nontrivial, and it must rely on an appeal to an analogy.

■ 6. Discuss the strength of the following argument by analogy:

Mercury is like water in that they are both liquids. Water seeks its own level, so the mercury in that thermometer will, too, if you break it open.

7. Discuss the quality of the reasoning in this argument. Is it valid, sound, fair, etc.?

You have to be a lesbian to be a feminist, but the film “Still Killing Us Softly” doesn't promote lesbianism, so the film is antifeminist.

8. Which of the following passages contain arguments that are inductions by appeal to a typical example?

a. This piece of copper is a typical example of copper. All copper conducts electricity. Therefore, this piece of copper does, too.
b. Let me make this appeal one more time, but it's the last time. If you want to keep your roof from leaking next winter, you’ve got to buy our Number One roof treatment. It has worked for all our customers, so it will work for you, too.
c. Woody Allen's “Annie Hall” was a comedy, so his films are probably all comedies, don't you think?
d. Our polling indicates that very few black Canadians can name one famous black American who lived in the nineteenth century. Their best guess for an example of a black American was Huey Newton. Newton was black, but he was a Black Panther organizer in Oakland, California in the 1970s, not in the nineteenth century.

9. Do some independent research and then write a short essay explaining to what extent the flow of electricity in a wire is analogous to the flow of water in a pipe.

10. Write a short essay explaining to what extent the operation of a family is and isn’t analogous to the workings of a country.

11. After receiving another student's answer to the previous question, write a short essay evaluating the student's answer.

12. State the implicit analogy used in the following argument:

You wouldn’t think it’s right to attack your neighbor across the street, so it is immoral for any country to attack its neighbor.

■ 13. State the implicit analogy used in the following argument:

There's no challenge in defeating Princeton in baseball. Would you take candy from a baby for the challenge of it?

■ 14. Choose the letter of the ranking that goes from strongest argument to weakest:

(1) Pele scored 10, 9, and 11 goals respectively in his last three games, so he will score 10 goals next game.

(2) Pele scored 10, 9, and 11 goals respectively in his last three games, so he will score 9 to 11 goals next game.

(3) Pele scored 10, 9, and 11 goals respectively in his last three games, so he scored an average of 10 goals in his last three games.

a. 123
b. 321
c. 213
d. 312
e. 231

15. During this year's soccer season, our team has lost all three of its games against Princeton University. It's a good bet that tomorrow's game against them will also be one big tragedy.

Consider the following changes to the above argument. Would each change (separately) be likely to strengthen, weaken, or not affect the argument?

a. Meredith, who is Princeton's best player, played in all three of the previous games, but she won't be playing tomorrow.
b. Helen, who is our team's best player, played in all three of the previous games, but she won't be playing tomorrow.
c. The last three games against Princeton were played on our field, and the next one will be, too.
d. The last three games against Princeton were played in different places: on our field, Princeton's, and the local community college's.
e. One of the games was played during a high wind, and the other two were played during a cold drizzle, but the weather prediction for tomorrow is warm, sunny, and calm.
f. During the past three games you have bet on the results and won, but this time you are not going to bet.

■ 16. Lady Theresa claims to be a psychic and to have perceptive abilities beyond those of most other people. She was tested in a laboratory once for her ability to guess which queen is missing from an ordinary deck of fifty-two playing cards, each containing four different queens. A friend of Lady Theresa was surprised to learn that she correctly identified the missing card only 50 percent of the time; she expected her to have a 100 percent success rate.

a. If in future card tests the experimenter were to have a professional magician specializing in card tricks observe Lady Theresa and help detect any cheating, should this make a 50% success rate more believable or less believable?
b. If not 50 percent, then what score should you expect the average, non-psychic person to get on the card tests?
c. The experimenter says that Lady Theresa's 50 percent is not statistically significant. Why do you suppose it isn't significant, and what do you recommend doing to determine whether her ability on these card tests is significantly better than the average person's ability?

