1. © The McGraw-Hill Companies, Inc., 2000 Business and Finance College Principles of Statistics Lecture 10 aaed EL Rabai week 12- 2010

2. © The McGraw-Hill Companies, Inc., 2000 lecture 10 Correlation and Regression

3. © The McGraw-Hill Companies, Inc., 2000 4-2 Outline 11-1 Introduction 11-2 Scatter Plots 11-3 Correlation 11-4 Regression

4. © The McGraw-Hill Companies, Inc., 2000 4-3 Outline 11-5 Coefficient of Determination and Standard Error of Estimate

5. © The McGraw-Hill Companies, Inc., 2000 11-4 Objectives Draw a scatter plot for a set of ordered pairs. Find the correlation coefficient. Test the hypothesis H 0 :  = 0. Find the equation of the regression line.

6. © The McGraw-Hill Companies, Inc., 2000 11-5 Objectives Find the coefficient of determination. Find the standard error of estimate.

7. © The McGraw-Hill Companies, Inc., 2000 11-6 11-2 Scatter Plots A scatter plot (x, y) x y A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable, x, and the dependent variable, y.

8. © The McGraw-Hill Companies, Inc., 2000 11-7 11-2 Scatter Plots - 11-2 Scatter Plots - Example Construct a scatter plot for the data obtained in a study of age and systolic blood pressure of six randomly selected subjects. The data is given on the next slide.

9. © The McGraw-Hill Companies, Inc., 2000 11-8 11-2 Scatter Plots - 11-2 Scatter Plots - Example

10. © The McGraw-Hill Companies, Inc., 2000 11-9 11-2 Scatter Plots - 11-2 Scatter Plots - Example Positive Relationship

11. © The McGraw-Hill Companies, Inc., 2000 11-10 11-2 Scatter Plots - 11-2 Scatter Plots - Other Examples Negative Relationship

12. © The McGraw-Hill Companies, Inc., 2000 11-11 11-2 Scatter Plots - 11-2 Scatter Plots - Other Examples No Relationship

13. © The McGraw-Hill Companies, Inc., 2000 11-12 11-3 Correlation Coefficient correlation coefficient The correlation coefficient computed from the sample data measures the strength and direction of a relationship between two variables. Sample correlation coefficient, r. Population correlation coefficient, 

14. © The McGraw-Hill Companies, Inc., 2000 11-13 11-3 Range of Values for the Correlation Coefficient  Strong negative relationship Strong positive relationship No linear relationship

15. © The McGraw-Hill Companies, Inc., 2000 11-14 11-3 Formula for the Correlation Coefficient r                  r nxyxy nxxnyy          2 2 2 2 Where n is the number of data pairs

16. © The McGraw-Hill Companies, Inc., 2000 11-15 11-3 Correlation Coefficient - 11-3 Correlation Coefficient - Example (Verify) correlation coefficient Compute the correlation coefficient for the age and blood pressure data.

17. © The McGraw-Hill Companies, Inc., 2000 Business and Finance College Principles of Statistics Lecture 11 aaed EL Rabai week 11- 2010

18. © The McGraw-Hill Companies, Inc., 2000 11-16 11-3 The Significance of the Correlation Coefficient population correlation coefficient The population correlation coefficient, , is the correlation between all possible pairs of data values (x, y) taken from a population.

19. © The McGraw-Hill Companies, Inc., 2000 11-17 11-3 The Significance of the Correlation Coefficient H 0 :  = 0 H 1 :   0 This tests for a significant correlation between the variables in the population.

20. © The McGraw-Hill Companies, Inc., 2000 11-22 The scatter plot for the age and blood pressure data displays a linear pattern. We can model this relationship with a straight line. This regression line is called the line of best fit or the regression line. The equation of the line is y = a + bx. 11-4 Regression

21. © The McGraw-Hill Companies, Inc., 2000 11-23 11-4 Formulas for the Regression Line 11-4 Formulas for the Regression Line y = a + bx.                      a yxxxy nxx b n xy nxx               2 2 2 2 2 Where a is the y intercept and b is the slope of the line. 

22. © The McGraw-Hill Companies, Inc., 2000 11-24 11-4 11-4 Example Find the equation of the regression line for the age and the blood pressure data. Substituting into the formulas give a = 81.048 and b = 0.964 (verify). Hence, y = 81.048 + 0.964x. ainterceptb slope Note, a represents the intercept and b the slope of the line.

23. © The McGraw-Hill Companies, Inc., 2000 11-25 11-4 11-4 Example y = 81.048 + 0.964x

24. © The McGraw-Hill Companies, Inc., 2000 11-26 11-4 Using the Regression Line to Predict The regression line can be used to predict a value for the dependent variable (y) for a given value of the independent variable (x). Caution: Caution: Use x values within the experimental region when predicting y values.

25. © The McGraw-Hill Companies, Inc., 2000 11-27 11-4 11-4 Example Use the equation of the regression line to predict the blood pressure for a person who is 50 years old. Since y = 81.048 + 0.964x, then y = 81.048 + 0.964(50) = 129.248  129.2 Note that the value of 50 is within the range of x values. 

26. © The McGraw-Hill Companies, Inc., 2000 11-28 11-5 Coefficient of Determination and Standard Error of Estimate coefficient of determination The coefficient of determination, denoted by r 2, is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable.

27. © The McGraw-Hill Companies, Inc., 2000 11-29 11-5 Coefficient of Determination and Standard Error of Estimate r 2 is the square of the correlation coefficient. coefficient of nondetermination The coefficient of nondetermination is (1 – r 2 ). Example: If r = 0.90, then r 2 = 0.81.