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# Slides by JOHN LOUCKS St. Edward’s University - PowerPoint PPT Presentation

Slides by JOHN LOUCKS St. Edward’s University. Chapter 14, Part A Simple Linear Regression. Simple Linear Regression Model. Least Squares Method. Coefficient of Determination. Simple Linear Regression. Managerial decisions often are based on the

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JOHN

LOUCKS

St. Edward’s

University

Chapter 14, Part ASimple Linear Regression

• Simple Linear Regression Model

• Least Squares Method

• Coefficient of Determination

• Managerial decisions often are based on the

relationship between two or more variables.

• Regression analysis can be used to develop an

equation showing how the variables are related.

• The variable being predicted is called the dependent

variable and is denoted by y.

• The variables being used to predict the value of the

dependent variable are called the independent

variables and are denoted by x.

• Simple linear regression involves one independent

variable and one dependent variable.

• The relationship between the two variables is

approximated by a straight line.

• Regression analysis involving two or more

independent variables is called multiple regression.

• 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:

y = b0 + b1x +e

where:

• b0 and b1 are called parameters of the model,

• e is a random variable called the error term.

• The simple linear regression equation is:

E(y) = 0 + 1x

• Graph of the regression equation is a straight line.

• b0 is the y intercept of the regression line.

• b1 is the slope of the regression line.

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

E(y)

x

Simple Linear Regression Equation

• Positive Linear Relationship

Regression line

Intercept

b0

Slope b1

is positive

E(y)

x

Simple Linear Regression Equation

• Negative Linear Relationship

Intercept

b0

Regression line

Slope b1

is negative

E(y)

x

Simple Linear Regression Equation

• No Relationship

Intercept

b0

Regression line

Slope b1

is 0

Estimated Simple Linear Regression Equation

• The estimated simple linear regression equation

• The graph is called the estimated regression line.

• b0 is the y intercept of the line.

• b1 is the slope of the line.

x y

x1 y1

. .

. .

xnyn

Estimated

Regression Equation

Sample Statistics

b0, b1

Estimation Process

Regression Model

y = b0 + b1x +e

Regression Equation

E(y) = b0 + b1x

Unknown Parameters

b0, b1

b0 and b1

provide estimates of

b0 and b1

yi = estimated value of the dependent variable

for the ith observation

Least Squares Method

• Least Squares Criterion

where:

yi = observed value of the dependent variable

for the ith observation

_

x = mean value for independent variable

y = mean value for dependent variable

Least Squares Method

• Slope for the Estimated Regression Equation

where:

xi = value of independent variable for ith

observation

yi = value of dependent variable for ith

observation

• y-Intercept for the Estimated Regression Equation

• Example: Reed Auto Sales

Reed Auto periodically has a special week-long sale.

As part of the advertising campaign Reed runs one or

more television commercials during the weekend

preceding the sale. Data from a sample of 5 previous

sales are shown on the next slide.

• Example: Reed Auto Sales

Number of

Number of

Cars Sold (y)

1

3

2

1

3

14

24

18

17

27

Sx = 10

Sy = 100

• Slope for the Estimated Regression Equation

• y-Intercept for the Estimated Regression Equation

• Estimated Regression Equation

Scatter Diagram & Estimated Regression Equation

• Excel Worksheet (showing data)

Using Excel’s Chart Tools for

Scatter Diagram & Estimated Regression Equation

• Producing a Scatter Diagram

Step 1 Select cells B1:C6

Step 2 Click the Insert tab on the Excel ribbon

Step 3 In the Charts group, click Scatter

Step 4 When the list of scatter diagram subtypes appears:

Click Scatter with only Markers

Step 5 In the Chart Layouts group, click Layout 1

Step 6 Select the Chart Title and replace it with Reed

Auto Sales Estimated Regression Equation

Scatter Diagram & Estimated Regression Equation

• Producing a Scatter Diagram

Step 7 Select the Horizontal Axis Title and replace it

Step 8 Select the Vertical Axis Title and replace it with

Cars Sold

Step 9 Right click on the legend and click Delete

Step 10 Position the mouse pointer over any Vertical

Axis Major Gridline in the scatter diagram and

right-click to display a list of options and then

choose Delete

Scatter Diagram & Estimated Regression Equation

• Adding the Trendline

Step 11 Position the mouse pointer over any data point

in the scatter diagram and right-click to display

a list of options

Step 12 Choose Add Trendline

Step 13 When the Format Trendline dialog box appears:

Select Trendline Options and then

Choose Linear from the Trend/Regression

Type list

Choose Display Equation on chart

Click Close

Using Excel’s Chart Tools for

Scatter Diagram & Estimated Regression Equation

• Relationship Among SST, SSR, SSE

SST = SSR + SSE

where:

SST = total sum of squares

SSR = sum of squares due to regression

SSE = sum of squares due to error

• The coefficient of determination is:

r2 = SSR/SST

where:

SSR = sum of squares due to regression

SST = total sum of squares

r2 = SSR/SST = 100/114 = .8772

The regression relationship is very strong; 87.72%

of the variability in the number of cars sold can be

explained by the linear relationship between the

number of TV ads and the number of cars sold.

Coefficient of Determination

• Displaying the Coefficient of Determination

Step 1 Position the mouse pointer over any data point

in the scatter diagram and right-click to display

a list of options

Step 2 Choose Add Trendline

Step 3 When the Trendline dialog box appears:

Select Trendline Options and then

Choose Display R-squared value on chart

Click Close

Using Excel to Compute the

Coefficient of Determination

b1 = the slope of the estimated regression

equation

Sample Correlation Coefficient

The sign of b1 in the equation is “+”.

Sample Correlation Coefficient

rxy = +.9366