1. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter Eight Linear Regression

2. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 2 Linear Regression Linear regression is a statistical procedure that uses relationships to predict unknown Y scores based on the X scores from a correlated variable.

3. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 3 Linear Regression Line The linear regression line is the straight line that summarizes the linear relationship in the scatterplot by, on average, passing through the center of the Y scores at each X.

4. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 4 The symbol stands for a predicted Y score Each is our best prediction of the Y score at a corresponding X, based on the linear relationship that is summarized by the regression line Predicted Y Scores

5. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 5 Slope and Intercept The slope is a number that indicates how slanted the regression line is and the direction in which it slants The Y intercept is the value of Y at the point where the regression line intercepts, or crosses, the Y axis (that is, when X equals 0)

6. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 6 Regression Lines Having Different Slopes and Y Intercepts

7. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 7 The Linear Regression Equation The linear regression equation indicates the predicted Y values are equal to the slope ( b ) times a given X value and this product then is added to the Y intercept ( a )

8. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 8 Computing the Slope The formula for the slope ( b ) is Since we usually first compute r, the values of the elements of this formula already are known

9. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 9 Computing the Y -Intercept The formula for the Y -intercept ( a ) is

10. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 10 Summary of Computations for the Linear Regression Equation

11. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 11 Describing Errors in Prediction

12. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 12 The variance of the Y scores around is the average squared difference between the actual Y scores and their corresponding predicted scores The formula for defining the variance of the Y scores around is The Variance of Y Scores Around

13. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 13 The standard error of the estimate is the clearest way to describe the “average” error when using to predict Y scores. The Standard Error of the Estimate

14. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 14 Interpreting the Standard Error of the Estimate

15. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 15 Assumption 1 of Linear Regression The first assumption of linear regression is that the data are homoscedastic

16. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 16 Homoscedasticity Homoscedasticity occurs when the Y scores are spread out to the same degree at every X

17. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 17 Heteroscedasticity Heteroscedasticity occurs when the spread in Y is not equal throughout the relationship

18. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 18 Assumption 2 of Linear Regression The second assumption of linear regression is that the Y scores at each X form an approximately normal distribution

19. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 19 Scatterplot Showing Normal Distribution of Y Scores at Each X

20. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 20 The Strength of a Relationship and Prediction Error

21. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 21 As the strength of the relationship increases, the actual Y scores are closer to their corresponding scores, producing less prediction error and smaller values of and. The Strength of a Relationship

22. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 22 Scatterplot of a Strong Relationship

23. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 23 Scatterplot of a Weak Relationship

24. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 24 Computing the Proportion of Variance Accounted For

25. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 25 Proportion of Variance Accounted For The proportion of variance accounted for is the proportional improvement in the accuracy of our predictions produced by using a relationship to predict Y scores, compared to our accuracy when we do not use the relationship.

26. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 26 When we do not use the relationship, we use the overall mean of the Y scores as everyone’s predicted Y The error here is the difference between the actual Y scores and the we predict they got When we do not use the relationship to predict scores, our error is Proportion of Variance Accounted For

27. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 27 When we do use the relationship, we use the corresponding as determined by the linear regression equation as our predicted value The error here is the difference between the actual Y scores and the that we predict they got When we do use the relationship to predict scores, our error is Proportion of Variance Accounted For

28. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 28 The computational formula for the proportion of variance in Y that is accounted for by a linear relationship with X is. Proportion of Variance Accounted For

29. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 29 The computational formula for the proportion of variance in Y that is not accounted for by a linear relationship with X is. Proportion of Variance Not Accounted For

30. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 30 X Y 18 26 36 45 51 63 Example 1 For the following data set, calculate the linear regression equation.

31. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 31 Example 1 Linear Regression Equation

32. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 32 Example 1 Linear Regression Equation

33. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 33 Example 1 Linear Regression Equation

34. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 34 Example 2 Predicted Y Value Using the linear regression equation from example 1, determine the predicted Y score for X = 4.

35. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 35 X Y 18 26 36 45 51 63 Example 3 Using the same data set, calculate the standard error of the estimate.

36. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 36 Example 3 Standard Error of the Estimate

37. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 37 X Y 18 26 36 45 51 63 Example 4 Using the same data set, calculate the proportion of variance accounted for and the proportion of variance not accounted for.

38. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Chapter 8 - 38 The proportion of variance in Y that is accounted for by a linear relationship with X is. The proportion of variance in Y that is not accounted for by a linear relationship with X is. Example 4—Proportion of Variance Accounted For

39. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms coefficient of alienation coefficient of determination criterion variable heteroscedasticity homoscedasticity linear regression equation Chapter 8 - 39 linear regression line multiple correlation coefficient multiple regression equation predicted Y score predictor variable proportion of the variance accounted for

40. © 2011 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use. Key Terms (Cont’d) Chapter 8 - 40 slope standard error of the estimate variance of the Y scores around Y intercept