2. REGRESSION Predict future scores on Y based on measured scores on X Predictions are based on a correlation from a sample where both X and Y were measured

3. The Regression Equation Equation is linear: y’ = bx + a y’ = predicted score on y x = measured score on x b = slope a = y-intercept

4. The Regression Line X Y low high o o o o o o o o o o o o o o o o o o o o o o o o o o o

5. slope: y-intercept:

6. Example: Compute the regression equation for predicting Exam 2 scores from Exam 1 scores, using these sample data. StudentExam1Exam2 19786 28295 37479 48995 59390

7. STEP 1: Compute the slope (b) STEP 2: Compute the y-intercept (a) STEP 3: Write the regression equation.

8. Using the Equation If Exam1 = 82: If Exam1 = 90:

9. Least-Squares Solution Draw the regression line to minimize squared error in prediction. Error in prediction = difference between predicted y and actual y Positive and negative errors are both important

10. The Equation with Standard Scores Replace x and y with z X and z Y : z Y = bz X + a and the y-intercept becomes 0: z Y = bz X and the slope becomes r: z Y = rz X

11. Assumptions for Using Regression Linear relationship between variables Normal distributions homoscedasticity - y scores are spread out the same degree for every x score Predict for the same population from which you sampled

12. Standard Error of the Estimate Index of how far off predictions are expected to be Larger r means smaller standard error Average distance of y scores from predicted y scores

13. Calculating Standard Error of the Estimate

14. Example: The r between Exam 1 and Exam 2 was +.36. The standard deviation for Exam 2 was 6.03. What is the standard error of the estimate for predicting Exam 2 scores (y) from Exam 1 scores (x) ?