© The McGraw-Hill Companies, Inc., Chapter 11 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. Find a prediction interval.

© 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., The Significance of the Correlation Coefficient population corelation coefficient The population corelation 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., Formula for the t tests for the Correlation Coefficient t n r withdfn    

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

© The McGraw-Hill Companies, Inc., 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 = and t = – Step 3: Step 3: Compute the test value. t = (verify) Example

© The McGraw-Hill Companies, Inc., 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 Example

© 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) =  129. 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.

© The McGraw-Hill Companies, Inc., 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.

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

© The McGraw-Hill Companies, Inc., Standard Error of Estimate Standard Error of Estimate - Example From the regression equation, y = x 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).

© The McGraw-Hill Companies, Inc., Prediction Interval prediction interval A prediction interval is an interval constructed about a predicted y value, y, for a specified x value.

© The McGraw-Hill Companies, Inc., Prediction Interval For given  value, we can state with (1 –  )100% confidence that the interval will contain the actual mean of the y values that correspond to the given value of x.

© The McGraw-Hill Companies, Inc., Formula for the Prediction Interval about a Value y  )(1 1 2 xxn Xxn n est sty      )(1 1 2 xxn Xxn n sty     2..  nfdwith

© The McGraw-Hill Companies, Inc., Prediction interval Prediction interval - Example A researcher collects the data shown on the next slide and determines that there is a significant relationship between the age of a copy machine and its monthly maintenance cost. The regression equation is y = x. Find the 95% prediction interval for the monthly maintenance cost of a machine that is 3 years old.

© The McGraw-Hill Companies, Inc., Prediction Interval Prediction Interval - Example A1$62 B2$78 C3$70 D4$90 E4$93 F6$103

© The McGraw-Hill Companies, Inc., Step 1: Step 1: Find  x,  x 2 and.  x = 20,  x 2 = 82, Step 2: Step 2: Find y for x = 3. y = (3) = Step 3: Step 3: Find s est s est = 6.48 as shown in previous example Prediction Interval Prediction Interval - Example

© The McGraw-Hill Companies, Inc., Step 4: Step 4: Substitute in the formula and solve. t  /2 = 2.776, d.f. = 6 – 2 = 4 for 95% < y < (verify) Hence, one can be 95% confident that the interval < y < contains the actual value of y Prediction Interval Prediction Interval - Example