AP STATISTICSAP STATISTICS

LESSON 3 – 3 (DAY 2)LESSON 3 – 3 (DAY 2)

The role of r2 in regression

Essential Question: Essential Question:

How is the rHow is the r22 used to determine the used to determine the reliability of a linear regression line?reliability of a linear regression line?

To calculate r2.

To find the SST, the SSE and find the r2

from them.

Definitions and AbbreviationsDefinitions 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

ExercisesExercises

Small rSmall r22 and Large r and Large r22

Page 158: Example 3.10 SMALL r2

Page 160: Example 3.11 LARGE r2

rr2 2 in Regressionin 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

Facts about Least-squares Facts about Least-squares RegressionsRegressions

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.

Facts of RegressionFacts of Regression(continued)(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.