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ACTIVITY 12.2: Color and Perceived Taste

# ACTIVITY 12.2: Color and Perceived Taste

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Chapter Review Exercises

607

TERM OR FORMULA

COMMENT

Two-way frequency table (contingency table)

A rectangular table used to summarize a categorical data

set; two-way tables are used to compare several populations on the basis of a categorical variable or to determine

if an association exists between two categorical variables.

X 2 test for homogeneity

The hypothesis test performed to determine whether category proportions are the same for two or more populations or treatments.

X 2 test for independence

The hypothesis test performed to determine whether an

association exists between two categorical variables.

Chapter Review Exercises 12.35 - 12.45

Each observation in a random sample of 100

bicycle accidents resulting in death was classified according to the day of the week on which the accident occurred. Data consistent with information given on the

web site www.highwaysafety.com are given in the following table

12.35

Day of Week

Sunday

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Frequency

14

13

12

15

14

17

15

Color

First Peck Frequency

Blue

Green

Yellow

Red

16

8

6

3

Do the data provide evidence of a color preference? Test

using a ϭ .01.

In November 2005, an international study to

assess public opinion on the treatment of suspected terrorists was conducted (“Most in U.S., Britain, S. Korea

12.37

Based on these data, is it reasonable to conclude that the

proportion of accidents is not the same for all days of the

week? Use a ϭ .05.

The color vision of birds plays a role in their

foraging behavior: Birds use color to select and avoid

certain types of food. The authors of the article “Colour

12.36

Avoidance in Northern Bobwhites: Effects of Age, Sex,

and Previous Experience” (Animal Behaviour [1995]:

519–526) studied the pecking behavior of 1-day-old bobwhites. In an area painted white, they inserted four pins

with different colored heads. The color of the pin chosen

on the bird’s first peck was noted for each of 33 bobwhites, resulting in the accompanying table.

Data set available online

and France Say Torture Is OK in at Least Rare Instances,” Associated Press, December 7, 2005). Each

individual in random samples of 1000 adults from each

of nine different countries was asked the following question: “Do you feel the use of torture against suspected

terrorists to obtain information about terrorism activities

is justified?” Responses consistent with percentages given

in the article for the samples from Italy, Spain, France,

the United States, and South Korea are summarized in

the table at the top of the next page. Based on these data,

is it reasonable to conclude that the response proportions

are not the same for all five countries? Use a .01 significance level to test the appropriate hypotheses.

Video Solution available

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608

Chapter 12

The Analysis of Categorical Data and Goodness-of-Fit Tests

Response

Country

Never

Rarely

Sometimes

Often

Not

Sure

Italy

Spain

France

United

States

South

Korea

600

540

400

360

140

160

250

230

140

140

200

270

90

70

120

110

30

90

30

30

100

330

470

60

40

12.40

Each boy in a sample of Mexican American

males, age 10 to 18, was classified according to smoking

status and response to a question asking whether he likes

to do risky things. The following table is based on data

given in the article “The Association Between Smoking

and Unhealthy Behaviors Among a National Sample

of Mexican-American Adolescents” (Journal of School

Health [1998]: 376–379):

Smoking Status

According to Census Bureau data, in 1998 the

California population consisted of 50.7% whites, 6.6%

blacks, 30.6% Hispanics, 10.8% Asians, and 1.3% other

ethnic groups. Suppose that a random sample of 1000

students graduating from California colleges and universities in 1998 resulted in the accompanying data on

ethnic group. These data are consistent with summary

statistics contained in the article titled “Crumbling Public

12.38

School System a Threat to California’s Future (Investor’s Business Daily, November 12, 1999).

Ethnic Group

Number in Sample

White

Black

Hispanic

Asian

Other

679

51

77

190

3

Nonsmoker

45

36

46

153

Assume that it is reasonable to regard the sample as a

random sample of Mexican-American male adolescents.

a. Is there sufficient evidence to conclude that there is

an association between smoking status and desire to

do risky things? Test the relevant hypotheses using

a ϭ .05.

b. Based on your conclusion in Part (a), is it reasonable

to conclude that smoking causes an increase in the

desire to do risky things? Explain.

The article “Cooperative Hunting in Lions:

The Role of the Individual” (Behavioral Ecology and

Sociobiology [1992]: 445–454) discusses the different

12.41

Do the data provide evidence that the proportion of

students graduating from colleges and universities in

California for these ethnic group categories differs from

the respective proportions in the population for California? Test the appropriate hypotheses using a ϭ .01.

12.39 Criminologists have long debated whether there

is a relationship between weather and violent crime. The

author of the article “Is There a Season for Homicide?”

(Criminology [1988]: 287–296) classified 1361 homicides according to season, resulting in the accompanying

data. Do these data support the theory that the homicide

rate is not the same over the four seasons? Test the relevant hypotheses using a significance level of .05.

