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Chapter 6 Prediction, Residuals, Influence Some remarks: Residual = Observed Y – Predicted Y Residuals are errors.

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Chapter 6 Prediction, Residuals, Influence Example: X: Age in months Y: Height in inches X: 18 19 20 21 22 23 24 Y: 29.9 30.3 30.7 3131.38 31.45 31.9

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Chapter 6 Prediction, Residuals, Influence Linear Model: Height = 25.2 +.271 * Age Examples Age = 24 months, Observed Height = 31.9 Predicted Height = 31.704 Residual = 31.9 – 31.704 =.196

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Chapter 6 Prediction, Residuals, Influence Age = 30 years months Predicted Height ~ 10 ft!! Residual = BIG! Be aware of Extrapolation!

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Chapter 6 Prediction, Residuals, Influence

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Chapter 7 Correlation and Coefficient of Determination How strong is the linear relationship between two quantitative variables X and Y?

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Chapter 7 Correlation and Coefficient of Determination Answer: Use scatterplots Compute the correlation coefficient, r. Compute the coefficient of determination, r^2.

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Chapter 7 Correlation and Coefficient of Determination Properties of Correlation coefficient r is a number between -1 and 1 r = 1 or r = -1 indicates a perfect correlation case where all data points lie on a straight line r > 0 indicates positive association r < 0 indicates negative association r value does not change when units of measurement are changed (correlation has no units!) Correlation treats X and Y symmetrically. The correlation of X with Y is the same as the correlation of Y with X

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Chapter 7 Correlation and Coefficient of Determination r is an indicator of the strength of linear relationship between X and Y strong linear relationship for r between.8 and 1 and -.8 and -1: moderate linear relationship for r between.5 and.8 and -.5 and -.8: weak linear relationship for r between.-.5 and.5 It is possible to have an r value close to 0 and a strong non-linear relationship between X and Y. r is sensitive to outliers.

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Chapter 7 Correlation and Coefficient of Determination

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How do we compute r? r = Sxy/(Sqrt(Sxx)*Sqrt(Syy)) Example: X: 6 10 14 19 21 Y: 5 3 7 8 12 Compute: Sxy = 72, Sxx = 154 and Syy = 46 Hence r = 72/(Sqrt(154)*Sqrt(46)) =.855

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Chapter 7 Correlation and Coefficient of Determination r^2: Coefficient of Determination r^2 is between 0 and 1. The closer r^2 is to 1, the stronger the linear relationship between X and Y r^2 does not change when units of measurement are changed r^2 measures the strength of linear relatioship

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Chapter 7 Correlation and Coefficient of Determination Some Remarks Quantitative variable condition: Do not apply correlation to categorical variables Correlation can be misleading if the relationship is not linear Outliers distort correlation dramatically. Report corrlelation with/without outliers.

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Chapter 7 Correlation and Coefficient of Determination

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