1. Chapter 12a Simple Linear Regression
Simple Linear Regression Model
Least Squares Method
Coefficient of Determination
Model Assumptions

2. Regression may be the most widely used statistical technique in the social and natural sciences—as well as in business

3. 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:
y = b0 + b1x +e
where:
b0 and b1 are called parameters of the model,
e is a random variable called the error term.

4. 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.

5. Simple Linear Regression Equation
Positive Linear Relationship
E(y) x
Regression line
Intercept b0
Slope b1
is positive

6. Simple Linear Regression Equation
Negative Linear Relationship
E(y) x
Intercept b0
Regression line
Slope b1
is negative

7. Simple Linear Regression Equation
No Relationship
E(y) x
Regression line
Intercept b0
Slope b1
is 0

8. 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.
is the estimated value of y for a given x value.

9. Estimation Process Regression Model y = b0 + b1x +e
Regression Equation
E(y) = b0 + b1x
Unknown Parameters
b0, b1
Sample Data:
x y
x y1
xn yn
b0 and b1
provide estimates of
Estimated
Regression Equation
Sample Statistics
b0, b1

10. 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

11. Least Squares Method
Slope for the Estimated Regression Equation

12. Least Squares Method y-Intercept for the Estimated Regression Equation
where:
xi = value of independent variable for ith
observation
yi = value of dependent variable for ith
observation _
x = mean value for independent variable _
y = mean value for dependent variable
n = total number of observations

13. Example: Reed Auto Sales
Simple Linear Regression
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.

14. Example: Reed Auto Sales
Simple Linear Regression
Number of
TV Ads
Number of
Cars Sold 1 3 2 14 24 18 17 27

15. Estimated Regression Equation
Slope for the Estimated Regression Equation
y-Intercept for the Estimated Regression Equation
Estimated Regression Equation

16. Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation
Formula Worksheet (showing data)

17. Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation
Producing a Scatter Diagram
Step 1 Select cells B1:C6
Step 2 Select the Chart Wizard
Step 3 When the Chart Type dialog box appears:
Choose XY (Scatter) in the Chart type list
Choose Scatter from the Chart sub-type display
Click Next >
Step 4 When the Chart Source Data dialog box appears
Click Next >

18. Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation
Producing a Scatter Diagram
Step 5 When the Chart Options dialog box appears:
Select the Titles tab and then
Delete Cars Sold in the Chart title box
Enter TV Ads in the Value (X) axis box
Enter Cars Sold in the Value (Y) axis box
Select the Legend tab and then
Remove the check in the Show Legend box
Click Next >

19. Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation
Producing a Scatter Diagram
Step 6 When the Chart Location dialog box appears:
Specify the location for the new chart
Select Finish to display the scatter diagram

20. Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation
Adding the Trendline
Step 1 Position the mouse pointer over any data
point and right click to display the Chart
menu
Step 2 Choose the Add Trendline option
Step 3 When the Add Trendline dialog box appears:
On the Type tab select Linear
On the Options tab select the Display
equation on chart box
Click OK

21. Scatter Diagram and Trend Line

22. Coefficient of Determination
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

23. Coefficient of Determination
The coefficient of determination is:
r2 = SSR/SST
where:
SSR = sum of squares due to regression
SST = total sum of squares

24. Coefficient of Determination
r2 = SSR/SST = 100/114 =
The regression relationship is very strong; 88%
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.

25. Using Excel to Compute the Coefficient of Determination
Producing r 2
Step 1 Position the mouse pointer over any data
point in the scatter diagram and right click
Step 2 When the Chart menu appears:
Choose the Add Trendline option
Step 3 When the Add Trendline dialog box appears:
On the Options tab, select the Display
R-squared value on chart box
Click OK

26. Using Excel to Compute the Coefficient of Determination
Value Worksheet (showing r 2)

27. Sample Correlation Coefficient
where:
b1 = the slope of the estimated regression
equation

28. Sample Correlation Coefficient
The sign of b1 in the equation is “+”.
rxy =

29. Assumptions About the Error Term e
1. The error  is a random variable with mean of zero.
2. The variance of  , denoted by  2, is the same for
all values of the independent variable.
3. The values of  are independent.
4. The error  is a normally distributed random
variable.