Regression Chapter 6 I Introduction to Regression Figure 1. Girl’s basketball team (Data from Ch. 5, Table 1)

II Criterion for the Line of Best Fit A. Predicting Y from X 2. Line of best fit minimizes the sum of the squared prediction errors

3. Errors in predicting Y from X

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5. Illustration of Y intercept, aY.X, and slope of the best fitting line, bY.X

Table 1. Height and Weight of Girl’s Basketball Team 1 7.0 140 .64 289 13.6 2 6.5 130 .09 49 2.1 3 6.5 140 .09 289 5.1 4 6.5 130 .09 49 2.1 5 6.5 120 .09 9 –0.9 6 6.0 120 .04 9 0.6 7 6.0 130 .04 49 –1.4 8 6.0 110 .04 169 2.6 9 5.5 100 .49 529 16.1 10 5.5 110 .49 169 9.1

B. Computation of Line of Best Fit: Predicting Y from X  

1. Predicted weight for girl whose height is Xi = 6.5 C. Predicting X from Y  

1. Error in predicting X from Y

2. Predicted height for girl whose weight is Yi = 130 D. Comparison of Two Regression Equations

E. Two Regression Lines

F. Relationships Between r and the Two Regression Slopes

G. Predicted Value of Yi When r = 0 1. Alternative form of the regression equation

A. Comparison of SY.X & Standard Deviation (S) III Standard Error of Estimate (SY.X) A. Comparison of SY.X & Standard Deviation (S)

B. Alternative Formula for SY.X 1. Maximum value of SY.X occurs when r = 0 2. Minimum value of SY.X occurs when r = 1

2. Descriptive Application of SY.X Figure 2. Approximately 68.27% of the Y scores fall within Yi ± SY.X

IV. Assumptions Associated with Regression IV Assumptions Associated with Regression and the Standard Error of Estimate A. Regression 1. Relationship between X and Y is linear 2. X and Y are quantitative variables B. Standard Error of Estimate 1. Relationship between X and Y is linear 3. Homoscedasticity

V Multiple Regression A. Regression Equation for k Predictors B. Example with n = 5 Subjects and k = 2 Predictors

Observed Predictor Predictor Predicted Prediction Table 2. Multiple Regression Example with Two Predictors Observed Predictor Predictor Predicted Prediction Subject Score One Two Score Error __________________________________________________ 1 3 4 3 3.90 -0.90 2 1 2 6 1.02 -0.02 3 2 1 4 1.70 0.30 4 4 6 5 3.75 0.25 5 6 5 1 5.63 0.37 ___________________________________________________

C. Multiple regression equation D. Simple Regression Equations

Table 3. Correlation Matrix for Data in Table 1 ______________________________________ Variable Variable Y X1 X2 Y 1.000 .777 –.797 X1 1.000 –.338 X2 1.000

E. Regression Plane for Data in Table 2 Figure 3. (a) Predicted scores fall on the surface of the plane (b) Prediction errors fall above or below the surface of the plane

VI Multiple Correlation (R) A. Multiple Correlation for Data in Table 2

1. R2 for the multiple correlation data with two B. Coefficient of Multiple Determination (R2) 1. R2 for the multiple correlation data with two predictors is R2 = (.962)2 = .93 2. Coefficient of determination for the best predictor, X2, is r2 = (–.797)2 = .64 3. Coefficient of determination for the worst predictor, X1, is r2 = (.777)2 = .60 C. The problem of multicollinearity