Chapter 12a Simple Linear Regression

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Chapter 12a Simple Linear Regression

• Simple Linear Regression Model
• Least Squares Method
• Coefficient of Determination
• Model Assumptions

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

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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.

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Simple Linear Regression Equation

• Positive Linear Relationship

Regression line
Intercept b0
Slope b1 is positive
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Simple Linear Regression Equation

• Negative Linear Relationship

Regression line
Intercept b0
Slope b1 is negative
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Simple Linear Regression Equation

• No Relationship

Regression line
Intercept b0
Slope b1 is 0
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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.

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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
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Least Squares Method

• Least Squares Criterion

where yi observed value of the dependent
variable for the ith observation
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Least Squares Method

• Slope for the Estimated Regression Equation

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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
n total number of observations
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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.

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Example Reed Auto Sales

• Simple Linear Regression

Number of TV Ads
Number of Cars Sold
1 3 2 1 3
14 24 18 17 27
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Estimated Regression Equation

• Slope for the Estimated Regression Equation
• y-Intercept for the Estimated Regression Equation
• Estimated Regression Equation

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Using Excel to Develop a Scatter Diagram
andCompute the Estimated Regression Equation

• Formula Worksheet (showing data)

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Using Excel to Develop a Scatter Diagram
andCompute the Estimated Regression Equation

• Producing a Scatter Diagram

Step 1 Select cells B1C6
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 gt
Step 4 When the Chart Source Data dialog box
appears Click Next gt
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Using Excel to Develop a Scatter Diagram
andCompute 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 gt
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Using Excel to Develop a Scatter Diagram
andCompute 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
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Using Excel to Develop a Scatter Diagram
andCompute 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
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Scatter Diagram and Trend Line
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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
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Coefficient of Determination

• The coefficient of determination is

r2 SSR/SST
where SSR sum of squares due to
regression SST total sum of squares
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Coefficient of Determination
r2 SSR/SST 100/114 .8772
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.
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Using Excel to Computethe 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
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Using Excel to Computethe Coefficient of
Determination

• Value Worksheet (showing r 2)

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Sample Correlation Coefficient
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Sample Correlation Coefficient
rxy .9366
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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.