17. Is mathematical induction a particular kind of inductive argument?

18. Comment on the strength of these inductive arguments:

a. Our lunar module landed on Saturn's closest moon and found the surface everywhere to be powdery down to two inches. Therefore, the surface of Saturn itself is covered everywhere with two inches of powder.
b. The chemical 3,4,5-trimethoxylate benzaldehyde killed David and his son when they drank it, so it will kill anybody.

19. Create your own multiple-choice question, with answer, about induction by appeal to a typical example. Make the question realistic, unambiguous, and the appropriate level of difficulty for students in your own class.

20. Create your own essay question, with answer, about induction by appeal to a typical example. Make the question realistic, unambiguous, and the appropriate level of difficulty for students in your own class.

21. For the problem of deciding whether a vaccine manufactured from chicken eggs will be effective against the common cold, would you say that a healthy sixty-two-year-old female designer of anti-tank weapons for the Boeing Corporation in Seattle, Washington would be a sufficiently typical member of the target population such that if the vaccine works on her it would work on anybody? Why? Mention any relevant background knowledge you have about diversity.

■ 22. State the conclusion of the following inductive argument, and then describe the argument's structure:

David was caught cheating on his history homework when he was in high school, and now you want to hire him to work the cash register in our office? Get serious. A leopard doesn't change its spots. ^[354]

23. Which of the following is the only one argument that relies on an induction from the past to the future? State the conclusions of all the arguments.

a. Joey's leopard had spots in the past and it will have spots in the future. So, a leopard doesn't change its spots.
b. Yesterday there was a full jar of jelly beans on that shelf. This morning there is a half-empty jar. Somebody took some last night, right?
c. When you bought that goldfish, who ended up taking care of it, me or you? Now you want to buy a guinea pig, and you expect me to believe that you will take care of it. No thanks.
d. You’ve got to buy either the goldfish or the guinea pig. My older sister told us that the goldfish is cheaper to buy and to feed, although it is also a little less fun to play with. So, let's buy the guinea pig, not the goldfish.

24. The following passage describes a scientific experiment. It then makes an induction from the past to the future.

We showed the person who claimed to be a psychic a deck of regular playing cards in which one card had been removed. The psychic was shown the backs of the cards but was not allowed to touch the cards. During the twenty times we tested the psychic, he correctly guessed which card was missing from the deck over 50 percent of the time. Therefore, he will get it right more than half of the time on the next twenty times we perform the test. Would the above argument be improved, weakened, or unaffected if

a. The phrase twenty times we tested is changed to twenty-four times we tested.
b. The phrases missing from and half are replaced by not in and 50 percent, respectively.
c. The word half is changed to three-quarters.
d. The psychic was quite comfortable in the past tests but will be made uncomfortable in the future tests.
e. In the previous tests a magician trained in card tricks was present to observe the psychic and to help the experimenter discover cheating that would invalidate the experiment, but in future tests the magician will not be present.
f. In the past tests the experimenter knew which card was missing, but in the future even the experimenter won't know the answer at the time the question is asked of the psychic.
g. The past tests and results were duplicated by an independent and trustworthy research organization.
h. Instead of all past twenty tests having been performed on a single day in a single lab, they were spread across fourteen days in seven different labs.

■ 25. John is a part-time cotton farmer in Alabama who has tried for four years to get a decent crop on his small plot. Every year he's had so much damage from pests that he hasn't made a decent profit. He concludes that next year's results will be just as bad. Would the strength of his argument be improved, weakened, or unaffected if he next year John will be adding alfalfa clippings as a fertilizer to his crop? Why?

26. Consider the character of this passage:

Medieval war is like a chess game because there are knights battling on horseback, kings at the center of attention, powerful queens, bishops who support the king, and so forth.

In the passage, there is an argument

a. by disanalogy
b. by appeal to a typical example.
c. by analogy that is not an appeal to a typical example.
d. whose conclusion is an analogy.

27. Criticize the following argument by analogy by using the technique of pointing out disanalogies:

Government budgets are like personal budgets in so many ways. Since you can't last long when your own budget is in the red, you shouldn't permit deficit spending by your government.

