1. Correlation and Regression
Chapter 9
Correlation and Regression
Larson/Farber 4th ed.

2. Correlation Correlation A relationship between two variables.
The data can be represented by ordered pairs (x, y)
x is the independent (or explanatory) variable
y is the dependent (or response) variable
Larson/Farber 4th ed.

3. Correlation
A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables. x 2 4 –2
– 4 y 6
Example: x 1 2 3 4 5 y
– 4
– 2
– 1
Larson/Farber 4th ed.

4. Types of Correlation As x increases, y tends to decrease.
As x increases, y tends to increase.
Negative Linear Correlation
Positive Linear Correlation x y x y
No Correlation
Nonlinear Correlation
Larson/Farber 4th ed.

5. Various Types of Relations in a Scatter Diagram
© 2010 Pearson Prentice Hall. All rights reserved

6. © 2010 Pearson Prentice Hall. All rights reserved

7. Example: Constructing a Scatter Plot
A marketing manager conducted a study to determine whether there is a linear relationship between money spent on advertising and company sales. The data are shown in the table. Display the data in a scatter plot and determine whether there appears to be a positive or negative linear correlation or no linear correlation.
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

8. Solution: Constructing a Scatter Plotx y
Advertising expenses
(in thousands of dollars)
Company sales
Appears to be a positive linear correlation. As the advertising expenses increase, the sales tend to increase.
Larson/Farber 4th ed.

9. Example: Constructing a Scatter Plot Using Technology
Old Faithful, located in Yellowstone National Park, is the world’s most famous geyser. The duration (in minutes) of several of Old Faithful’s eruptions and the times (in minutes) until the next eruption are shown in the table. Using a TI-83/84, display the data in a scatter plot. Determine the type of correlation.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

10. Solution: Constructing a Scatter Plot Using Technology
Enter the x-values into list L1 and the y-values into list L2.
Use Stat Plot to construct the scatter plot.
STAT > Edit…
STATPLOT 1 5 50
100
From the scatter plot, it appears that the variables have a positive linear correlation.
Larson/Farber 4th ed.

11. Correlation Coefficient
A measure of the strength and the direction of a linear relationship between two variables.
The symbol r represents the sample correlation coefficient.
A formula for r is
The population correlation coefficient is represented by ρ (rho).
n is the number of data pairs
Larson/Farber 4th ed.

12. Correlation Coefficient
The range of the correlation coefficient is -1 to 1. -1 1
If r = -1 there is a perfect negative correlation
If r is close to 0 there is no linear correlation
If r = 1 there is a perfect positive correlation
Larson/Farber 4th ed.

13. © 2010 Pearson Prentice Hall. All rights reserved

14. Linear Correlation Strong negative correlationx y x y
r = 0.91
r = 0.88
Strong negative correlation
Strong positive correlation x y x y
r = 0.42
r = 0.07
Weak positive correlation
Nonlinear Correlation
Larson/Farber 4th ed.

15. Example: Finding the Correlation Coefficient
Calculate the correlation coefficient for the advertising expenditures and company sales data. What can you conclude?
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

16. Example: Using Technology to Find a Correlation Coefficient
Use a technology tool to calculate the correlation coefficient for the Old Faithful data. What can you conclude?
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

17. Solution: Using Technology to Find a Correlation Coefficient
To calculate r, you must first enter the DiagnosticOn command found in the Catalog menu
STAT > Calc
r ≈ suggests a strong positive correlation.
Larson/Farber 4th ed.

18. Using a Table to Test a Population Correlation Coefficient ρ
Once the sample correlation coefficient r has been calculated, we need to determine whether there is enough evidence to decide that the population correlation coefficient ρ is significant at a specified level of significance.
Use Table 11 in Appendix B.
If |r| is greater than the critical value, there is enough evidence to decide that the correlation coefficient ρ is significant.
Larson/Farber 4th ed.

19. Example: Using a Table to Test a Population Correlation Coefficient ρ
Using the Old Faithful data, you used 25 pairs of data to find r ≈ Is the correlation coefficient significant? Use α = 0.05.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

20. Correlation and Causation
The fact that two variables are strongly correlated does not in itself imply a cause-and-effect relationship between the variables.
If there is a significant correlation between two variables, you should consider the following possibilities.
Is there a direct cause-and-effect relationship between the variables?
Does x cause y?
Larson/Farber 4th ed.

21. Correlation and Causation
Is there a reverse cause-and-effect relationship between the variables?
Does y cause x?
Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables?
Is it possible that the relationship between two variables may be a coincidence?
Larson/Farber 4th ed.

22. Section 9.2
Linear Regression
Larson/Farber 4th ed.

23. Section 9.2 Objectives Find the equation of a regression line
Predict y-values using a regression equation
Larson/Farber 4th ed.

24. Regression lines
After verifying that the linear correlation between two variables is significant, next we determine the equation of the line that best models the data (regression line).
Can be used to predict the value of y for a given value of x. x y
Larson/Farber 4th ed.

25. Regression Line Regression line (line of best fit)
The line for which the sum of the squares of the residuals is a minimum.
The equation of a regression line for an independent variable x and a dependent variable y is
ŷ = ax + b
y-intercept
Predicted y-value for a given x-value
Slope
Larson/Farber 4th ed.

26. Example: Finding the Equation of a Regression Line
Find the equation of the regression line for the advertising expenditures and company sales data.
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
Larson/Farber 4th ed.

27. Solution: Finding the Equation of a Regression Line
To sketch the regression line, use any two x-values within the range of the data and calculate the corresponding y-values from the regression line. x
Advertising expenses
(in thousands of dollars)
Company sales y
Larson/Farber 4th ed.

28. Example: Using Technology to Find a Regression Equation
Use a technology tool to find the equation of the regression line for the Old Faithful data.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65
Larson/Farber 4th ed.

29. Solution: Using Technology to Find a Regression Equation
100 50 1 5
Larson/Farber 4th ed.

30. Example: Predicting y-Values Using Regression Equations
The regression equation for the advertising expenses (in thousands of dollars) and company sales (in thousands of dollars) data is ŷ = x Use this equation to predict the expected company sales for the following advertising expenses. (Recall from section 9.1 that x and y have a significant linear correlation.)
1.5 thousand dollars
1.8 thousand dollars
2.5 thousand dollars
Larson/Farber 4th ed.

31. Solution: Predicting y-Values Using Regression Equations
ŷ = x
1.5 thousand dollars
ŷ =50.729(1.5) ≈
When the advertising expenses are $1500, the company sales are about $180,155.
1.8 thousand dollars
ŷ =50.729(1.8) ≈
When the advertising expenses are $1800, the company sales are about $195,373.
Larson/Farber 4th ed.

32. Solution: Predicting y-Values Using Regression Equations
2.5 thousand dollars
ŷ =50.729(2.5) ≈
When the advertising expenses are $2500, the company sales are about $230,884.
Prediction values are meaningful only for x-values in (or close to) the range of the data. The x-values in the original data set range from 1.4 to 2.6. So, it would
not be appropriate to use the regression line to predict
company sales for advertising expenditures such as 0.5 ($500) or 5.0 ($5000).
Larson/Farber 4th ed.