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Slides by JOHN LOUCKS St. Edward’s University

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Slides by

JOHN

LOUCKS

St. Edward’s

University

- Simple Linear Regression Model

- Least Squares Method

- Coefficient of Determination

Simple Linear Regression

- 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

- 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.

Simple Linear Regression Equation

- 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

- 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

- is the estimated value of y for a given x value.

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.

Sample Data:

x y

x1 y1

. .

. .

xnyn

Estimated

Regression Equation

Sample Statistics

b0, b1

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 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

- Slope for the Estimated Regression Equation

where:

xi = value of independent variable for ith

observation

yi = value of dependent variable for ith

observation

Least Squares Method

- y-Intercept for the Estimated Regression Equation

Simple Linear Regression

- 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.

Simple Linear Regression

- Example: Reed Auto Sales

Number of

TV Ads (x)

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

Using Excel’s Chart Tools for

Scatter Diagram & Estimated Regression Equation

- Excel Worksheet (showing data)

continue

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

Using Excel’s Chart Tools for

Scatter Diagram & Estimated Regression Equation

- Producing a Scatter Diagram

Step 7 Select the Horizontal Axis Title and replace it

with TV Ads

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

Using Excel’s Chart Tools for

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

Reed Auto Sales Estimated Regression Line

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

Coefficient of Determination

- The coefficient of determination is:

r2 = SSR/SST

where:

SSR = sum of squares due to regression

SST = total sum of squares

Coefficient of Determination

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.

Using Excel to Compute the

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

Reed Auto Sales Estimated Regression Line

Using Excel to Compute the

Coefficient of Determination

where:

b1 = the slope of the estimated regression

equation

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

rxy = +.9366