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

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

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

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

© The McGraw-Hill Companies, Inc., 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.

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

© The McGraw-Hill Companies, Inc., 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.

© The McGraw-Hill Companies, Inc., Scatter Plots 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.

© The McGraw-Hill Companies, Inc., Scatter Plots Scatter Plots - Example

© The McGraw-Hill Companies, Inc., Scatter Plots Scatter Plots - Example Positive Relationship

© The McGraw-Hill Companies, Inc., Scatter Plots Scatter Plots - Other Examples Negative Relationship

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

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

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

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

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

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

© The McGraw-Hill Companies, Inc., 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.

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

© The McGraw-Hill Companies, Inc., 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 Regression

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

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

© The McGraw-Hill Companies, Inc., Example y = x

© The McGraw-Hill Companies, Inc., 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.

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

© The McGraw-Hill Companies, Inc., 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.

© The McGraw-Hill Companies, Inc., 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.