1. Linear Regression 1 Section 9.2

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

3. Regression lines 3 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 4 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 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 Predicted y- value for a given x- value Slope y-intercept

6. The Equation of a Regression Line 6 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 7 Find the equation of the regression line for the advertising expenditures and company sales data. Advertising expenses, ($1000), x Company sales ($1000), y 2.4225 1.6184 2.0220 2.6240 1.4180 1.6184 2.0186 2.2215

8. Solution: Finding the Equation of a Regression Line 8 xyxyx2x2 y2y2 2.4225 1.6184 2.0220 2.6240 1.4180 1.6184 2.0186 2.2215 540 294.4 440 624 252 294.4 372 473 5.76 2.56 4 6.76 1.96 2.56 4 4.84 50,625 33,856 48,400 57,600 32,400 33,856 34,596 46,225 Σx = 15.8Σy = 1634Σxy = 3289.8Σx 2 = 32.44Σy 2 = 337,558 Recall from section 9.1:

9. Solution: Finding the Equation of a Regression Line 9 Σx = 15.8Σy = 1634Σxy = 3289.8Σx 2 = 32.44Σy 2 = 337,558 Equation of the regression line

10. Solution: Finding the Equation of a Regression Line 10 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. x Advertising expenses (in thousands of dollars) Company sales (in thousands of dollars) y

11. Example: Using Technology to Find a Regression Equation 11 Use a technology tool to find the equation of the regression line for the Old Faithful data. Duration x Time, y Duration x Time, y 1.8563.7879 1.82583.8385 1.9623.8880 1.93564.189 1.98574.2790 2.05574.389 2.13604.4389 2.3574.4786 2.37614.5389 2.82734.5586 3.13764.692 3.27774.6391 3.6577

12. Solution: Using Technology to Find a Regression Equation 12 5 50 100 1

13. Example: Predicting y-Values Using Regression Equations 13 The regression equation for the advertising expenses (in thousands of dollars) and company sales (in thousands of dollars) data is ŷ = 50.729x + 104.061. Use this equation to predict the expected company sales for the following advertising expenses. (Recall from section 9.1 that x and y have a significant linear correlation.) 1. 1.5 thousand dollars 2. 1.8 thousand dollars 3. 2.5 thousand dollars

14. Solution: Predicting y-Values Using Regression Equations 14 ŷ = 50.729x + 104.061 1. 1.5 thousand dollars When the advertising expenses are $1500, the company sales are about $180,155. ŷ =50.729(1.5) + 104.061 ≈ 180.155 2.1.8 thousand dollars When the advertising expenses are $1800, the company sales are about $195,373. ŷ =50.729(1.8) + 104.061 ≈ 195.373

15. Solution: Predicting y-Values Using Regression Equations 15 3. 2.5 thousand dollars When the advertising expenses are $2500, the company sales are about $230,884. ŷ =50.729(2.5) + 104.061 ≈ 230.884 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 1.4 to 2.6. So, it would not be appropriate to use the regression line to predict company sales for advertising expenditures such as 0.5 ($500) or 5.0 ($5000).

16. Section 9.2 Summary 16 Found the equation of a regression line Predicted y-values using a regression equation