Devore J.L., Berk K.N. Modern Mathematical Statistics with Applications

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(e.g., constant pressure and a fixed angle). The

accompanying table shows the two lifetimes

(min) obtained for each brand–surface combina-

tion. In addition,

PPP

ijk

¼ 11; 499; 492 and

ij

¼ 22; 982; 552.

Writing Surface

Brand

of Pen

123x

i..

1 709, 659 713, 726 660, 645 4112

2 668, 685 722, 740 692, 720 4227

3 659, 685 666, 684 678, 750 4122

4 698, 650 704, 666 686, 733 4137

.j.

5413 5621 5564 16,598

Carry out an appropriate ANOVA, and state your

conclusions.

54. The accompanying data was obtained in an experi-

ment to investigate whether compressive strength

of concrete cylinders depends on the type of cap-

ping material used or variability in different batches

(“The Effect of Type of Capping Material on the

Compressive Strength of Concrete Cylinders,”

Proceedings ASTM, 1958: 1166–1186). Each num-

ber is a cell total (x

ij.

)basedonK ¼ 3 observations.

Batch

Capping

Material

12345

1 1847 1942 1935 1891 1795

2 1779 1850 1795 1785 1626

3 1806 1892 1889 1891 1756

In addition,

PPP

ijk

¼ 16;815;853 and

ij

¼ 50;443;409. Obtain the ANOVA

table and then test at level .01 the hypotheses

versus H

, H

versus H

, and H

versus

, assuming that capping is a fixed effect and

batches is a random effect.

55. a. Show that Eð

i

 X



Þ¼a

, so that X

i

 X



is an unbiased estimator for a

(in the fixed

effects model).

b. With

¼ X

ij

 X

i

 X

j

þ X



, show that

is an unbiased estimator for g

(in the fixed

effects model).

56. Show how a 100(1 – a)% t CI for a

 a

can be

obtained. Then compute a 95% interval for a

– a

using the data from Example 11.16. [Hint: With

y ¼ a

– a

, the result of Exercise 55(a) indicates

how to obtain

y. Then compute V(

y) and s

and

obtain an estimate of s

by using

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

MSE

to esti-

mate s (which identifies the appropriate number

of df).]

57. When both factors are random in a two-way

ANOVA experiment with K replications per

combination of factor levels, the expected

mean squares are E MSEðÞ¼s

; E MSAðÞ¼s

þ JKs

, E MSB

ðÞ

¼ s

þ Ks

þ IKs

, and

E MSABðÞ¼s

þ Ks

a. What F ratio is appropriate for testing

: s

¼ 0 versus H

: s

> 0?

b. Answer part (a) for testing H

: s

¼ 0 versus

: s

> 0 and H

: s

¼ 0 versus

: s

> 0

Supplementary Exercises (58–70)

58. An experiment was carried out to compare flow

rates for four different types of nozzle.

a. Sample sizes were 5, 6, 7, and 6, respectively,

and calculations gave f ¼ 3.68. State and test

the relevant hypotheses using a ¼ .01.

b. Analysis of the data using a statistical com-

puter package yielded P-value ¼ .029. At

level .01, what would you conclude, and

why?

59. The article “Computer-Assisted Instruction Aug-

mented with Planned Teacher/Student Contacts”

(J. Exper. Ed., Winter 1980–1981: 120–126)

compared five different methods for teaching

descriptive statistics. The five methods were tra-

ditional lecture and discussion (L/D), pro-

grammed textbook instruction (R), programmed

text with lectures (R/L), computer instruction

(C), and computer instruction with lectures

(C/L). Forty-five students were randomly

608

CHAPTER 11 The Analysis of Variance

assigned, 9 to each method. After completing

the course, the students took a 1-h exam. In

addition, a 10-minute retention test was adminis-

tered 6 weeks later. Summary quantities are

given.

Exam Retention Test

Method



i



i

L/D 29.3 4.99 30.20 3.82

R 28.0 5.33 28.80 5.26

R/L 30.2 3.33 26.20 4.66

C 32.4 2.94 31.10 4.91

C/L 34.2 2.74 30.20 3.53

The grand mean for the exam was 30.82, and the

grand mean for the retention test was 29.30.

a. Does the data suggest that there is a difference

among the five teaching methods with respect

to true mean exam score? Use a ¼ .05.

b. Using a .05 significance level, test the null

hypothesis of no difference among the true

mean retention test scores for the five differ-

ent teaching methods.

