1. © 2001 Prentice-Hall, Inc.Chap 14-1 BA 201 Lecture 23 Correlation Analysis And Introduction to Multiple Regression (Data)Data

2. © 2001 Prentice-Hall, Inc. Chap 14-2 Topics Correlation - Measuring the Strength of the Association The Multiple Regression Model

3. © 2001 Prentice-Hall, Inc. Chap 14-3 Correlation Example: Mid-term Scores Here is an Excel Workbook that contains the correlation analysis between Mid-term Scores and each of the various components of this course.

4. © 2001 Prentice-Hall, Inc. Chap 14-4 Purpose of Correlation Analysis Correlation Analysis is Used to Measure Strength of Association (Linear Relationship) Between 2 Numerical Variables Only Strength of the Relationship is Concerned No Causal Effect is Implied Population Correlation Coefficient  (Rho) is Used to Measure the Strength between the Variables

5. © 2001 Prentice-Hall, Inc. Chap 14-5 Purpose of Correlation Analysis Sample Correlation Coefficient r is an Estimate of  and is Used to Measure the Strength of the Linear Relationship in the Sample Observations (continued)

6. © 2001 Prentice-Hall, Inc. Chap 14-6 r =.6r = 1 Sample of Observations from Various r Values Y X Y X Y X Y X Y X r = -1 r = -.6r = 0

7. © 2001 Prentice-Hall, Inc. Chap 14-7 Features of  and r Unit Free Range between -1 and 1 The Closer to -1, the Stronger the Negative Linear Relationship The Closer to 1, the Stronger the Positive Linear Relationship The Closer to 0, the Weaker the Linear Relationship

8. © 2001 Prentice-Hall, Inc. Chap 14-8 Sample Correlation Coefficient: Example You wish to examine the relationship between the annual sales of produce stores and their sizes in square footage. Sample data for 7 stores were obtained. Find the sample correlation coefficient. Annual Store Square Sales Feet($1000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

9. © 2001 Prentice-Hall, Inc. Chap 14-9 Solution: Produce Stores In PHStat, the sample correlation coefficient (r ) is the “signed” multiple R The sign of r is the same as the sign of the estimated slope coefficient The absolute value of r is the same as the value of multiple R From Excel Printout

10. © 2001 Prentice-Hall, Inc. Chap 14-10 Test if There is a Linear Relationship Hypotheses H 0 :  = 0 (No Correlation) H 1 :  0 (Correlation) Test Statistic

11. © 2001 Prentice-Hall, Inc. Chap 14-11 Example: Produce Stores Is there any evidence of linear relationship between Annual Sales of a store and its Square Footage at.05 level of significance? H 0 :  = 0 (No association) H 1 :   0 (Association)  .05 n = 7 df  7 - 2 = 5 Solution:

12. © 2001 Prentice-Hall, Inc. Chap 14-12 Example: Produce Stores Solution 02.5706-2.5706.025 Reject.025 Critical Value(s): Conclusion: There is evidence of a linear relationship at 5% level of significance Decision: Reject H 0 The value of the t statistic is exactly the same as the t statistic value for test on the slope coefficient r 0 t

13. © 2001 Prentice-Hall, Inc. Chap 14-13 Simple Linear Regression in PHStat In Excel, use PHStat | Regression | Simple Linear Regression … EXCEL Spreadsheet of Regression Sales on Footage

14. © 2001 Prentice-Hall, Inc. Chap 14-14 Population Y-intercept Population slopesRandom Error The Multiple Regression Model Relationship between 1 dependent & 2 or more independent variables is a linear function Dependent (Response) variable for sample Independent (Explanatory) variables for sample model Residual

15. © 2001 Prentice-Hall, Inc. Chap 14-15 Simple Linear Regression Model Revisited Y X Observed Value

16. © 2001 Prentice-Hall, Inc. Chap 14-16 Population Multiple Regression Model Bivariate model (2 Independent Variables: X 1 and X 2 )

17. © 2001 Prentice-Hall, Inc. Chap 14-17 Sample Multiple Regression Model Bivariate model Sample Regression Plane

18. © 2001 Prentice-Hall, Inc. Chap 14-18 Multiple Linear Regression Equation Too complicated by hand! Ouch!

19. © 2001 Prentice-Hall, Inc. Chap 14-19 Multiple Regression Model: Example ( 0 F) Develop a model for estimating heating oil used for a single family home in the month of January based on average temperature and amount of insulation in inches.

20. © 2001 Prentice-Hall, Inc. Chap 14-20 Sample Multiple Regression Equation: Example Excel Output For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5.437 gallons, holding insulation constant. For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20.012 gallons, holding temperature constant.

21. © 2001 Prentice-Hall, Inc. Chap 14-21 Interpretation of Estimated Coefficients Slope (b i ) Estimated that the average value of Y changes by b i for each 1 unit increase in X i holding all other variables constant (ceterus paribus) Example: If b 1 = -2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X 1 ) given the inches of insulation (X 2 ) Y-Intercept (b 0 ) The estimated average value of Y when all X i = 0

22. © 2001 Prentice-Hall, Inc. Chap 14-22 Multiple Regression in PHStat PHStat | Regression | Multiple Regression … EXCEL spreadsheet for the heating oil example.

23. © 2001 Prentice-Hall, Inc. Chap 14-23 Summary Discussed Correlation - Measuring the Strength of the Association Developed The Multiple Regression Model