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Lecture 8

Lecture 8 (week 10)

7 Pages
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Department
Quantitative Methods
Course Code
QMS 202
Professor
Clare Chua

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3/17/2011
1
Simple Linear Regression
Chapter 13
Page 578
Regression Analysis
Regression Analysis(RA) is a statistical forecasting model that is
concerned with describing and evaluating the relationshipbetween a given
variable (usually called the dependent variable, denoted as Y) and one or
more other variable (usually known as the independent/exploratory
variable, denoted as X) .
RA can predict the outcome of a given key business indicator (dependent
variable) based on the interactions of other related business drivers
(independent /exploratory variables)
Analyzes the relationship between two variables, X and Y.
The
relationship
can be described as afunction of alinear (
straight
line
)
The
relationship
can
be
described
as
a
function
of
a
linear
(
straight
-
line
)
equation <called linear regression> “simple linear regression”
Y=Eo+E1X
Eo
E1
linear regression linelinear regression lineY
X
intercept
slope
Example
1. For elementary school children, it is possible to predict
a student’s reading ability level by measuring the height
of the student.
Reading_ability= f(height) <Simple Linear Regression>
2. If we assume the value of an automobile decreases by a
constant amount each year after its purchase, and for
each mile it is driven, the following linear function would
One X to predict Y
predict its value (the dependent variable on the left side
of the equal sign) as a function of the two independent
variables which are age and miles:
Car_value = f(age, miles) <Multiple Linear Regression>
where Car_value, the dependent variable, is the value
of the car, age is the age of the car, and miles is the
number of miles that the car has been driven.
MORE than One X to predict Y
Learning Objectives
1. How to use regression analysis to predict
the value of a dependent variable
based on an independent variable
2. The meanin
g
of the re
g
ression
gg
coefficients, boand b
3. To make inferences about the slope and
correlation coefficient
4. To estimate mean values and predict
individual values
Dependent Variable (Notation: Y)
The variable you wish to predict
Independent Variable (Notation: X)
Variable used to make the prediction
Simple Linear Regression
–A singlesingle numerical independent variable X is used to predict the
numerical dependent variable Y
Multiple Regression
–Use severalseveral independent variables to predict a numerical
de
p
endent variable Y.
p
One variable XCalled the independent or explanatory
variable
can be used
to
explain” (forecast, predict….)
a second
variable
YCalled the dependent or response variable
When changes in the variable X leads to predictable change in the variable Y
then we sayX can be used to explain Y”
Example
For elementary school children, it is possible
to predict a student’s reading ability level
by measuring the height of the student.
In the regression terms, we could say that
for elementary school children
there isadirect relationship between a
-
there
is
a
direct
relationship
between
a
student’s height and reading ability level”
or
- “a student’s reading level ability can be
explained (forecasted) by the student’s
height”
www.notesolution.com
3/17/2011
2
Does this mean that we should conclude
that taller children are better readers?
NO!
Since for the elementary school children
height is usually a good predictor of age
•a
e is a
ood
redictor of
rade level
and finally reading level will be strongly related
to grade level
So while height may allow us to predict reading
level for elementary school students we should
not conclude that we should stretch out the
students to make them better readers.
Simple Linear Regression
Regression analysis allows you to identify the type of type of
relationshiprelationship that exists between a dependent variable
(X) and an independent variable (Y)
The simplest relationship is the straight line or linear
relationship
YYSee Figure 13.2 page 579
(textbook) for the different
types of relationships
x
Eo
Slope=E1
x
Eo
Slope=E1
Positive (Direct) Linear RelationshipNegative (Inverse) Linear
Relationship
..
..
.
..
.
.
..
.
.
.
.
TYPES OF RELATIONSHIPS
Regression analysis allows you to identify the type of relationshiptype of relationship
that exists between a dependent variable (X) and an independent
variable (Y)
The simplest relationship is the straight line or linear relationship
Y
“Rule of Thumb” concepts
XYX,Y
Type of
Relationship
Increases Increases Move in the
same
Direct
relationship
d
d
x
Eo
No Relationship
..
.
...
.
.
same
direction
relationship
d
ecreases
d
ecreases
Increases decreases Move in the
opposite
direction
Inverse
relationship
decreases Increases
Increases Cannot tellNo
apparent
relationship
No
relationship
decreases Cannot tell
To predict the straight-line (linear) model
y
X
Eo
Slope=E1
called the regression coefficient
Recall that when the graph of y versus x tends to be in the straight line, we
say that there is a linear relationship
...
.
.
..
0xy10
X
Y intercept Slope of the line
Random error in Y for each
observation i that occurs. It is also called
the residual
The equation of the population least squares line:
The equation of the sample least squares line:
exbby10
Note: In general we will work with the sample
least square line
y
X
Eo
Slope=E1
..
.
..
.
.
The e
q
uation of the
p
o
p
ulation least s
q
uares line:
For a given x, there
are usually many
values of y
0xy10
exbby10
qpp q
The equation of the sample least squares line:
xbby
ˆ10
The equation of the forecasting line is:
is the predicted value of average of y for a given x Given x
Multiple y values
To predict the straight-line (linear) model
Y
X
bo
Slope=b1
Recall that when the graph of y versus x tends to be in the straight line, we
say that there is a linear relationship
...
.
.
..xbby
ˆ10
X
In practice, there are often many y values for a particular x value.
In some cases, we concentrate on explaining / forecasting”
www.notesolution.com

