Linear Regression 9.2 486 CHAPTER 9 Correlation and Regression What You Should Learn How to find the equation of a regression line How to predict y-values using a regression equation Regression Lines Applications of Regression Lines Regression Lines After verifying that the linear correlation between two variables is significant, the next step is to determine the equation of the line that best models the data. This line is called a regression line, and its equation can be used to predict the value of y for a given value of x. Although many lines can be drawn through a set of points, a regression line is determined by specific criteria. Consider the scatter plot and the line shown below. For each data point, di represents the difference between the observed y@value and the predicted y@value for a given x@value. These differences are called residuals and can be positive, negative, or zero. When the point is above the line, di is positive. When the point is below the line, di is negative. When the observed y@value equals the predicted y@value, di = 0. Of all possible lines that can be drawn through a set of points, the regression line is the line for which the sum of the squares of all the residuals Σdi 2 Sum of the squares of the residuals is a minimum. x d1 d2 d3 d4 d5 di Predicted y-value Observed y-value d = (observed y-value) − (predicted y-value) For a given x-value, y A regression line, also called a line of best fit, is the line for which the sum of the squares of the residuals is a minimum. DEFINITION In algebra, you learned that you can write an equation of a line by finding its slope m and y@intercept b. The equation has the form y = mx + b. Recall that the slope of a line is the ratio of its rise over its run and the y@intercept is the y@value of the point at which the line crosses the y@axis. It is the y@value when x = 0. For instance, the graph of y = 2x + 1 is shown in the figure at the right. The slope of the line is 2 and the y@intercept is 1. In algebra, you used two points to determine the equation of a line. In statistics, you will use every point in the data set to determine the equation of the regression line. x 1 1 2 3 4 5 6 2 3 4 5 6 y = 2x + 1 2 1 m = = 2 b = 2(0) + 1 = 1 y 2 1 Study Tip When determining the equation of a regression line, it is helpful to construct a scatter plot of the data to check for outliers, which can greatly influence a regression line. You should also check for gaps and clusters in the data. For help with slope-intercept form of the equation of a line, see Integrated Review at MyLab Statistics
RkJQdWJsaXNoZXIy NjM5ODQ=