1. © The McGraw-Hill Companies, Inc., 2000 CorrelationandRegression Further Mathematics - CORE

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

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

4. © The McGraw-Hill Companies, Inc., 2000 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.

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

6. © The McGraw-Hill Companies, Inc., 2000 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.

7. © The McGraw-Hill Companies, Inc., 2000 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.

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

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

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

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

12. © The McGraw-Hill Companies, Inc., 2000 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, 

13. © 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

14. © 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

15. © 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.

16. © 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.

17. © 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.

18. © The McGraw-Hill Companies, Inc., 2000 11-18 11-3 Formula for the t-tests for the Correlation Coefficient t n r with d fn     2 1 2 2..

19. © The McGraw-Hill Companies, Inc., 2000 11-19 11-3 11-3 Example Test the significance of the correlation coefficient for the age and blood pressure data. Use  = 0.05 and r = 0.897. Step 1: Step 1: State the hypotheses. H 0 :  = 0 H 1 :  0

20. © The McGraw-Hill Companies, Inc., 2000 11-20 Step 2: Step 2: Find the critical values. Since  = 0.05 and there are 6 – 2 = 4 degrees of freedom, the critical values are t = +2.776 and t = –2.776. Step 3: Step 3: Compute the test value. t = 4.059 (verify). 11-3 11-3 Example

21. © The McGraw-Hill Companies, Inc., 2000 11-21 Step 4: Step 4: Make the decision. Reject the null hypothesis, since the test value falls in the critical region (4.059 > 2.776). Step 5: Step 5: Summarize the results. There is a significant relationship between the variables of age and blood pressure. 11-3 11-3 Example

22. © 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

23. © 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. 

24. © 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.

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

26. © 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.

27. © 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. 

28. © 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.

29. © 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.

30. © The McGraw-Hill Companies, Inc., 2000 11-30 11-5 Coefficient of Determination and Standard Error of Estimate standard error of estimate The standard error of estimate, denoted by s est, is the standard deviation of the observed y values about the predicted y values. The formula is given on the next slide.

31. © The McGraw-Hill Companies, Inc., 2000 11-31 11-5 Formula for the Standard Error of Estimate   s yy n or s yaybxy n est          2 2 2 2

32. © The McGraw-Hill Companies, Inc., 2000 11-32 11-5 Standard Error of Estimate - 11-5 Standard Error of Estimate - Example From the regression equation, y = 55.57 + 8.13x and n = 6, find s est. Here, a = 55.57, b = 8.13, and n = 6. Substituting into the formula gives s est = 6.48 (verify).