1. Regression

2. A regression line attempts to predict one variable based on the relationship with another variable (its correlation). The regression line is placed so that the error (distance from the line to each data point) is minimized. The placement of the regression line minimizes the total squared predictive error. (That way there are no negative values.) Chapter 7-Regression2

3. Prediction What is predictive error? The amount of error associated with the placement of a best fitting regression line. The placement of the regression line: Minimizes the total predictive error, and Minimizes the total squared predictive error 3Chapter 7-Regression

4. Progress Check 7.1 Chapter 7-Regression4 a)Predict the approximate rate of inflation, given an unemployment rate of 5 percent. b)Predict the approximate rate of inflation, given an unemployment rate of 15 percent. 5 10 15 20 20 15 10 5

5. Estimating regression and relationship Witte web demonstration Chapter 7-Regression5

6. Least Squares Regression Equation Y = bX + a Where Y = the predicted value X = a known value b = slope of the line a = Y-intercept Ability to predict an outcome for a variable, given a regression line and a value of a second paired variable. Chapter 7-Regression6

7. Calculating Least Squares Regression Page 159 1. Calculate SS x, SS y, and r for the data. 2. Substitute numbers into the formula below SS y b = SS x 3. Find the mean for X and mean for Y 4. Solve for a a = Y-(b)(X) 5. Solve for the predicted value Y’ = (b)(X) + a Chapter 7-Regression7 (r) √

8. Standard error of estimate This represents a special kind of standard deviation that reflects the magnitude of predictive error. It is the difference between known values and predicted values based on the regression equation. It is how much we over/under estimate a value based on the regression equation, which is related to the strength of the correlation. Chapter 7-Regression8

9. Calculation of standard error of estimate Page 162 Square root of the quantity of sum of squares for Y times one minus r squared divided by n minus 2. (n-2 because 2 paired variable results in n-2 degrees of freedom) SS y (1-r 2 ) S y|x = n-2 Chapter 7-Regression9 √

10. Assumptions Use of regression equation requires that the underlying relationship be linear. Use of the standard error of estimate assumes that except for chance, the dots in the original scatterplot will be dispersed equally about all segments of the regression line. (homoscedasticity) Chapter 7-Regression10

11. Progress Check 7.2 Page 160-1 a) Determine the least squares equation for predicting weekly reading time from educational level. b) Faith’s education level is 15. What is her predicted reading time? c) Keegan’s educational level is 11. What is his predicted reading time? Chapter 7-Regression11 Educational Level (X)Weekly Reading Time (Y) X = 13Y = 8 SS x = 25SS y = 50 R =.30

12. Calculate Standard Error of Estimate Calculate the Standard Error of Estimate using the data in 7.2 on page 160 Chapter 7-Regression12 Educational Level (X)Weekly Reading Time (Y) X = 13Y = 8 SS x = 25SS y = 50 R =.30

13. Correlation, Prediction, Error XYX2X2 Y2Y2 X*Y 54 46 14 23 45 24 Chapter 7-Regression13 X - mean number of cases of influenza in a month among employees Y - mean number of bacteria x 1 million on a door knob on front door

14. Regression toward the mean Extreme scores on multiple trials will tend toward the mean. The regression fallacy: accepting that regression toward the mean is real, rather than a chance effect. Chapter 7-Regression14

15. Regression toward the mean Tversky and Kahnemann Study of Israeli Air Force pilots in 1974 Some trainees were praised after good landings, while others were reprimanded after bad landings. On their next landings, praised trainees did more poorly and reprimanded trainees did better. Conclusion: Praise hinders but a reprimand helps performance! Chapter 7-Regression15