Season

Winter

Spring

Summer

Fall

328

334

372

327

Likes Risky Things

Doesn’t Like Risky Things

Smoker

Data set available online

roles taken by lionesses as they attack and capture prey.

The authors were interested in the effect of the position

in line as stalking occurs; an individual lioness may be in

the center of the line or on the wing (end of the line) as

the role of the lioness was also considered. A lioness

could initiate a chase (be the first one to charge the prey),

or she could participate and join the chase after it has

been initiated. Data from the article are summarized in

the accompanying table.

Role

Position

Initiate Chase

Participate in Chase

Center

Wing

28

66

48

41

Is there evidence of an association between position and

role? Test the relevant hypotheses using a ϭ .01. What

assumptions about how the data were collected must be

true for the chi-square test to be an appropriate way to

analyze these data?

Video Solution available

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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.

Chapter Review Exercises

12.42 The authors of the article “A Survey of Parent

Attitudes and Practices Regarding Underage Drinking”

(Journal of Youth and Adolescence [1995]: 315–334)

would like to know whether male and female inmates

differ with respect to type of offense.

conducted a telephone survey of parents with preteen

and teenage children. One of the questions asked was

“How effective do you think you are in talking to your

children about drinking?” Responses are summarized in

the accompanying 3 ϫ 2 table. Using a significance level

of .05, carry out a test to determine whether there is an

association between age of children and parental

response.

Age of Children

Response

Preteen

Teen

126

44

51

149

41

26

Very Effective

Somewhat Effective

Not at All Effective or Don’t Know

The article “Regional Differences in Attitudes

Toward Corporal Punishment” (Journal of Marriage

and Family [1994]: 314–324) presents data resulting

12.43

from a random sample of 978 adults. Each individual in

the sample was asked whether he or she agreed with the

following statement: “Sometimes it is necessary to discipline a child with a good, hard spanking.” Respondents

were also classified according to the region of the United

States in which they lived. The resulting data are summarized in the accompanying table. Is there an association between response (agree, disagree) and region of

residence? Use a 5 .01.

Response

Region

Northeast

West

Midwest

South

Agree

Disagree

130

146

211

291

59

42

52

47

Jail inmates can be classified into one of the

following four categories according to the type of crime

committed: violent crime, crime against property, drug

offenses, and public-order offenses. Suppose that random samples of 500 male inmates and 500 female inmates are selected, and each inmate is classified according to type of offense. The data in the accompanying

table are based on summary values given in the article

“Profile of Jail Inmates” (USA Today, April 25, 1991). We

12.44

Data set available online

609

Gender

Type of Crime

Male

Female

Violent

Property

Drug

Public-Order

117

150

109

124

66

160

168

106

a. Is this a test of homogeneity or a test of

independence?

b. Test the relevant hypotheses using a significance

level of .05.

Drivers born under the astrological sign of

Capricorn are the worst drivers in Australia, according to

an article that appeared in the Australian newspaper The

Mercury (October 26, 1998). This statement was based

on a study of insurance claims that resulted in the following data for male policyholders of a large insurance

company.

12.45

Astrological Sign

Aquarius

Aries

Cancer

Capricorn

Gemini

Leo

Libra

Pisces

Sagittarius

Scorpio

Taurus

Virgo

Number of

Policyholders

35,666

37,926

38,126

54,906

37,179

37,354

37,910

36,677

34,175

35,352

37,179

37,718

a. Assuming that it is reasonable to treat the male policyholders of this particular insurance company as a

random sample of male insured drivers in Australia,

are the observed data consistent with the hypothesis

that the proportion of male insured drivers is the

same for each of the 12 astrological signs?

b. Why do you think that the proportion of Capricorn

policyholders is so much higher than would be expected if the proportions are the same for all astrological signs?

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.

610

Chapter 12 The Analysis of Categorical Data and Goodness-of-Fit Tests

c. Suppose that a random sample of 1000 accident

claims submitted to this insurance company is selected and each claim classified according to the astrological sign of the driver. (The accompanying table is consistent with accident rates given in the

article.)

Astrological Sign

Aquarius

Aries

Cancer

Capricorn

Gemini

Leo

Libra

Pisces

Sagittarius

Observed Number

in Sample

85

83

82

88

83

83

83

82

81

Astrological Sign

Scorpio

Taurus

Virgo

Observed Number

in Sample

85

84

81

Test the null hypothesis that the proportion of accident

claims submitted by drivers of each astrological sign is

consistent with the proportion of policyholders of each

sign. Use the given information on the distribution of

policyholders to compute expected frequencies and then

carry out an appropriate test.

Continued

Data set available online

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.

