Rigochi owners expressed the lowest satisfaction with their SUVs. So Rigochi
owners are less satisfied with their cars than other SUV owners.
Analysis Here we know about the 816 people who were surveyed in Cincinnatti.
They are the sample. The conclusion is about all SUV owners everywhere, and they
constitute the population.
Is the generalization good? That is, is the argument good? Unstated premises
are needed about how the study was conducted. Is there any reason that we should
think that these 816 people are like all SUV owners everywhere?
Example 2 I should build my house with the bedroom facing this direction to catch
the morning sun.
Analysis We believe we know where the sun will rise in the future based on where
we see it rise today. The sample is all the times in the past when the sun rose: We
know that the point where the sun rises varies slightly from season to season, but is
roughly east. The population is all times the sun has risen or will rise, which we
think will be in roughly the same direction.
Example 3 Of potential customers surveyed, 72% said that they liked "very much"
the new green color that Yoda plans to use for its cars. So about 72% of all potential
customers will like it.
Analysis The sample is the group of potential customers interviewed, and the
population is all potential customers.
Sometimes the generalization we want and we're entitled to isn't "all," but
"most," or "72%": The same proportion of the whole as in the sample will have the property. This is a statistical generalization.
EXERCISES for Section A 281
Example 4 Every time unemployment goes past 6% there's a call for restricting
immigration. I read that the forecast is for unemployment to reach 6.2% this month.
So I hope that Juan can get the visa for his wife real soon.
Analysis Here the unstated conclusion is that there will be a call for restricting
immigration. The reason given is in the past that's happened when there's been the
same rate of unemployment. The sample is all times in the past that the unemploy-
ment rate rose above 6%, and the population is all times it has risen or will rise.
We need to know how to judge whether the examples are sufficient for the
generalization: Do they have enough in common with the situation now? Are there
enough examples?
Example 5 The doctor tells you to fast from 10 p.m. Then at 10 a.m. she gives you
glucose to drink. Forty-five minutes later she takes some of your blood and has it
analyzed. She concludes you don't have diabetes.
Analysis The sample is the blood the doctor took. It's a very small sample
compared to the amount of blood you have in your body, but the doctor is confident
that it is representative of all your blood.
Example 6 You go to the city council meeting with a petition signed by all the
people who live on your block requesting that a street light be put in. Addressing the
city council, you say, "Everyone on this block wants a street light here."
Analysis You're not generalizing here: There's no argument from some to more,
since the sample equals the population.
Exercises for Section A
Here's some of Tom's work on identifying generalizations.
282 CHAPTER 14 Generalizing
Should we try the new Mexican restaurant on Sun Street? I heard it was pretty good.
Generalization! (yes/no) Yes.
Sample: People who told him it was good.
Population: It will be good food for him, too.
A generalization is an argument, right. "But the sampk. and the population aren't
claims— they're groups. Ihe sampk here is the times that other people have eaten there
(and reported that it was good). Ihe population is all times anyone has or will eat there.
It's a past to future generalization.
For Exercises 1-13 answer the following questions:
Generalization? (yes/no)
Sample:
Population:
1. German shepherds have a really good temperament. I know, because lots of my friends
and my sister have one.
2. Maria: Look! That dry cleaner broke a button on my blouse again. I'm going to go
over there and complain.
3. Suzy: I hear you got one of those MP3 players from Hirangi.
Maria: Yeah, and I wish I'd never gotten one. It's always breaking down.
Suzy: Well, I won't get one then, since they're probably all the same.
4. Maria to Suzy: Don't bother to ask Tom to do the dishes. My brother's a football player
and no football player will do the dishes.
5. Suzy: Guys are such nitwits.
Zoe: What do you mean?
Suzy: Like, they can't even tell when you're down. Emotionally, they're clods.
Besides, they just want a girl for her body.
Zoe: How do you know?
Suzy: Duh, it's like a cheerleader like me isn't going to have a lot of dates?
6. Lee: Are you taking Spot for a walk?
Dick: No. I'm getting the leash because I have to take him to the vet, and it will be
hard to get him to go. Every time I take him to the vet he seems to know it
before we get in the car.
7. Manuel: Are those refried beans?
Maria: Yes.
Manuel: I can't believe you'd cook those for dinner. Don't you remember I had terrible
indigestion the last time you made them?
8. Maria: Do you know of a good dry cleaner other than Ricardo's?
Zoe: The one in the plaza north of campus is pretty good. They've always done
O.K. with the stuff I take them.
