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2: Probability as a Basis for Making Decisions

2: Probability as a Basis for Making Decisions

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6.2

313

Probability as a Basis for Making Decisions

T A B L E 6 .1 Age and Gender Distribution

AGE

Gender

Under 17

17–20

21–24

25–30

31–39

40 and

Older

Male

Female

.006

.004

.15

.18

.16

.14

.08

.10

.04

.04

.01

.09

Suppose that you are told that a 43-year-old student from this university is waiting to meet you. You are asked to decide whether the student is male or female. How

would you respond? What if the student had been 27 years old? What about 33 years

old? Are you equally confident in all three of your choices?

A reasonable response to these questions could be based on the probability information in Table 6.1. We would decide that the 43-year-old student was female. We

cannot be certain that this is correct, but we can see that someone in the 40-and-over

age group is much less likely to be male than female. We would also decide that the

27-year-old was female. However, we would be less confident in our conclusion than

we were for the 43-year-old student. For the age group 31–39, the proportion of

males and the proportion of females are equal, so we would think it equally likely that

a 33-year-old student would be male or female. We could decide in favor of male (or

female), but with little confidence in our conclusion; in other words, there is a good

chance of being incorrect.

E X A M P L E 6 . 6 Can You Pass by Guessing?

A professor planning to give a quiz that consists of 20 true–false questions is interested

in knowing how someone who answers by guessing would do on such a test. To investigate, he asks the 500 students in his introductory psychology course to write the

numbers from 1 to 20 on a piece of paper and then to arbitrarily write T or F next to

each number. The students are forced to guess at the answer to each question, because

they are not even told what the questions are. These answer sheets are then collected

and graded using the key for the quiz. The results are summarized in Table 6.2.

T A B L E 6 .2 Quiz “Guessing” Distribution

Number of

Correct

Responses

Number of

Students

Proportion

of Students

0

1

2

3

4

5

6

7

8

9

10

0

0

1

1

2

8

18

37

58

81

88

.000

.000

.002

.002

.004

.016

.036

.074

.116

.162

.176

Number of

Correct

Responses

Number of

Students

Proportion

of Students

11

12

13

14

15

16

17

18

19

20

79

61

39

18

7

1

1

0

0

0

.158

.122

.078

.036

.014

.002

.002

.000

.000

.000

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

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314

Chapter 6 Probability

Because probabilities are long-run proportions, an entry in the “Proportion of

Students” column of Table 6.2 can be considered an estimate of the probability

of correctly guessing a specific number of responses. For example, a proportion of

.122 (or 12.2%) of the 500 students got 12 of the 20 correct when guessing. We

then estimate the long-run proportion of guessers who would get 12 correct to be

.122, and we say that the probability that a student who is guessing will get 12 correct is (approximately) .122.

Let’s use the information in Table 6.2 to answer the following questions.

1. Would you be surprised if someone who is guessing on a 20-question true–

false quiz got only 3 correct? The approximate probability of a guesser getting

3 correct is .002. This means that, in the long run, only about 2 in 1000 guessers would score exactly 3 correct. This would be an unlikely outcome, and we

would consider its occurrence surprising.

2. If a score of 15 or more correct is required to receive a passing grade on the

quiz, is it likely that someone who is guessing will be able to pass? The longrun proportion of guessers who would pass is the sum of the proportions for all

the passing scores (15, 16, 17, 18, 19, and 20). Then,

probability of passing Ϸ .014 1 .002 1 .002 1 .000 1 .000 1 .000 5 .018

It would be unlikely that a student who is guessing would be able to pass.

3. The professor actually gives the quiz, and a student scores 16 correct. Do you

think that the student was just guessing? Let’s begin by assuming that the student

was guessing and determine whether a score as high as 16 is a likely or an unlikely

occurrence. Table 6.2 tells us that the approximate probability of getting a score

at least as high as this student’s score is

probability of scoring 16 or higher < .002 1 .002 1 .000 1 .000 1 .000

5 .004

That is, in the long run, only about 4 times in 1000 would a guesser score 16 or

higher. This would be rare. There are two possible explanations for the observed

score: (1) The student was guessing and was really lucky, or (2) the student was

not just guessing. Given that the first explanation is highly unlikely, a more plausible choice is the second explanation. We would conclude that the student was

not just guessing at the answers. Although we cannot be certain that we are correct in this conclusion, the evidence is compelling.

