LeastSquares Regression Regression Line Model n It has

Least-Squares Regression

Regression Line (Model) n It has the form y = a + bx, n where b is the slope, the amount by which y changes when x increases by 1 unit n where 0 a is the intercept, the value of y when x =

n Slope: n Intercept: n The line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible.

Linear Regression n Purpose: To predict the value of a difficult to measure variable, Y, based on an easy to measure variable, X. n Examples n n n Predict state revenues Predict GPA based on SAT predict reaction time from blood alcohol level

Extrapolation n Extrapolation is the use of a regression line for prediction far outside the range of values of the independent variable x that you used to obtain the line. Such predictions are not accurate. n GRE consideration? Be Careful!

Interpreting Results n The regression line always passes through the point n The slope ‘says’ that along the regression line, a change of one standard deviation in x corresponds to a change of r standard deviations in y n When r = 1 or – 1 n the change in standard units is the same

Variation n The R squared value, , is the % of the variation of Y explained by the model. n The higher the value, the better the model.

No Straight Line? n What if the scatterplot shows a straight line model is not appropriate? n n Examples n n n Might see if some function of y is approximately linear in some function of x. Plot y versus ln(x) Plot 1/y versus 1/x If so, fit straight line model in terms of new variables.

Example n Let’s use the alcoholic beverage and recall data n How can we tell if it is reasonable to fit a linear regression model? n Let’s run the analysis and interpret the results