SECTION B What Is a Good Sample! 283
9. Don't go to Seattle in December. It rains there all the time then.
10. Dogs can be trained to retrieve a newspaper.
11. I want to marry a Japanese guy. They're hard-working and really family-oriented.
12. You don't have to worry about getting the women's gymnastic team in your van—
I saw them at the last meet, and they're small enough to fit in.
13. From our study it appears that bald men are better husbands.
14. Write down three examples of generalizations you have heard or made in the last week
and one example of a claim that sounds like a generalization but isn't. See if your
classmates can pick out the one that isn't. For the generalizations, ask a classmate to
identify the sample and the population.
B. What is a Good Sample?
1. How you can go wrong
Tom's sociology professor has assigned him to conduct a survey to find out the
attitudes of students on campus about sex before marriage. "That's easy," Tom
thinks, "I'll just ask some of my friends. They're typical, aren't they?"
So he asks all his friends he can reach on Tuesday whether they think sex before
marriage is a great idea or not. Twenty of the twenty-eight say "Yes," while eight
say "No." That was easy.
Tom takes the results to his professor and she asks why he thinks his friends are
typical. "Typical? I guess they are," Tom responds. But aren't they mostly your
age? And the same sex as you? How many are gay? How many are married? And
is twenty-eight really enough to generalize from? And what about that question
"Is sex before marriage a great idea or not?" A bit biased?
O.K., it wasn't such a good job. Back to the drawing board. Tom brainstorms
with some of his friends and figur
es he'll ask 100 students as they leave the student
union one question, "Do you approve of sexual intercourse before marriage?"
He goes to the student union at 4 p.m. on Wednesday, asks the students, and
finds that 83 said "No," while 17 said "Yes." That's different from what he
expected, but what the heck, this is science, and science can't be wrong. There was
no bias in the question, and surely those 100 students are typical.
Tom presents the results to his professor,
and she suggests that perhaps he should find
out what was going on at the student union
that day. . . . It seems the campus Bible
society was having a big meeting there that
let out about 4 p.m. Maybe this survey won't
give a good generalization.
So Tom and two friends get together,
and at 9 a.m., 1 p.m., and 6 p.m. they station
284 CHAPTER 14 Generalizing
themselves outside the student union, the administration offices, and the big
classroom building. Each is to ask the first 20 people who come by just two
questions: "Are you a student here?" and "Do you approve of sexual intercourse
before marriage?"
They get 171 people saying they are students, with 133 saying "Yes" and 38
saying "No" to the second question. That's a lot of responses with no evident bias in
the sampling. Tom's sure his professor will be happy this time.
Tom tells his professor what they've done, and she asks, "Why do you think your
sample is representative? Why do you think it's big enough?"
Tom's puzzled. It's big enough. Surely 170 out of 20,000 students is a lot, isn't
it? How many could she expect us to interview? We're just human.
And representative? What does she mean? "We didn't do anything to get a
bias," he says. "But are those students typical?" she asks. "Is not doing anything to get a bias enough to ensure your sample is representative?"
2. Representative samples
Tom's first two attempts to survey students about their attitudes towards sex before
marriage used clearly unrepresentative samples. But his third attempt? Can we be
sure he has a sample that is just like the population, one that is representative?
Representative sample A sample is representative if no one subgroup of the
whole population is represented more than its proportion in the population.
A sample is biased if it is not representative.
Tom's method was haphazard sampling: Choosing the sample with no
intentional bias. Possibly the sample is representative. Maybe not. But we don't
have any good reason to believe that it is representative. There is, however, a way
we can choose a sample that is very likely to get us a representative sample.
Random sampling A sample is chosen randomly if at every choice
there is an equal chance for any one of the remaining members of
the population to be picked.
If you assign a number to each student, write the numbers on slips of paper, put
them in a fishbowl, and draw one number out at a time, that's probably going to be a
random selection. But there's a chance that slips with longer numbers will have
more ink and fall to the bottom of the bowl when you shake it. Or the slips aren't all
the same size. So typically to get a random selection we use tables of random
numbers prepared by mathematicians. Most spreadsheet programs for home
SECTION B What Is a Good Sample! 285
computers can now generate tables of random numbers. So for Tom's survey he
could get a list of all students; if the first number on the table is 413, he'd pick the
413th student on the list; if the second number is 711, he'd pick the 711th student on
the list; and so on, until he has a sample that's big enough.