60. Numerous factors contribute to the smooth run-

ning of an electric motor (“Increasing Market

Share Through Improved Product and Process

Design: An Experimental Approach,” Qual.

Engrg., 1991: 361–369). In particular, it is desir-

able to keep motor noise and vibration to a mini-

mum. To study the effect that the brand of

bearing has on motor vibration, five different

motor bearing brands were examined by instal-

ling each type of bearing on different random

samples of six motors. The amount of motor

vibration (measured in microns) was recorded

when each of the 30 motors was running. The

data for this study follows. State and test the

relevant hypotheses at significance level .05,

and then carry out a multiple comparisons analy-

sis if appropriate.

Mean

Brand 1: 13.1 15.0 14.0 14.4 14.0 11.6 13.68

Brand 2: 16.3 15.7 17.2 14.9 14.4 17.2 15.95

Brand 3: 13.7 13.9 12.4 13.8 14.9 13.3 13.67

Brand 4: 15.7 13.7 14.4 16.0 13.9 14.7 14.73

Brand 5: 13.5 13.4 13.2 12.7 13.4 12.3 13.08

61. An article in the British scientific journal Nature

(“Sucrose Induction of Hepatic Hyperplasia in

the Rat,” August 25, 1972: 461) reports on an

experiment in which each of five groups consist-

ing of six rats was put on a diet with a different

carbohydrate. At the conclusion of the experi-

ment, the DNA content of the liver of each rat

was determined (mg/g liver), with the following

results:

Carbohydrate



i

Starch 2.58

Sucrose 2.63

Fructose 2.13

Glucose 2.41

Maltose 2.49

a. Assuming also that

¼ 183:4, is the

true average DNA content affected by the

type of carbohydrate in the diet? Construct

an ANOVA table and use a .05 level of sig-

nificance.

b. Construct a t CI for the contrast

y ¼ m

ðm

þ m

Þ=4

which measures the difference between the

average DNA content for the starch diet and

the combined average for the four other diets.

Does the resulting interval include zero?

c. What is b for the test when true average DNA

content is identical for three of the diets and

falls below this common value by 1 standard

deviation (s) for the other two diets?

62. Four laboratories (1–4) are randomly selected

from a large population, and each is asked to

make three determinations of the percentage of

methyl alcohol in specimens of a compound

taken from a single batch. Based on the accom-

panying data, are differences among laboratories

a source of variation in the percentage of methyl

alcohol? State and test the relevant hypotheses

using significance level .05.

1: 85.06 85.25 84.87

2: 84.99 84.28 84.88

3: 84.48 84.72 85.10

4: 84.10 84.55 84.05

Supplementary Exercises 609

63. The critical flicker frequency (cff) is the highest

frequency (in cycles/sec) at which a person can

detect the flicker in a flickering light source. At

frequencies above the cff, the light source appears

to be continuous even though it is actually flicker-

ing. An investigation carried out to see whether

true average cff depends on iris color yielded the

following data (based on the article “The Effects

of Iris Color on Critical Flicker Frequency,” J.

Gen. Psych., 1973: 91–95):

Iris Color

1. Brown 2. Green 3. Blue

26.8 26.4 25.7

27.9 24.2 27.2

23.7 28.0 29.9

25.0 26.9 28.5

26.3 29.1 29.4

24.8 28.3

25.7

24.5

856

i

204.7 134.6 169.0



i

25.59 26.92 28.17

n ¼ 19 x



¼ 508:3

a. State and test the relevant hypotheses at signif-

icance level .05 by using the F table to obtain

an upper and/or lower bound on the P-value.

[Hint:

¼ 13; 659:67 and

CF ¼ 13; 598:36.]

b. Investigate differences between iris colors

with respect to mean cff.

64. Recall from Section 11.2 that if c

, c

, ..., c

are

numbers satisfying Sc

¼ 0 then Sc

¼ c

 +c

is called a contrast in the m

’s. Notice

that with c

¼ 1, c

¼1, c

¼  ¼ c

¼ 0,

¼ m

– m

, which implies that every pair-

wise difference between m

’s is a contrast (so is,

e.g., m

– .5m

). A method attributed to

Scheffe

gives simultaneous CIs with simultaneous

confidence level 100(1 – a)% for all possible

contrasts (an infinite number of them!). The inter-

val for Sc



i



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ðI  1ÞF

a;I1;nI

MSE

Using the critical flicker frequency data of Exer-

cise 63, calculate the Scheffe

intervals for the

contrasts m

– m

, m

– m

, m

– m

, and

.5m

+ .5 m

– m

(the last contrast compares

blue to the average of brown and green). Which

contrasts differ significantly from 0, and why?

