1. Section 9.2 Linear Regression

2. Section 9.2 Objectives Find the equation of a regression line Predict y-values using a regression equation

3. Regression lines After verifying that the linear correlation between two variables is significant, next we determine the equation of the line that best models the data (regression line). Can be used to predict the value of y for a given value of x. x y

4. Residuals Residual The difference between the observed y-value and the predicted y-value for a given x-value on the line. For a given x-value, d i = (observed y-value) – (predicted y-value) x y }d1}d1 }d2}d2 d3{d3{ d4{d4{ }d5}d5 d6{d6{ Predicted y-value Observed y-value

5. Regression line (line of best fit) The line for which the sum of the squares of the residuals is a minimum. The equation of a regression line for an independent variable x and a dependent variable y is ŷ = mx + b Regression Line Predicted y-value for a given x- value Slope y-intercept

6. The Equation of a Regression Line ŷ = mx + b where is the mean of the y-values in the data is the mean of the x-values in the data The regression line always passes through the point

7. Example: Finding the Equation of a Regression Line Find the equation of the regression line for the gross domestic products and carbon dioxide emissions data. GDP (trillions of $), x CO 2 emission (millions of metric tons), y 1.6428.2 3.6828.8 4.91214.2 1.1444.6 0.9264.0 2.9415.3 2.7571.8 2.3454.9 1.6358.7 1.5573.5

8. Solution: Finding the Equation of a Regression Line xyxyx2x2 y2y2 1.6428.2685.122.56183,355.24 3.6828.82983.6812.96686,909.44 4.91214.25949.5824.011,474,281.64 1.1444.6489.061.21197,669.16 0.9264.0237.60.8169,696 2.9415.31204.378.41172,474.09 2.7571.81543.867.29326,955.24 2.3454.91046.275.29206,934.01 1.6358.7573.922.56128,665.69 1.5573.5860.252.25328,902.25 Σx = 23.1Σy = 5554Σxy = 15,573.71Σx 2 = 67.35 Σy 2 = 3,775,842.76 Recall from section 9.1:

9. Solution: Finding the Equation of a Regression Line Σx = 23.1Σy = 5554 Σxy = 15,573.71 Σx 2 = 67.35Σy 2 = 3,775,842.76 Equation of the regression line

10. Solution: Finding the Equation of a Regression Line To sketch the regression line, use any two x-values within the range of the data and calculate the corresponding y-values from the regression line.

11. Example: Predicting y-Values Using Regression Equations The regression equation for the gross domestic products (in trillions of dollars) and carbon dioxide emissions (in millions of metric tons) data is ŷ = 196.152x + 102.289. Use this equation to predict the expected carbon dioxide emissions for the following gross domestic products. (Recall from section 9.1 that x and y have a significant linear correlation.) 1. 1.2 trillion dollars 2. 2.0 trillion dollars 3. 2.5 trillion dollars

12. Solution: Predicting y-Values Using Regression Equations ŷ = 196.152x + 102.289 1.1.2 trillion dollars When the gross domestic product is $1.2 trillion, the CO 2 emissions are about 337.671 million metric tons. ŷ =196.152(1.2) + 102.289 ≈ 337.671 2.2.0 trillion dollars When the gross domestic product is $2.0 trillion, the CO 2 emissions are 494.595 million metric tons. ŷ =196.152(2.0) + 102.289 = 494.593

13. Solution: Predicting y-Values Using Regression Equations 3.2.5 trillion dollars When the gross domestic product is $2.5 trillion, the CO 2 emissions are 592.669 million metric tons. ŷ =196.152(2.5) + 102.289 = 592.669 Prediction values are meaningful only for x-values in (or close to) the range of the data. The x-values in the original data set range from 0.9 to 4.9. So, it would not be appropriate to use the regression line to predict carbon dioxide emissions for gross domestic products such as $0.2 or $14.5 trillion dollars.

14. Section 9.2 Summary Found the equation of a regression line Predicted y-values using a regression equation