4. What score on the quiz would it take to convince us that a student was not just

guessing? We would be convinced that a student was not just guessing if his or

her score was high enough that it was unlikely that a guesser would have been

able to do as well. Consider the following approximate probabilities (computed

from the entries in Table 6.2):

Score

Approximate Probability

20

19 or better

18 or better

17 or better

16 or better

15 or better

14 or better

13 or better

.000

.000 1 .000 5 .000

.000 1 .000 1 .000 5 .000

.002 1 .000 1 .000 1 .000 5 .002

.002 1 .002 1 .000 1 .000 1 .000 5 .004

.014 1 .002 1 .002 1 .000 1 .000 1 .000 5 .018

.036 1 .014 1 .002 1 .002 1 .000 1 .000 1 .000 5 .054

.078 1 .036 1 .014 1 .002 1 .002 1 .000 1 .000 1 .000 5 .132

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

6.2

315

Probability as a Basis for Making Decisions

We might say that a score of 14 or higher is reasonable evidence that someone is not

guessing, because the approximate probability that a guesser would score this high is

only .054. Of course, if we conclude that a student is not guessing based on a quiz

score of 14 or higher, there is a risk that we are incorrect (about 1 in 20 guessers

would score this high by chance). About 13.2% of the time, a guesser will score 13

or more correct. This would happen by chance often enough that most people would

not rule out the student being a guesser.

Examples 6.5 and 6.6 show how probability information can be used to make a

decision. This is a primary goal of statistical inference. Later chapters look more formally at the problem of drawing a conclusion based on available but often incomplete

information and then assessing the reliability of such a conclusion.

E X E RC I S E S 6 . 1 5 - 6 . 1 8

6.15 Is ultrasound a reliable method for determining

the gender of an unborn baby? Consider the following

data on 1000 births, which are consistent with summary

values that appeared in the online Journal of Statistics

Education (“New Approaches to Learning Probability

in the First Statistics Course” [2001]):

Actual Gender Is Female

Actual Gender Is Male

Ultrasound

Predicted

Female

Ultrasound

Predicted

Male

432

130

48

390

Do you think that a prediction that a baby is male and a

prediction that a baby is female are equally reliable? Explain, using the information in the table to calculate estimates of any probabilities that are relevant to your

conclusion.

6.16 Researchers at UCLA were interested in whether

working mothers were more likely to suffer workplace

injuries than women without children. They studied

1400 working women, and a summary of their findings

was reported in the San Luis Obispo Telegram-Tribune

(February 28, 1995). The information in the following

table is consistent with summary values reported in the

article:

Children

Under 6

Children,

but None

Under 6

32

68

56

156

368

232

644

1244

400

300

700

1400

No

Children

Injured on the

Job in 1989

Not Injured

on the Job

in 1989

Total

The researchers drew the following conclusion: Women

with children younger than age 6 are much more likely

to be injured on the job than childless women or mothers with older children. Provide a justification for the

researchers’ conclusion. Use the information in the table

to calculate estimates of any probabilities that are relevant to your justification.

6.17 A Gallup Poll conducted in November 2002 examined how people perceived the risks associated with

smoking. The following table summarizes data on smoking status and perceived risk of smoking that is consistent

Perceived Risk

Very

Somewhat Not Too Not at All

Smoking Status Harmful Harmful

Harmful Harmful

Current Smoker 60

Former Smoker 78

Never Smoked 86

Data set available online

Total

30

16

10

5

3

2

1

2

1

Video Solution available

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

316

Chapter 6 Probability

Assume that it is reasonable to consider these data representative of the U.S. adult population. Consider the following conclusion: Current smokers are less likely to

view smoking as very harmful than either former smokers or those who have never smoked. Provide a justification for this conclusion. Use the information in the table

to calculate estimates of any probabilities that are relevant to your justification.