65. Four types of mortars—ordinary cement mortar

(OCM), polymer impregnated mortar (PIM), resin

mortar (RM), and polymer cement mortar

(PCM)—were subjected to a compression test to

measure strength (MPa). Three strength observa-

tions for each mortar type are given in the article

“Polymer Mortar Composite Matrices for Mainte-

nance-Free Highly Durable Ferrocement” (J. Fer-

rocement, 1984: 337–345) and are reproduced

here. Construct an ANOVA table. Using a .05

significance level, determine whether the data sug-

gests that the true mean strength is not the same for

all four mortar types. If you determine that the true

mean strengths are not all equal, use Tukey’s

method to identify the significant differences.

OCM: 32.15 35.53 34.20

PIM: 126.32 126.80 134.79

RM: 117.91 115.02 114.58

PCM: 29.09 30.87 29.80

66. In single-factor ANOVA, suppose the x

’s are

“coded” by y

¼ cx

+d. How does the value of

the F statistic computed from the y

’s compare to

the value computed from the x

’s? Justify your

assertion.

67. In Example 11.10, subtract x

i

from each observa-

tion in the ith sample (i ¼ 1, ..., 6) to obtain a set

of 18 residuals. Then construct a normal probabil-

ity plot and comment on the plausibility of the

normality assumption.

68. The results of a study on the effectiveness of line

drying on the smoothness of fabric were summar-

ized in the article “Line-Dried vs. Machine-Dried

Fabrics: Comparison of Appearance, Hand, and

Consumer Acceptance” (Home Econ. Res. J.,

1984: 27–35). Smoothness scores were given for

nine different types of fabric and five different

drying methods: (1) machine dry, (2) line dry,

(3) line dry followed by 15-min tumble, (4) line

dry with softener, and (5) line dry with air move-

ment. Regarding the different types of fabric as

blocks, construct an ANOVA table.

a. Using a .05 significance level, test to see

whether there is a difference in the true mean

smoothness score for the drying methods.

610

CHAPTER 11 The Analysis of Variance

b. Make a plot like Figure 11.8 with fabric on the

horizontal axis. Discuss the result of part (a) in

terms of the plot.

c. Did the two methods involving the dryer yield

significantly smoother fabric compared to the

other three?

Drying method

Fabric 12345

Crepe 3.3 2.5 2.8 2.5 1.9

Double knit 3.6 2.0 3.6 2.4 2.3

Twill 4.2 3.4 3.8 3.1 3.1

Twill mix 3.4 2.4 2.9 1.6 1.7

Terry 3.8 1.3 2.8 2.0 1.6

Broadcloth 2.2 1.5 2.7 1.5 1.9

Sheeting 3.5 2.1 2.8 2.1 2.2

Corduroy 3.6 1.3 2.8 1.7 1.8

Denim 2.6 1.4 2.4 1.3 1.6

69. The water absorption of two types of mortar used

to repair damaged cement was discussed in the

article “Polymer Mortar Composite Matrices for

Maintenance-Free, Highly Durable Ferrocement”

(J. Ferrocement, 1984: 337–345). Specimens of

ordinary cement mortar (OCM) and polymer

cement mortar (PCM) were submerged for vary-

ing lengths of time (5, 9, 24, or 48 h), and water

absorption (% by weight) was recorded. With

mortar type as factor A (with two levels) and

submersion period as factor B (with four levels),

three observations were made for each factor level

combination. Data included in the article was used

to compute the sums of squares, which were SSA

¼ 322.667, SSB ¼ 35.623, SSAB ¼ 8.557, and

SST ¼ 372.113. Use this information to construct

an ANOVA table. Test the appropriate hypotheses

at a .05 significance level.

70. Four plots were available for an experiment to

compare clover accumulation for four different

sowing rates (“Performance of Overdrilled Red

Clover with Different Sowing Rates and Initial

Grazing Managements,” New Zeal. J. Exper.

