Linear Regression

1. Problems where one variable is used to predict the behavior of a second variable are called regression problems. If a linear function or line is used to approximate the data, then the technique is referred to as linear regression. A statistical method used to determine a unique linear function or line is based on least squares. Linear Regression

2. When determining the least-squares line, calculators often compute a real number r, called the correlation coefficient, where –1≤r≤1. When r is positive and near 1, low x-values correspond to low y-values and high x-values correspond to high y-values. For example, there is a positive correlation between years of education x and income y. More years of education correlate with higher income: positive correlation. Linear Regression

3. r = 1 0 < r < 1 Linear Regression

4. r = –1 –1 < r < 0 Linear Regression

5. If r ≈ 0, then there is little or no correlation between the data points. In this case, a linear function does not provide a suitable model. r ≈ 0 Linear Regression

6. Find the line of least-squares fit for the data points (1, 1), (2, 3), and (3, 4). What is the correlation coefficient? Plot the data and graph the line. Example 7 Determining a line of least-squares fit Solution Begin by entering the three data points into the STAT EDIT menu.

7. Select the LinReg option from the STAT CALC menu. The line (linear function) of least-squares is given by the formula Example 7 Determining a line of least-squares fit Solution (continued) Begin by entering the three data points into the STAT EDIT menu.

8. The correlation coefficient is r ≈ 0.98.Since r ≠ 1, the line does not provide an exact model of the data. Example 7 Determining a line of least-squares fit Solution (continued)