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Chapter Eight

Chapter Eight

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Chapter Eight

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  1. Chapter Eight Linear Regression

  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. Chapter 8 - 2

  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. Chapter 8 - 3

  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 Chapter 8 - 4

  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) Chapter 8 - 5

  6. Regression Lines Having Different Slopes and Y Intercepts Chapter 8 - 6

  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) Chapter 8 - 7

  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 Chapter 8 - 8

  9. Computing the Y-Intercept The formula for the Y-intercept (a) is Chapter 8 - 9

  10. Summary of Computations for the Linear Regression Equation Chapter 8 - 10

  11. Describing Errors in Prediction Chapter 8 - 11

  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 Chapter 8 - 12

  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 Chapter 8 - 13

  14. Interpreting the Standard Error of the Estimate Chapter 8 - 14

  15. Assumption 1 of Linear Regression The first assumption of linear regression is that the data are homoscedastic Chapter 8 - 15

  16. Homoscedasticity Homoscedasticity occurs when the Y scores are spread out to the same degree at every X Chapter 8 - 16

  17. Heteroscedasticity Heteroscedasticity occurs when the spread in Y is not equal throughout the relationship Chapter 8 - 17

  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 Chapter 8 - 18

  19. Scatterplot Showing Normal Distribution of Y Scores at Each X Chapter 8 - 19

  20. The Strength of a Relationship and Prediction Error Chapter 8 - 20

  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 Chapter 8 - 21

  22. Scatterplot of a Strong Relationship Chapter 8 - 22

  23. Scatterplot of a Weak Relationship Chapter 8 - 23

  24. Computing the Proportion of Variance Accounted For Chapter 8 - 24

  25. Proportion of Variance Accounted For The proportion of varianceaccounted 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. Chapter 8 - 25

  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 Chapter 8 - 26

  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 Chapter 8 - 27

  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 Chapter 8 - 28

  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 Chapter 8 - 29

  30. Example 1 For the following data set, calculate the linear regression equation. Chapter 8 - 30

  31. Example 1Linear Regression Equation Chapter 8 - 31

  32. Example 1Linear Regression Equation Chapter 8 - 32

  33. Example 1Linear Regression Equation Chapter 8 - 33

  34. Example 2Predicted Y Value Using the linear regression equation from example 1, determine the predicted Y score for X = 4. Chapter 8 - 34

  35. Example 3 Using the same data set, calculate the standard error of the estimate. Chapter 8 - 35

  36. Example 3Standard Error of the Estimate Chapter 8 - 36

  37. Example 4 Using the same data set, calculate the proportion of variance accounted for and the proportion of variance not accounted for. Chapter 8 - 37

  38. Example 4—Proportion of Variance Accounted For 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 . Chapter 8 - 38

  39. Key Terms • linear regression line • multiple correlation coefficient • multiple regression equation • predicted Y score • predictor variable • proportion of the variance accounted for coefficient of alienation coefficient of determination criterion variable heteroscedasticity homoscedasticity linear regression equation Chapter 8 - 39

  40. Key Terms (Cont’d) slope standard error of the estimate variance of the Y scores around Y intercept Chapter 8 - 40