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## Chapter Eight

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**Chapter Eight**Linear Regression**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**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**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**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**Regression Lines Having Different Slopes and Y Intercepts**Chapter 8 - 6**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**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**Computing the Y-Intercept**The formula for the Y-intercept (a) is Chapter 8 - 9**Summary of Computations for the Linear Regression Equation**Chapter 8 - 10**Describing Errors in Prediction**Chapter 8 - 11**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**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**Interpreting the Standard**Error of the Estimate Chapter 8 - 14**Assumption 1 of Linear Regression**The first assumption of linear regression is that the data are homoscedastic Chapter 8 - 15**Homoscedasticity**Homoscedasticity occurs when the Y scores are spread out to the same degree at every X Chapter 8 - 16**Heteroscedasticity**Heteroscedasticity occurs when the spread in Y is not equal throughout the relationship Chapter 8 - 17**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**Scatterplot Showing Normal Distribution of Y Scores at Each**X Chapter 8 - 19**The Strength of a Relationship**and Prediction Error Chapter 8 - 20**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**Scatterplot of a Strong Relationship**Chapter 8 - 22**Scatterplot of a Weak Relationship**Chapter 8 - 23**Computing the Proportion of**Variance Accounted For Chapter 8 - 24**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**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**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**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**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**Example 1**For the following data set, calculate the linear regression equation. Chapter 8 - 30**Example 1Linear Regression Equation**Chapter 8 - 31**Example 1Linear Regression Equation**Chapter 8 - 32**Example 1Linear Regression Equation**Chapter 8 - 33**Example 2Predicted Y Value**Using the linear regression equation from example 1, determine the predicted Y score for X = 4. Chapter 8 - 34**Example 3**Using the same data set, calculate the standard error of the estimate. Chapter 8 - 35**Example 3Standard Error of the Estimate**Chapter 8 - 36**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**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**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**Key Terms (Cont’d)**slope standard error of the estimate variance of the Y scores around Y intercept Chapter 8 - 40