1. Warm Up Feel free to share data points for your activity. Determine if the direction and strength of the correlation is as agreed for this class, for the given r value. 1. r=-.89 2. r=.5 3. r=-.2 4. r=0 5. r=-0.72 AP Statistics, Section 3.3, Part 1 1

2. Warm Up Determine if the direction and strength of the correlation is as agreed for this class, for the given r value. 1. r=-.89 2. r=.5 3. r=-.2 4. r=0 5. r=-0.72 AP Statistics, Section 3.3, Part 1 2

3. Section 3.3 Linear Regression AP Statistics

4. AP Statistics, Section 3.3, Part 1 4 Linear Regression It would be great to be able to look at multi- variable data and reduce it to a single equation that might help us make predictions “What would be the predicted number of wins for a team with a 4.0 ERA?” WinsWins Team ERA 30 Major League teams in 2003

5. AP Statistics, Section 3.3, Part 1 5 Linear Regression Model equation  y is for exact value y is for predicted value

6. AP Statistics, Section 3.3, Part 1 6 The Least-Square Regression Finds the best fit line by trying to minimize the areas formed by the difference of the real data from the predicted data.

7. AP Statistics, Section 3.3, Part 1 7 The Least-Square Regression (LSRL) Finds the best fit line by trying to minimize the areas formed by the difference of the real data from the values predicted by the model. Residual = y-y  residuals=0 Largest residual may be an outlier

8. AP Statistics, Section 3.3, Part 1 8 The Least-Square Regression Statisticians use a slightly different version of “slope- intercept” form. There is one point guaranteed to be on the LSRL. (x,y) Know for test equations for test

9. AP Statistics, Section 3.3, Part 1 9 The Least-Square Regression (using the formula sheet) Statisticians use a slightly different version of “slope- intercept” form. There is one point guaranteed to be on the LSRL. (x,y)

10. AP Statistics, Section 3.3, Part 1 10 Predicting Model To put the regression line on the graph use the Statistics:Eq:RegEQ from the Vars menu to put the Y 1 equation. Then you can use Trace or Table or Y 1 to find response values that correspond to particular experimental values.

11. AP Statistics, Section 3.3, Part 1 11 Fact about least-square regression Make sure you know which is the explanatory (x) variable and which is the response (y) variable.  Switching them gets a different regression line.

12. AP Statistics, Section 3.3, Part 1 12 Fact about least-square regression Regression line always goes through the point (x, y) The coefficient of correlation (r) explains the strength of the linear relationship The square of the correlation (r 2 ) is the variation in the values of y that is explained by x. ___%(r 2 ) of the variation of ______ (y) is explained by _____ (x). Coefficient of explanation

13. AP Statistics, Section 3.3, Part 1 13 r 2 “coefficient of explanation” In the regression of ERA vs. WINS, we find a r 2 value of.4512 We say “45% of the variation in WINS can be explained by ERA”

14. AP Statistics, Section 3.3, Part 1 14 Outliers vs. Influential Data An outlier is an observation outside the overall pattern If an observation is influential it has a large effect on the regression line. Removing the observation markedly changes the calculation. Outliers Influential points

15. AP Statistics, Section 3.3, Part 1 15 Outliers vs. Influential Data Child 19 and 18 are outliers Child 18 is an influential point  From solid to dashed line

16. AP Statistics, Section 3.3, Part 1 16 Residuals It is important to note that the observed value almost never match the predicted values exactly The difference between the observed value and predicted has a special name: residual Observed Value (y): 5.3 ERA, 43 Wins Predicted Value ( ): 5.3 ERA 67.03 Wins Residual: 43-67.03=-24.03

17. AP Statistics, Section 3.3, Part 1 17 Residuals Observed Value (y): 5.3 ERA, 43 Wins Predicted Value ( ): 5.3 ERA 67.03 Wins Residual: 43-67.03=-24.03

18. AP Statistics, Section 3.3, Part 1 18 Residual Plots You can plot the residuals to see if the there is any trends with the quality of the predictive model

19. AP Statistics, Section 3.3, Part 1 19 Residual Plots This residual shows no tendencies. It is equally bad throughout. This suggests that the original relationship is linear.

20. AP Statistics, Section 3.3, Part 1 20 Not Linear

21. AP Statistics, Section 3.3, Part 1 21 “Well Distributed”=Linear

22. AP Statistics, Section 3.3, Part 1 22 Assignment Section 3.3 Exercises: 3.38(table 1 is on pg.127), 3.40, 3.42, 3.43, 3.46, 3.47, 3.49, 3.53, 3.55, 3.57, 3.61 Chapter 3 Test on

23. AP Statistics, Section 3.3, Part 1 23 Assignment Section 3.3 Exercises: 3.38(table 1 is on pg.127), 3.40, 3.42, 3.43, 3.46, 3.47, 3.49, 3.53, 3.55, 3.57, 3.61 Chapter 3 Test on

24. AP Statistics, Section 3.3, Part 1 24 Assignments to end the chapter Chapter Review: 3.63, 3.67, 3.71, 3.73, 3.75, 3.77 Summary Chapter 3 Test

25. AP Statistics, Section 3.3, Part 1 25 Assignments to end the chapter Chapter 3 Activity (due ?) Chapter Review: 3.63, 3.67, 3.71, 3.73, 3.75, 3.77 Summary Chapter 3 Test

26. AP Statistics, Section 3.3, Part 1 26 Assignments to end the chapter Section 3.3 Exercises: 3.38(table 1 is on pg.127), 3.40, 3.42, 3.43, 3.46, 3.47, 3.49, 3.53, 3.55, 3.57, 3.61 (due Friday) Chapter 3 Activity (due Friday) Chapter Review: 3.63, 3.67, 3.71, 3.73, 3.75, 3.77(?) Summary (?) Chapter 3 Test on ? (we will start chapter 4 ?)