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Description
3/17/2011 Regression Analysis Regression Analysis(RA)is a statistical forecasting model that is concerned with describing and evaluating theip between a given variable (usually called the dependent variable, denoted as Y) and one or more other variable (usually known as thent/exploratory variable, denoted as X) . RA can predict the outcome of a given key business indicator (dependent Simple Linear Regression variablebased on the interactions of other related business drivers (independent /exploratory variables) Analyzes the relationship between two variables, X and Y. The eatonship an bedessced asa uncconofa nearrstaghttne))in equation simple linear regression Chapter 13 Y Y=- +- X lnearegresson lnein o 1 Page 578 -1slope -o X intercept Example 1. For elementary school children, it is possible to predict a students reading ability level by measuring the height Learning Objectives of the student. One X to predict Y Reading_ability = f(height) 1. How to use regression analysis to predict the value of a dependent variable 2. If we assume the value of an automobile decreases by a constant amount each year after its purchase, and for based on an independent variable each mile it is driven, the following linear function would 2. The meanin g of the reg gression predict its value (the dependent variable on the left side coefficients, b and b of the equal sign) as a function of the two independent o variables which are age and miles: 3. To make inferences about the slope and MORE than One X to predict Y Car_value = f(age, miles) correlation coefficient where Car_value, the dependent variable, is the value 4. To estimate mean values and predict of the car, age is the age of the car, and miles is the individual values number of miles that the car has been driven. Dependent Variable (Notation: Y) The variable you wish to predict Example Independent Variable (Notation: X) For elementary school children, it is possible Variable used to make the prediction Simple Linear Regression to predict a students reading ability level A single numerical independent variable X is used to predict the by measuring the height of the student. numerical dependent variable Y In the regression terms, we could say that Multiple Regression Use sevveall independent variables to predict a numerical for elementary school children dependent variable Y. -therre is aa diirecttrellattonnshhip bbettweeeen aa students height and reading ability level One variable X Called the independent or explanatory variable or can be used explain (forecast, predict.) to - a students reading level ability can be a second Y Called the dependent or response variable explained (forecasted) by the students variable height When changes in the variable X leads to predictable change in the variable Y then we say X can be used to explain Y 1 www.notesolution.com3/17/2011 Simple Linear Regression Does this mean that we should conclude Regression analysis allows you to identify the type off that taller children are better readers? relatonnship that exists between a dependent variable NO! (X) and an independent variable (Y) The simplest relationship is the straight line or linear Since for the elementary school children relationship height is usually a good predictor of age Y Y See Figure 13.2 page 579 types of relationshipsferent a ge is a ggood predicp tor of grade level g and finally reading level will be strongly related . to grade level . . . -o . . So while height may allow us to predict reading . . . level for elementary school students we should . . . . not conclude that we should stretch out the Slope=1 . . -o Slope=1 students to make them better readers. x Positive (Direct) Linear RelationNegative (Inverse) Linear Relationship TYPES OF RELATIONSHIPS Recall that when the graph of y versus x tends to be in the straight line, we Regression analysis allows you to identype offeatonsshpip say that there is a linear relationship that exists between a dependent variable (X) and an independent y To predict the straight-line (linear) model variable (Y) . The simplest relationship is the straight line or linear relationship . . . Rule of Thumb concepts . . Slope=1 X Y ,Y XType of
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