Why is random sampling better? Suppose that of the 20,000 students at your
school, 500 are gay males. Then the chance that one student picked at random
would be a gay male is 500/20 0 0 0 = 1/40 . If you were to pick 300 students at
random, the chance that half of them would be gay is very small. It is very likely,
however, that 7 or 8 (I/40 of 300) will be gay males.
Or suppose that roughly 50% of the students at your school are female. Then
each time you choose a student at random there's a roughly 50% chance the person
will be female. And if you randomly choose a sample of 300 students the chance is
very high that about 50% will be female.
The law of large numbers says, roughly, that if the probability of something
occurring is X percent, then over the long run the percentage of times that happens
will be about X percent. For example, the probability of a flip of a fair coin landing
heads is 50%. So, though you may get a run of 8 tails, then 5 heads, then 4 tails, then
36 heads to start, in the long run, repeating the flipping, if the coin is fair, eventually
the number of heads will tend toward 50%.
If you choose a large sample randomly, the chance is very high that it will be
representative. That's because the chance of any one subgroup being over-
represented is small—not nonexistent, but small. It doesn't matter if you know
anything about the composition of the population in advance. After all, to know
how many homosexuals there are, and how many married women, and how many
Muslims, and how many . . . you'd need to know almost everything about the
population in advance. But that's what you use surveys to find out.
286 CHAPTER 14 Generalizing
With a random sample we have good reason to believe the sample is
representative. A sample chosen haphazardly may give a representative sample
but you have no good reason to believe it will be representative.
Weak Argument
Strong Argument
Sample is chosen haphazardly.
Sample is chosen randomly.
Therefore,
Therefore,
The sample is representative.
The sample is representative.
Lots of ways the sample could
Very unlikely that the sample
be biased.
is biased.
The classic example that haphazard sampling needn't work, even with an
enormous sample, is the poll done in 1936 by Literary Digest. The magazine
mailed out 10,000,000 ballots asking who the person would vote for in the 1936
presidential election. They received 2,300,000 back. With that huge sample the
magazine confidently predicted that Alf Landon would win. Roosevelt received 60%
of the vote, one of the biggest wins ever. What went wrong? The magazine selected
its sample from lists of it own subscribers and telephone and automobile owners.
In 1936 that was the wealthy class. And the wealthy folks preferred Alf Landon.
The sample wasn't representative of all voters.
In any case, we can't always get a perfectly representative sample. Of 400
voters in Mississippi that are chosen randomly, 6 are traveling out of the state, 13
have moved with no forwarding address,. . . you can't locate them all. Like being
vague, the right question to ask is: Does the sample seem too biased to be reliable?
Beware of selective attention:
II seems that buttered (oast always lands
the wrong side down because you notice-
or remember—when it does.
Exercises for Section B
1. What is a re
presentative sample?
2. Explain why a good generalization is unlikely to be valid.
3. a. What is the law of large numbers?
b. How does it justify random sampling as giving unbiased samples?
4. Why does the phone ring more often when you're in the shower?
SECTION C When is a Generalization Good? 287
5. Which of the following seem too biased to be reliable, and why?
a. To determine the average number of people in your city who played tennis last
week, interview women only.
b. To determine what kind of cat food is purchased most often, interview only people
who are listed in the telephone directory.
c. To determine what percentage of women think that more women should be doctors,
poll female students as they leave their classes at your school.
d. To determine whether to buy grapes at the supermarket, pick a grape from the bunch
you're interested in and taste it.
6. a. Suppose you want to find out whether people in your city believe that there are
enough police officers. Give four characteristics of people that could bias the
survey. That is, list four subgroups of the population that you would not want to
have represented out of proportion to their actual percentages in the population,
b. Now list four characteristics that you feel would not matter for giving bias.
7. A professor suggested the best way to get a sample is to make sure that for the relevant
characteristics, for example, gender, age, ethnicity, income,. . ., we know that the
sample has the same proportion as in the population as a whole. Why won't that work?
8. One of Dr. E's students was a blackjack dealer at a casino and heard a player say,
"I ran a computer simulation of this system 1,000 times and made money. So why
didn't I win today playing for real?" Can you explain it?
9. Is every randomly chosen sample representative? Explain.
C. When Is a Generalization Good?
1. Sample size
I've got a couple of Chinese students in my classes. They're both hard-
working and get good grades. I suppose that all Chinese are like that.
That's generalizing from too small a sample—the way stereotypes begin. It's a
hasty generalization using anecdotal evidence.
But how big does a sample have to be? To estimate what percentage of
students at your school approve of sex before marriage, is it enough to ask 5? 2 5 ?
Richard L Epstein Page 36