1. Correlation and Regression
11-1
Chapter 11
Correlation and Regression

2. Outline 11-1 Introduction 11-2 Scatter Plots 11-3 Correlation
11-4 Regression

3. Outline
11-3
11-5 Coefficient of Determination and Standard Error of Estimate

4. Objectives Draw a scatter plot for a set of ordered pairs.
11-4
Draw a scatter plot for a set of ordered pairs.
Find the correlation coefficient.
Test the hypothesis H0:  = 0.
Find the equation of the regression line.

5. Objectives Find the coefficient of determination.
11-5
Find the coefficient of determination.
Find the standard error of estimate.
Find a prediction interval.

6. 11-2 Scatter Plots
11-6
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. 11-2 Scatter Plots - Example
11-7
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. 11-2 Scatter Plots - Example
11-8

9. 11-2 Scatter Plots - Example
11-9
Positive Relationship

10. 11-2 Scatter Plots - Other Examples
11-10
Negative Relationship

11. 11-2 Scatter Plots - Other Examples
11-11
11-2 Scatter Plots - Other Examples 7 6 5 4 3 2 1 x y 7 6 5 4 3 2 1 X Y
No Relationship

12. 11-3 Correlation Coefficient
11-12
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. 11-3 Range of Values for the Correlation Coefficient
11-13
Strong negative
relationship
No linear
relationship
Strong positive
relationship   

14. 11-3 Formula for the Correlation Coefficient r
11-14       n  xy   x  y r                n  x  x 2 2 n  y  2 y 2
Where n is the number of data pairs

15. 11-3 Correlation Coefficient - Example (Verify)
11-15
Compute the correlation coefficient for the age and blood pressure data.

16. 11-3 The Significance of the Correlation Coefficient
11-16
The population corelation coefficient, , is the correlation between all possible pairs of data values (x, y) taken from a population.

17. 11-3 The Significance of the Correlation Coefficient
11-17
H0: = H1:  0
This tests for a significant correlation between the variables in the population.

18. 11-3 Formula for the t tests for the Correlation Coefficient
11-18  n 2  t  1 r 2  
with d . f . n 2

19. 11-3 Example
11-19
Test the significance of the correlation coefficient for the age and blood pressure data. Use  = 0.05 and r =
Step 1: State the hypotheses.
H0: = H1:  0

20. 11-3 Example
11-20
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 = –2.776.
Step 3: Compute the test value t = (verify).

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

22. 11-4 Regression
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.

23. 11-4 Formulas for the Regression Line y  = a + bx.
11-23           y  x   2 x xy a        n  x  2 x 2       n  xy   x  y  b     n   x  2 2 x
Where a is the y  intercept and b is the slope of the line.

24. 11-4 Example
11-24
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.
Note, a represents the intercept and b the slope of the line.

25. 11-4 Example
11-25
y  = x

26. 11-4 Using the Regression Line to Predict
11-26
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: Use x values within the experimental region when predicting y values.

27. 11-4 Example
11-27
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.

28. 11-5 Coefficient of Determination and Standard Error of Estimate
11-28
The coefficient of determination, denoted by r2, is a measure of the variation of the dependent variable that is explained by the regression line and the independent variable.

29. 11-5 Coefficient of Determination and Standard Error of Estimate
11-29
r2 is the square of the correlation coefficient.
The coefficient of nondetermination is (1 – r2).
Example: If r = 0.90, then r2 = 0.81.

30. 11-5 Coefficient of Determination and Standard Error of Estimate
11-30
The standard error of estimate, denoted by sest, is the standard deviation of the observed y values about the predicted y  values.
The formula is given on the next slide.

31. 11-5 Formula for the Standard Error of Estimate
11-31    y  y 2   s 
est n 2 or  y  a  y  2 b  xy s  n 
est 2

32. 11-5 Standard Error of Estimate - Example
11-32
From the regression equation, y  = x and n = 6, find sest.
Here, a = 55.57, b = 8.13, and n = 6.
Substituting into the formula gives sest = 6.48 (verify).

33. 11-5 Prediction Interval
11-33
A prediction interval is an interval constructed about a predicted y value, y , for a specified x value.

34. 11-5 Prediction Interval
11-34
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.

35. 11-5 Formula for the Prediction Interval about a Value y
11-35 2 1 n ( x - X ) y - t s 1 + + a 2 2
est n 2 ( ) n å x - å x 2 1 n ( x - X ) y + t s 1 + + a 2
est n 2 2 ( ) n å x - å x
with d . f .  n  2

36. 11-5 Prediction interval - Example
11-36
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.

37. 11-5 Prediction Interval - Example
11-37 A 1
$62 B 2
$78 C 3
$70 D 4
$90 E 4
$93 F 6
$103

38. 11-5 Prediction Interval - Example
11-38
Step 1: Find x, x2 and . x = 20, x2 = 82,
Step 2: Find y  for x = 3. y  = (3) = 79.96
Step 3: Find sest sest = 6.48 as shown in previous example.

39. 11-5 Prediction Interval - Example
11-39
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.