Presentation on theme: "Chapter 12a Simple Linear Regression"— Presentation transcript:
1 Chapter 12a Simple Linear Regression Simple Linear Regression ModelLeast Squares MethodCoefficient of DeterminationModel 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 andan error term is called the regression model.The simple linear regression model is:y = b0 + b1x +ewhere: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 + 1xGraph 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 RelationshipE(y)xRegression lineInterceptb0Slope b1is positive
6 Simple Linear Regression Equation Negative Linear RelationshipE(y)xInterceptb0Regression lineSlope b1is negative
7 Simple Linear Regression Equation No RelationshipE(y)xRegression lineInterceptb0Slope b1is 0
8 Estimated Simple Linear Regression Equation The estimated simple linear regression equationThe 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 EquationE(y) = b0 + b1xUnknown Parametersb0, b1Sample Data:x yx y1xn ynb0 and b1provide estimates ofEstimatedRegression EquationSample Statisticsb0, b1
10 Least Squares Method Least Squares Criterion where: yi = observed value of the dependent variablefor the ith observation^yi = estimated value of the dependent variablefor the ith observation
11 Least Squares MethodSlope for the Estimated Regression Equation
12 Least Squares Method y-Intercept for the Estimated Regression Equation where:xi = value of independent variable for ithobservationyi = value of dependent variable for ithobservation_x = mean value for independent variable_y = mean value for dependent variablen = total number of observations
13 Example: Reed Auto Sales Simple Linear RegressionReed Auto periodically hasa special week-long sale.As part of the advertisingcampaign Reed runs one ormore television commercialsduring the weekend preceding the sale. Data from asample of 5 previous sales are shown on the next slide.
14 Example: Reed Auto Sales Simple Linear RegressionNumber ofTV AdsNumber ofCars Sold1321424181727
15 Estimated Regression Equation Slope for the Estimated Regression Equationy-Intercept for the Estimated Regression EquationEstimated 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 DiagramStep 1 Select cells B1:C6Step 2 Select the Chart WizardStep 3 When the Chart Type dialog box appears:Choose XY (Scatter) in the Chart type listChoose Scatter from the Chart sub-type displayClick Next >Step 4 When the Chart Source Data dialog box appearsClick Next >
18 Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation Producing a Scatter DiagramStep 5 When the Chart Options dialog box appears:Select the Titles tab and thenDelete Cars Sold in the Chart title boxEnter TV Ads in the Value (X) axis boxEnter Cars Sold in the Value (Y) axis boxSelect the Legend tab and thenRemove the check in the Show Legend boxClick Next >
19 Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation Producing a Scatter DiagramStep 6 When the Chart Location dialog box appears:Specify the location for the new chartSelect Finish to display the scatter diagram
20 Using Excel to Develop a Scatter Diagram and Compute the Estimated Regression Equation Adding the TrendlineStep 1 Position the mouse pointer over any datapoint and right click to display the ChartmenuStep 2 Choose the Add Trendline optionStep 3 When the Add Trendline dialog box appears:On the Type tab select LinearOn the Options tab select the Displayequation on chart boxClick OK
22 Coefficient of Determination Relationship Among SST, SSR, SSESST = SSR SSEwhere:SST = total sum of squaresSSR = sum of squares due to regressionSSE = sum of squares due to error
23 Coefficient of Determination The coefficient of determination is:r2 = SSR/SSTwhere:SSR = sum of squares due to regressionSST = 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 beexplained by the linear relationship between thenumber of TV ads and the number of cars sold.
25 Using Excel to Compute the Coefficient of Determination Producing r 2Step 1 Position the mouse pointer over any datapoint in the scatter diagram and right clickStep 2 When the Chart menu appears:Choose the Add Trendline optionStep 3 When the Add Trendline dialog box appears:On the Options tab, select the DisplayR-squared value on chart boxClick 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 regressionequation
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 forall values of the independent variable.3. The values of are independent.4. The error is a normally distributed randomvariable.