CHAPTER

13

Simple Linear

Regression and

Correlation:

Inferential

Methods

Regression and correlation were introduced in Chapter 5

as techniques for describing and summarizing bivariate

numerical data consisting of (x, y) pairs. For example,

consider a scatterplot of data on y 5 percentage of courses

taught by teachers with inappropriate or no license and

x 5 spending per pupil for a sample of Missouri public

school districts (“Is Teacher Pay Adequate?” Research

Arne Hodalic/Encyclopedia/Corbis

Working Papers Series, Kennedy School of Government, Harvard University, October 2005). A scatterplot of the data shows a surprising linear pattern. The sample correlation coefficient is r 5 .27, and the equation of

the least-squares line has a positive slope, indicating that school districts with higher

expenditures per student also tended to have a higher percentage of courses taught by

Make the most of your study time by accessing everything you need to succeed

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Step-by-step instructions for Minitab, Excel, TI-83/84, SPSS, and JMP

Video solutions to selected exercises

Data sets available for selected examples and exercises

Online quizzes

Flashcards

Videos

611

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.

612

Chapter 13 Simple Linear Regression and Correlation: Inferential Methods

teachers with an inappropriate license or no license. Could the pattern observed in

the scatterplot be plausibly explained by chance, or does the sample provide convincing evidence of a linear relationship between these two variables for school districts in

Missouri? If there is evidence of a meaningful relationship between these two variables, the regression line could be used as the basis for predicting the percentage of

teachers with inappropriate or no license for a school district with a specified expenditure per student or for estimating the average percentage of teachers with inappropriate or no license for all school districts with a specified expenditure per student. In

this chapter, we develop inferential methods for bivariate numerical data, including a

confidence interval (interval estimate) for a mean y value, a prediction interval for a

single y value, and a test of hypotheses regarding the extent of correlation in the entire

population of (x, y) pairs.

13.1

Simple Linear Regression Model

A deterministic relationship is one in which the value of y is completely determined by

the value of an independent variable x. Such a relationship can be described using

traditional mathematical notation, such as y 5 f (x) where f(x) is a specified function

of x. For example, we might have

y 5 f 1x2 5 10 1 2x

or

y 5 f 1x2 5 4 2 1102 2x

However, in many situations, the variables of interest are not deterministically related. For example, the value of y 5 first-year college grade point average is certainly

not determined solely by x 5 high school grade point average, and y 5 crop yield is

determined partly by factors other than x 5 amount of fertilizer used.

A description of the relationship between two variables x and y that are not deterministically related can be given by specifying a probabilistic model. The general

form of an additive probabilistic model allows y to be larger or smaller than f(x) by

a random amount e. The model equation is of the form

y 5 deterministic function of x 1 random deviation 5 f 1x2 1 e

Thinking geometrically, if e . 0, the corresponding point will lie above the graph of

y 5 f(x). If e , 0 the corresponding point will fall below the graph. If f (x) is a function used in a probabilistic model relating y to x and if observations on y are made for

various values of x, the resulting (x, y) points will be distributed about the graph of

f(x), some falling above it and some falling below it.

For example, consider the probabilistic model

f

y 5 50 2 10x 1 x 2 1 e

f (x)

The graph of the function y 5 50 2 10x 1 x 2 is shown as the orange curve in Figure 13.1. The observed point (4, 30) is also shown in the figure. Because

f 142 5 50 2 10 142 1 42 5 50 2 40 1 16 5 26

for the point (4, 30), we can write y 5 f 1x2 1 e, where e 5 4. The point (4, 30) falls

4 above the graph of the function y 5 50 2 10x 1 x 2.

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.

13.1

613

Simple Linear Regression Model

y

Observation (4, 30)

e=4

26

Graph of

y = 50 – 10x + x 2

FIGURE 13.1

A deviation from the deterministic part

of a probabilistic model.

x

4

Simple Linear Regression

The simple linear regression model is a special case of the general probabilistic model

in which the deterministic function f (x) is linear (so its graph is a straight line).

DEFINITION

The simple linear regression model assumes that there is a line with vertical

or y intercept a and slope b, called the population regression line. When a

value of the independent variable x is ﬁxed and an observation on the dependent variable y is made,

y ϭ a ϩ bx ϩ e

Without the random deviation e, all observed (x, y) points would fall exactly on

the population regression line. The inclusion of e in the model equation recognizes that points will deviate from the line by a random amount.

Figure 13.2 shows two observations in relation to the population regression line.

y

Observation when x = x1

(positive deviation)

Population regression

line (slope β)

e2

e1

Observation when x = x2

(negative deviation)

α = vertical

intercept

FIGURE 13.2

Two observations and deviations from

the population regression line.

x

0

0

x = x1

x = x2

Before we make an observation on y for any particular value of x, we are uncertain

about the value of e. It could be negative, positive, or even 0. Also, it might be quite

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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