5.5 Least-Squares Regression

Calculate the coefficients for the least-squares regression line model.

• The simple linear regression model is fit to the data by minimizing the sum of the squares of the residuals. Because of this, the resulting equation is often called the least-squares regression line (LSRL) and is calculated using technology. This regression line will pass through the point ()xy, .
• The slope of the regression line, b, is calculated using technology.
• The y-intercept of the regression line, a, is calculated using technology.
• In simple linear regression, the correlation coefficient, r, is calculated using technology.
• In simple linear regression, the square of the correlation coefficient, r2 , is called the coefficient of determination. The value of r2 is the proportion of variation in the response variable that is explained by the linear relationship with the explanatory variable.

Interpret coefficients for the least-squares regression line model.

• The coefficients of the least-squares regression line model (line of best fit) are the slope, b, and the y-intercept, a, because they are based on a sample of values.
• The slope of the least-squares regression line can be interpreted as the predicted increase or decrease in the response variable for a oneunit increase in the explanatory variable, and it should be interpreted in context.
• The y-intercept in the least-squares regression line is the predicted value of the response variable when the explanatory variable is equal to 0, and it should be interpreted in context. Sometimes, the y-intercept of the line does not have a reasonable interpretation in context because x =0 might be beyond the interval of x-values used to determine the regression line (extrapolation). At other times, the y-intercept of the line does not have a logical interpretation in context because it might be a negative value for a response variable that has no negative values, such as height.