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Understanding r2 in Linear Regression Analysis

Learn how to determine the reliability of a linear regression line using r2 - coefficient of determination. Explore SST, SSE, LSRL, and key facts about least-squares regressions.

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Understanding r2 in Linear Regression Analysis

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  1. AP STATISTICSLESSON 3 – 3 (DAY 2) The role of r2 in regression

  2. Essential Question: How is the r2 used to determine the reliability of a linear regression line? • To calculate r2. • To find the SST, the SSE and find the r2 from them.

  3. Definitions and Abbreviations r2 = coefficient of determination ( The proportion of the total sample variability that is explained by the least-squares regression of y on x. LSRL – Least squares regression line. SST – (Total Sum of Squares) SST = ∑ ( y – y )2 SSE – (Sum of squares of errors) SSE = ∑ ( y – ŷ)2

  4. ExercisesSmall r2 and Large r2 Page 158: Example 3.10 SMALL r2 Page 160: Example 3.11 LARGE r2

  5. r2 in Regression The coefficient of determination r2, is the fraction of the variation in the values of y that is explained by least-squares regression of y on x. r2 = SST - SSE SST

  6. Facts about Least-squares Regressions • Fact 1: The distinction between explanatory and response variable is essential in regression. • Fact 2: There is a close connection between correlation and the slope of the least-squares line. A change of one standard deviation of x corresponds to a change of r standard deviations in y.

  7. Facts of Regression(continued) • Fact 3. The least-squares regression line always passes through the point ( x, y ). • Fact 4. The square of the correlation, r2, is the fraction of the variation in the values of y that is explained by the least-squares regression of y on x.

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