Least-Squares Regression Section 3.3

Regression Line A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes. A regression line can be used to the predict the value of y for a given value of x. Regression requires that we have an explanatory variable and a response variable.

Mathematical Model The least-squares regression line, or LSRL, is a model for the data. If the data shows a linear trend, it would be appropriate to fit an LSRL to the data. For some data, a curve is a more appropriate model.

The least-squares regression line No line will pass exactly through all the points… The LSRL is used to predict y from x, so we want a line that is as close as possible to the points in the vertical direction. We want a regression line that makes the vertical distances of the points in a scatterplot from the line as small as possible.

LSRL The least-squares regression line of y on x is the line that makes the sum of the squares of the vertical distances of the data points from the line as small as possible. The equation is The slope is And intercept is

Slope The slope of a regression line is usually important for the interpretation of the data. The slope is the rate of change, the amount of change in y-hat when x increases by 1.

Facts about least-squares regression line In the regression setting you must know clearly which variable is explanatory. Because the slope of the LSRL involves the standard deviations of x and y, a change of one standard deviation in x corresponds to a change of r standard deviations in y.

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. When you report a regression, give r2 as a measure of how successful the regression was in explaining the response