Number of Units Secured During First

Attempt to Register

Priority

Group

0–3

4–6

7–9

10–12

More

Than 12

1

2

3

4

.01

.02

.04

.04

.01

.03

.06

.08

.06

.06

.06

.07

.10

.09

.06

.05

.07

.05

.03

.01

6.18 Students at a particular university use an online

registration system to select their courses for the next term.

There are four different priority groups, with students in

Group 1 registering first, followed by those in Group 2,

and so on. Suppose that the university provided the

accompanying information on registration for the fall semester. The entries in the table represent the proportion

of students falling into each of the 20 priority–unit

combinations.

6.3

Data set available online

a. What proportion of students at this university got

10 or more units during the first attempt to

register?

b. Suppose that a student reports receiving 11 units

during the first attempt to register. Is it more likely

that he or she is in the first or the fourth priority

group?

c. If you are in the third priority group next term, is it

likely that you will get more than 9 units during the

first attempt to register? Explain.

Video Solution available

Estimating Probabilities Empirically

and by Using Simulation

In the examples presented so far, reaching conclusions required knowledge of the

probabilities of various outcomes. In some cases, this is reasonable, and we know the

true long-run proportion of the time that each outcome will occur. In other situations, these probabilities are not known and must be determined. Sometimes probabilities can be determined analytically, by using mathematical rules and probability

properties, including the basic ones introduced in this chapter.

In this section, we change gears a bit and focus on an empirical approach to probability. When an analytical approach is impossible, impractical, or just beyond the

limited probability tools of the introductory course, we can estimate probabilities

empirically through observation or by using simulation.

Estimating Probabilities Empirically

It is fairly common practice to use observed long-run proportions to estimate probabilities. The process used to estimate probabilities is simple:

1. Observe a large number of chance outcomes under controlled circumstances.

2. By appealing to the interpretation of probability as a long-run relative frequency,

estimate the probability of an outcome by using the observed proportion of

occurrence.

This process is illustrated in Examples 6.7 and 6.8.

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

6.3 Estimating Probabilities Empirically and by Using Simulation

EXAMPLE 6.7

317

Fair Hiring Practices

The biology department at a university plans to recruit a new faculty member and

intends to advertise for someone with a Ph.D. in biology and at least 10 years of

college-level teaching experience. A member of the department expresses the belief

that requiring at least 10 years of teaching experience will exclude most potential applicants and will exclude far more female applicants than male applicants. The biology department would like to determine the probability that someone with a Ph.D.

in biology who is looking for an academic position would be eliminated from consideration because of the experience requirement.

A similar university just completed a search in which there was no requirement

for prior teaching experience but the information about prior teaching experience was

recorded. The 410 applications yielded the following data:

NUMBER OF APPLICANTS

Less Than 10 Years

of Experience

10 Years of

Experience or More

Total

178

99

277

112

21

133

290

120

410

Male

Female

Total

Let’s assume that the populations of applicants for the two positions can be regarded as

the same. We will use the available information to approximate the probability that an

applicant will fall into each of the four gender–experience combinations. The estimated

probabilities (obtained by dividing the number of applicants for each gender–experience

combination by 410) are given in Table 6.3. From Table 6.3, we calculate

estimate of P(candidate excluded) 5 .4341 1 .2415 5 .6756

We can also assess the impact of the experience requirement separately for male applicants and for female applicants. From the given information, we calculate

that the proportion of male applicants who have less than 10 years of experience is

178/290  5 .6138, whereas the corresponding proportion for females is 99/120 5

.8250. Therefore, approximately 61% of the male applicants would be eliminated by

the experience requirement, and about 83% of the female applicants would be

eliminated.

T A B L E 6 .3 Estimated Probabilities for Example 6.7

Male

Female

Less Than 10 Years

of Experience

10 Years of

Experience or More

.4341

.2415

.2732

.0512

These subgroup proportions—.6138 for males and .8250 for females—are examples of conditional probabilities. As discussed in Section 6.1, outcomes are dependent if the occurrence of one outcome changes our assessment of the probability that

the other outcome will occur. A conditional probability shows how the original probability changes in light of new information. In this example, the probability that a

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

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