Agric., 1984: 71–81). Since the four plots had

been grazed differently prior to the experiment

and it was thought that this might affect clover

accumulation, a randomized block experiment

was used with all four sowing rates tried on a

section of each plot. Use the given data to test

the null hypothesis of no difference in true mean

clover accumulation (kg DM/ha) for the different

sowing rates.

a. Test to see if the different sowing rates

make a difference in true mean clover accumu-

lation.

b. Make appropriate plots to go with your analy-

sis in (a): Make a plot like the one in Figure

11.8, make a normal plot of the residuals, and

plot the residuals against the predicted values.

Explain why, based on the plots, the assump-

tions do not appear to be satisfied for this data

set.

c. Repeat part (a) replacing the observations with

their natural logarithms.

d. Repeat the plots of (b) for the analysis in (c).

Do the logged observations appear to satisfy

the assumptions better?

e. Summarize your conclusions for this experi-

ment. Does mean clover accumulation increase

with increasing sowing rate?

Sowing Rate (kg/ha)

Plot 3.6 6.6 10.2 13.5

1 1155 2255 3505 4632

2 123 406 564 416

3 68 416 662 379

4 62 75 362 564

Bibliography

Miller, Rupert, Beyond ANOVA: The Basics of Applied

Statistics, Wiley, New York, 1986. An excellent

source of information about assumption checking

and alternative methods of analysis

Montgomery, Douglas, Design and Analysis of

Experiments (7th ed.), Wiley, New York, 2009.

An up-to-date presentation of ANOVA models

and methodology.

Kutner, Michael, Christopher Nachtsheim, John Neter,

and William Li, Applied Linear Statistical Models

(5th ed.), McGraw-Hill, New York, NY, 2005. The

second half of this book contains a well-presented

Bibliography 611

survey of ANOVA; the level is comparable to that

of the present text, but the discussion is more com-

prehensive, making the book an excellent reference

Ott, R. Lyman, and Michael Longnecker, An Introduc-

tion to Statistical Methods and Data Analysis

(6th ed.), Cengage, Belmont, CA, 2010. Includes

several chapters on ANOVA methodology that can

profitably be read by students desiring a nonmath-

ematical exposition; there is a good chapter on

various multiple comparison methods

612

CHAPTER 11 The Analysis of Variance

CHAPTER TWELVE

Regression

and Correlation

Introduction

The general objective of a regression analysis is to determine the relationship

between two (or more) variables so that we can gain information about one of

them through knowing values of the other(s). Much of mathematics is devoted to

studying variables that are

deterministically

related. Saying that

and

are related

in this manner means that once we are told the value of

, the value of

completely speciﬁed. For example, suppose we decide to rent a van for a day and

that the rental cost is $25.00 plus $.30 per mile driven. If we let

¼ the number

of miles driven and

¼ the rental charge, then

¼ 25 + .3

. If we drive the van

100 miles (

¼ 100), then

¼ 25 + .3(100) ¼ 55. As another example, if the

initial velocity of a particle is v

and it undergoes constant acceleration

, then

distance traveled ¼ y þ v

, where

¼ time.

There are many variables

and

that would appear to be related to each

other, but not in a deterministic fashion. A familiar example to many students is

given by variables

¼ high school grade point average (GPA) and

¼ college

GPA. The value of

cannot be determined just from knowledge of

, and two

different students could have the same

value but have very different

values.

Yet there is a tendency for those students who have high (low) high school GPAs

also to have high (low) college GPAs. Knowledge of a student’s high school GPA

should be quite helpful in enabling us to predict how that person will do in college.

Other examples of variables related in a nondeterministic fashion include

¼ age of a child and

¼ size of that child’s vocabulary,

¼ size of an engine in

cubic centimeters and

¼ fuel efﬁciency for an automobile equipped with that

engine, and

¼ applied tensile force and

¼ amount of elongation in a metal strip.

Regression analysis is the part of statistics that deals with investigation of the

relationship between two or more variables related in a nondeterministic fashion.

J.L. Devore and K.N. Berk, Modern Mathematical Statistics with Applications, Springer Texts in Statistics,

DOI 10.1007/978-1-4614-0391-3_12,

Springer Science+Business Media, LLC 2012

613

In this chapter, we generalize a deterministic linear relation to obtain a linear

probabilistic model for relating two variables

and

. We then develop procedures

for making inferences based on data obtained from the model, and obtain a

quantitative measure (the correlation coefﬁcient) of the extent to which the two

variables are related. Techniques for assessing the adequacy of any particular

regression model are then con sidered. We next introduce multiple regression

analysis as a way of relating

to two or more variables—for example, relating

fuel efﬁciency of an automobile to weight, engine size, number of cylinders, and

transmission type. The last section of the chapter shows how matrix algebra

techniques can be used to facilitate a concise and elegant development of regres-

sion procedures.