■ 28. 5Notice how different those two arguments are. All other things being equal, which is the stronger argument? Choose from one of the four suggested answers.

1. She’s not here, so she’s gone to the supermarket.
2. She’s not here, since she’s gone to the supermarket.

a. 1 is stronger because, if she’s gone to the supermarket, then she can’t be here provided we can assume that we aren’t in the supermarket, which is a very good assumption.
b. 2 is stronger because, if she’s not here, then she might be in any one of many different places. The supermarket could be one of those places. So, the conclusion follows, and this is a strong argument.
c. 2 is stronger because it is very probably ok to assume the word “here” does not refer to the supermarket, and it is fine to assume that people cannot be in two places at once. With those assumptions, the conclusion follows.
d. 1 is stronger because, if she’s not here, then there are many other places where she might be. The supermarket is one of them. So, the conclusion follows, and this is a strong argument.

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Solutions

Generalizing from a Sample

4 No. Such a sampling procedure won't guarantee a correct conclusion. Only a deductive argument will do that. Generalizing from sampling less than 100 percent of the population is always risky.

5 Getting a random sample is one of several methods that will help get a representative sample, but a representative sample can also be obtained by luck.

7 Answer (a).

9 Answer (c). 100% of the times in the past when she has taken his advice, she has been unhappy with the advice. She infers that this pattern will continue into the future and that she will also be unhappy again with the new advice. The property in question is the disappointing character of the film that is recommended by Ricardo.

11 This could well be true. Being unbiased only promotes the production of a representative sample; it won't guarantee it.

16 Answer (b).

18 Answer (b).

Other Types of Inductive Arguments

1 Conclusion: Winning an election depends on a great crime. A = amassing a fortune; B = winning an election; C = depends on a great crime.

2 An abortion clinic is like a nest of wasps in that both harm innocent persons. A nest of wasps deserves to be bombed (with pesticides). So, abortion clinics deserve to be bombed, too.

6 That argument is weak. Here is a much better argument:

All liquids seek their own level when not confined. Mercury is a liquid.

So, Mercury will seek its own level, too, when not confined. The first premise here follows from a scientific theory of liquids. The reason that the original argument is weak as it stands is that it is quite similar to the following inductively weak argument:

Mercury is like water in that they are both liquids. Water is a thirst-quenching liquid, so the mercury in that thermometer is, too. Mercury is actually poisonous, as are all metals.

13 Defeating Princeton in baseball would be like taking candy from a baby.

14 Answer (b). Argument 3 is deductively valid, unlike the other two arguments

16 Answer (a). This will make it more believable because it will make it more difficult for her to cheat or to get lucky. In short, you will have a better-designed experiment. (b) The answer is 25 percent, which is one queen out of four. (c) It probably wasn't significant, because so few tests were run. Maybe she guessed twice and was correct one of the two times. Do more tests.

22 Conclusion: David will continue to cheat if he is hired to work the cash register in our office. It's an argument from the past to the future passed on the idea that the past pattern of cheating will be likely to continue in the future. The comment about leopards is a common expression used to make the point that old patterns will continue to hold in the future. This passage could be analyzed as containing two arguments. First comes the argument mentioned in the preceding paragraph. The second argument has more implicit elements: its conclusion is that we shouldn't hire David. The argument contains implicit premises about it being unwise to hire people who cheat, especially if the job is to work a cash register. It is unclear, however, whether the second argument actually occurs; perhaps it doesn't and we are just guessing that the second argument is likely to be created or accepted by the arguer.

25 The argument probably will be weaker, because there is now less similarity between the past and the future in regard to a causally relevant characteristic. In particular, the fertilizer might make the crop more hardy and thus more resistant to the pest. Only if you knew that adding this fertilizer would tend to hurt the crop—say, by promoting pest growth—could you safely say that the argument would be strengthened. If you didn't know whether adding the fertilizer would help or hurt the crop, then just the fact that you know that adding it would be likely to affect the crop is reason enough to say the argument is weaker.

28 Answer (c).