LEASTSQUARES REGRESSION 3 2 Role of s and

  • Slides: 10
Download presentation
LEAST-SQUARES REGRESSION 3. 2 Role of s and r 2 in Regression

LEAST-SQUARES REGRESSION 3. 2 Role of s and r 2 in Regression

REVIEW FROM LAST WEEK • Plot Scatterplot • Find Correlation and LSRL • Graph

REVIEW FROM LAST WEEK • Plot Scatterplot • Find Correlation and LSRL • Graph LSRL • Look at Residual Plot to determine if LSRL is a good fit

Standard Deviation of the Residuals • “typical” prediction error when using the least-squares regression

Standard Deviation of the Residuals • “typical” prediction error when using the least-squares regression line. If we use a least-squares regression line to predict the values of a response variable y from an explanatory variable x, the standard deviation of the residuals (s) is given by This value gives the approximate size of a “typical” prediction error (residual).

Example • From the Truck example: Interpretation: When we use the LSRL to predict

Example • From the Truck example: Interpretation: When we use the LSRL to predict the price of a Ford F-150 using the number of miles it has been driven, our predictions will typically be off by about $5740.

The Coefficient of Determination • tells us how well the least-squares regression line predicts

The Coefficient of Determination • tells us how well the least-squares regression line predicts values of the response y. The coefficient of determination r 2 is the fraction of the variation in the values of y that is accounted for by the least-squares regression line of y on x. We can calculate r 2 using the following formula: r 2 tells us how much better the LSRL does at predicting values of y than simply guessing the mean y for each value in the dataset.

Example (Truck example) The equation of the least-squares regression line is and the coefficient

Example (Truck example) The equation of the least-squares regression line is and the coefficient of determination is r 2 = 0. 8150. Interpretation: About 81. 5% of the variation in price is accounted for by the linear model relating price to mileage.

Example, p. 180 •

Example, p. 180 •

Example: p. 180 (a) Calculate and interpret the residual for South Carolina, which scored

Example: p. 180 (a) Calculate and interpret the residual for South Carolina, which scored 30. 1 points per game and had 11 wins. The predicted amount of wins for South Carolina is The residual for South Carolina is South Carolina won 1. 60 more games than expected, based on the number of points they scored per game.

Example: p. 180 (b) Is a linear model appropriate for these data? Explain. No

Example: p. 180 (b) Is a linear model appropriate for these data? Explain. No obvious pattern in the residual plot, so the linear model is appropriate. (c) Interpret the value s = 1. 24. When using the least-squares regression line with x = points per game to predict y = the number of wins, we will typically be off by about 1. 24 wins. (d) Interpret the value r 2 = 0. 88. About 88% of the variation in wins is accounted for by the linear model relating wins to points per game.

HW: Due on Friday • p. 195 # 55, 78

HW: Due on Friday • p. 195 # 55, 78

4 3 LeastSquares Regression 1 Regression Lines LeastSquares4 3 LeastSquares Regression 1 Regression Lines LeastSquares
Ordinary LeastSquares Outline Linear regression Geometry of leastsquaresOrdinary LeastSquares Outline Linear regression Geometry of leastsquares
Nonlinear LeastSquares Regularized LeastSquares Why nonlinear The modelNonlinear LeastSquares Regularized LeastSquares Why nonlinear The model
LeastSquares regression line A regression line is aLeastSquares regression line A regression line is a
LEASTSQUARES REGRESSION 3 2 Least Squares Regression LineLEASTSQUARES REGRESSION 3 2 Least Squares Regression Line
Regression Line Definition The leastsquares regression line ofRegression Line Definition The leastsquares regression line of
Objectives 2 3 Leastsquares regression p Regression linesObjectives 2 3 Leastsquares regression p Regression lines
LEASTSQUARES REGRESSION 3 2 Interpreting Computer Regression OutputLEASTSQUARES REGRESSION 3 2 Interpreting Computer Regression Output
Regression Line Definition The leastsquares regression line ofRegression Line Definition The leastsquares regression line of