12.1

The Simple Linear and Logistic

Regression Models

The key idea in developing a probabilistic relationship between a dependent or

response variable y and an independent, explanatory,orpredictor variable x is to

realize that once the value of x has been fixed, there is still uncertainty in what the

resulting y value will be. That is, for a fixed value of x, we now think of the

dependent variable as being random. This random variable will be denoted by Y and

its observed value by y. For example, suppose an investigator plans a study to relate

y ¼ yearly energy usage of an industrial building (1000’s of BTUs) to x ¼ the shell

area of the building (ft

). If one of the buildings selected for the study has a shell

area of 25,000 ft

, the resulting energy usage might be 2,215,000 or it might be

2,348,000 or any one of a number of other possibilities. Since we don’t know a

priori what the value of energy usage will be (because usage is determined partly by

factors other than shell area), usage is regarded as a random variable Y.

We now relate the independent and dependent variables by an additive

model equation:

Y ¼ some particular deterministic function of x þ a random deviation

¼ f ðxÞþe

ð12:1Þ

The symbol e represents a random deviation or random “error” (random variable)

which is assumed to have mean value 0. Th is rv incorporates all variation in the

dependent variable due to factors other than x. Figure 12.1 shows the graph of a

particular f(x). Without the random deviation e, whenever x is fixed prior to making

an observation on the dependent variable, the resulting (x, y) point would fall

exactly on the graph. That is, y would be entirely determined by x. The role of

the random deviation e is to allow a non-deterministic relationship. Now if the

value of e is positive, the resulting (x, y) point falls above the graph of f(x), whereas

when e is negative, the resulting point falls below the graph. The assumption that e

has mean value 0 implies that we expect the point (x, y) to fall right on the graph,

but we virtually never see what we literally expect—the observed point will almost

always devi ate upward or downward from the graph.

614 CHAPTER 12 Regression and Correlation

How should the deterministic part of the model equation be selected?

Occasionally some sort of theoretical argument will suggest an appropriate choice

of f(x). However, in practice the specification of f(x) is almost always made by

obtaining sample data consisting of n (x, y) pairs. A picture of the resulting observa-

tions (x

, y

), (x

, y

), ...,(x

, y

), called a scatter plot, is then constructed. In this

scatter plot each (x

, y

) is represented as a point in a two-dimensional coordinate

system. The pattern of points in the plot should suggest an appropriate f(x).

Example 12.1 Visual and musculoskel etal problems associated with the use of visual display

terminals (VDTs) have become rather common in recent years. Some researchers

have focused on vertical gaze direction as a source of eye strain and irritation. This

direction is known to be closely related to ocular surface area (OSA), so a method

of measuring OSA is needed. The accompanying representative data on y ¼ OSA

(cm

) and x ¼ width of the palprebal fissure (i.e., the horizontal width of the eye

opening, in cm) is from the article “Analysis of Ocular Surface Area for Comfort-

able VDT Workstation Layout” (Ergonomics, 1996: 877–884). The order in which

observations were obtained was not given, so for convenience they are listed in

increasing order of x values.

Thus (x

, y

) ¼ (.40, 1.02), (x

, y

) ¼ (.57, 1.52), and so on. A MINITAB

scatter plot is shown in Figure 12.2; we used an option that produced a dotplot of both

the x values and y values individually along the right and top margins of the plot,

which makes it easier to visualize the distributions of the individual variables

positive

negative

(

x,y)

(

x,y)

Graph of

(

⎧

⎨

⎩

Figure 12.1 Observations resulting from the model equation (12.1)

i 1 23456789101112131415

.40 .42 .48 .51 .57 .60 .70 .75 .75 .78 .84 .95 .99 1.03 1.12

1.02 1.21 .88 .98 1.52 1.83 1.50 1.80 1.74 1.63 2.00 2.80 2.48 2.47 3.05

i 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

1.15 1.20 1.25 1.25 1.28 1.30 1.34 1.37 1.40 1.43 1.46 1.49 1.55 1.58 1.60

3.18 3.76 3.68 3.82 3.21 4.27 3.12 3.99 3.75 4.10 4.18 3.77 4.34 4.21 4.92

12.1 The Simple Linear and Logistic Regression Models 615

(histograms or boxplots are alternative options). Here are some things to notice about

the data and plot:

• Several observations have identical x values yet different y values (e.g.,

¼ x

¼ .75, but y

¼ 1.80 and y

¼ 1.74). Thus the value of y is not

determined solely by x but also by various other factors.

