Simple Linear Regression

Types of Regression Model

Regression Models

Simple(1 variable)

Multiple(2<variables)

Linear Non-LinearLinear Non-Linear

Simple Linear Regression Model

• The equation that describes how y is related to x and an error term is called the regression model.

• The simple linear regression model is:

0 and 1 are called parameters of the model.

– is a random variable called the error term.

y = 0 + 1x +

Simple Linear Regression Equation

The simple linear regression equation is:

EE((yy) = ) = 00 + + 11xx

• Graph of the regression equation is a straight line.

• 00 is the y intercept of the regression line.

• 11 is the slope of the regression line.

• E(y) is the expected value of y for a given x value.

Simple Linear Regression Equation Positive Linear RelationshipPositive Linear Relationship

EE((yy))

xx

Slope Slope 11

is positiveis positive

Regression lineRegression line

InterceptIntercept00

Simple Linear Regression EquationSimple Linear Regression Equation

Negative Linear RelationshipNegative Linear Relationship

EE((yy))

xx

Slope Slope 11

is negativeis negative

Regression lineRegression lineInterceptIntercept00

Simple Linear Regression EquationSimple Linear Regression Equation

No RelationshipNo Relationship

EE((yy))

xx

Slope Slope 11

is 0is 0

Regression lineRegression lineInterceptIntercept00

Estimated Simple Linear Regression EquationEstimated Simple Linear Regression Equation

The The estimated simple linear regression equationestimated simple linear regression equation is: is:

• The graph is called the estimated regression line.The graph is called the estimated regression line.• bb00 is the is the yy intercept of the line. intercept of the line.

• bb11 is the slope of the line. is the slope of the line.

• is the estimated value of is the estimated value of yy for a given for a given xx value. value.

0 1y b b x 0 1y b b x

yy

Estimation Process

Regression ModelRegression Modelyy = = 00 + + 11xx + +

Regression EquationRegression EquationEE((yy) = ) = 00 + + 11xx

Unknown ParametersUnknown Parameters00, , 11

Sample Data:Sample Data: x yx y

xx11 y y11

. .. . . .. . xxn n yynn

bb00 and and bb11

provide estimates ofprovide estimates of00 and and 11

EstimatedEstimatedRegression EquationRegression Equation

Sample StatisticsSample Statistics

bb00, , bb11

0 1y b b x 0 1y b b x

Simple Linear Regression Model

Armand’s Pizza Parlors is a chain of Italian-food restaurants located in a five-state area. Armand’s most successful locations are near college campuses. The managers believe that quarterly sales for these restaurants (denoted by y) are related positively to the size of the student population (denoted by x); that is, restaurants near campuses with a large student population tend to generate more sales than those located near campuses with a small student population. Using regression analysis, we can develop an equation showing how the dependent variable y is related to the independent variable x.

Least Squares MethodThe least squares method is a procedure for using sample data to find the estimated regression equation.

The slope of the estimated regression line

The y intercept of the estimated regression line

The estimated regression equation

Least Squares Method

• Least Squares Criterion

where:

yi = observed value of the dependent variable

for the ith observation

yi = estimated value of the dependent variable

for the ith observation

min (y yi i )2min (y yi i )2

^

The least squares method is a procedure for using sample data to find the estimated regression equation.