• There is a strong tend ency for y to increase as x increases. That is, larger values

of OSA tend to be associated with larger values of fissure width —a positive

relationship between the variables.

• It appears that the value of y could be predicted from x by finding a line that is

reasonably close to the points in the plot (the authors of the cited article

superimposed such a line on their plot). In other words, there is evidence of a

substantial (though not perfect) linear relationship between the two variables.

The horizontal and vertical axes in the scatter plot of Figure 12.2 intersect at

the point (0, 0). In many data sets, the values of x or y or the values of both variables

differ considerably from zero relative to the range(s) of the values. For example, a

study of how air conditioner efficiency is related to maximum daily outdoor

temperature might involve observations for temperatures ranging from 80



Fto

100



F. When this is the case, a more informative plot would show the appropriately

labeled axes intersecting at some point other than (0, 0).

Example 12.2 Forest growth and decline phenomena throughout the world have attracted consid-

erable public and scientific interest. The article “Relationships Among Crown

Condition, Growth, and Stand Nutrition in Seven Northern Vermont Sugarbushes”

(Canad. J. Forest Res., 1995: 386–397) included a scatter plot of y ¼ mean crown

dieback (%), one indicator of growth retardation, and x ¼ soil pH (higher pH

corresponds to more acidic soil), from which the following observations were taken:

x 3.3 3.4 3.4 3.5 3.6 3.6 3.7 3.7 3.8 3.8

y 7.3 10.8 13.1 10.4 5.8 9.3 12.4 14.9 11.2 8.0

x 3.9 4.0 4.1 4.2 4.3 4.4 4.5 5.0 5.1

y 6.6 10.0 9.2 12.4 2.3 4.3 3.0 1.6 1.0

Figure 12.2 Scatter plot from MINITAB for the data from Example 12.1, along

with dotplots of x and y values

■

616 CHAPTER 12 Regression and Correlation

Figure 12.3 shows two MINITAB scatter plots of this data. In Figure 12.3a,

MINITAB selected the scale for both axes. We obtained Figure 12.3b by specifying

minimum and maximum values for x and y so that the axes would intersect roughly

at the point (0, 0). The second plot is more crowded than the first one; such

crowding can make it more difficult to ascertain the general nature of any relation-

ship. For example, it can be more difficult to spot curvature in a crowded plot.

Large values of percentage dieback tend to be associated with low soil pH, a

negative or inverse relationship. Furthermore, the two variables appear to be at least

approximately linearly related, although the points would be spread out about any

straight line drawn through the plot.

■

A Linear Probabilistic Model

For a deterministic linear relationship y ¼ b

+ b

x, the slope coefficient b

is the

guaranteed increase in y when x increases by one unit and the intercept coefficient

is the value of y when x ¼ 0. A graph of y ¼ b

+ b

x is of course a straight line.

The slope gives the amount by which the line rises or falls when we move one unit

to the right, and the intercept is the height at which the line crosses the vertical axis.

For example, the line y ¼ 100 5x specifies an increase of 5 (i.e., a decrease of 5)

for each one-unit increase in x, and the vertical intercept of the line is 100. When a

scatter plot of bivariate data consisting of n (x, y) pairs shows a reasonably

substantial linear pattern, it is natural to specify f(x) in the model equation (12.1)

to be a linear function. Rather than assuming that the dependent variable itself is a

linear function of x, the model assumes that the expected value of Y is a linear

function of x. For any fixed x value, the observed value of Y will deviate by a

random amount from its expected value.

THE SIMPLE

LINEAR

REGRESSION

MODEL

There are parameters b

, b

, and s

such that for any fixed value of the

independent variable x, the dependent variable is related to x through the

model equation

Y ¼ b

þ b

x þ e

Figure 12.3 MINITAB scatter plots of data in Example 12.2

12.1 The Simple Linear and Logistic Regression Models 617