1. Simple linear regression and correlation Regression analysis is the process of constructing a mathematical model or function that can be used to predict or determine one variable by another variable. Correlation is a measure of the degree of relatedness of two variables. Dr. Ahmed M. Sultan1

2. Simple Regression Analysis bivariate (two variables) linear regression -- the most elementary regression model – dependent variable, the variable to be predicted, usually called Y – independent variable, the predictor or explanatory variable, usually called X Dr. Ahmed M. Sultan2

3. Airline Cost Data Number of Passengers X Cost ($1,000) Y 614.280 634.080 674.420 694.170 704.480 744.300 764.820 814.700 865.110 915.130 955.640 975.560 Dr. Ahmed M. Sultan3

4. Scatter Plot of Airline Cost Data Dr. Ahmed M. Sultan4

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6. Regression Models  Deterministic Regression Model Y =  0 +  1 X  Probabilistic Regression Model Y =  0 +  1 X +    0 and  1 are population parameters   0 and  1 are estimated by sample statistics b 0 and b 1

7. Equation of the Simple Regression Line

8. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-8 Least Squares Analysis

9. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-9 Least Squares Analysis

10. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-10 Solving for b 1 and b 0 of the Regression Line: Airline Cost Example (Part 1)

11. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-11 Solving for b 1 and b 0 of the Regression Line: Airline Cost Example (Part 2)

12. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-12 Graph of Regression Line for the Airline Cost Example

13. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.13-13 Airline Cost: Excel Summary Output SUMMARY OUTPUT Regression Statistics Multiple R0.94820033 R Square0.89908386 Adjusted R Square0.88899225 Standard Error0.17721746 Observations12 ANOVA dfSSMSFSignificance F Regression12.79803 89.0921792.7E-06 Residual100.314060.03141 Total113.11209 CoefficientsStandard Errort StatP-value Intercept1.569792780.338084.643220.0009175 Number of Passengers0.04070160.004319.438872.692E-06

14. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-14 Residual Analysis: Airline Cost Example

15. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-15 Excel Graph of Residuals for the Airline Cost Example

16. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-16 Nonlinear Residual Plot 0 X

17. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-17 Nonconstant Error Variance 0 X 0 X

18. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-18 Graphs of Nonindependent Error Terms 0 X 0 X

19. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-19 Healthy Residual Plot 0 X

20. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-20 Standard Error of the Estimate Sum of Squares Error Standard Error of the Estimate

21. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-21 Determining SSE for the Airline Cost Example

22. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-22 Standard Error of the Estimate for the Airline Cost Example Sum of Squares Error Standard Error of the Estimate Standard Error of the Estimate

23. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-23 Coefficient of Determination

24. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-24 Coefficient of Determination for the Airline Cost Example 89.9% of the variability of the cost of flying a Boeing 737 is accounted for by the number of passengers.

25. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-25 Hypothesis Tests for the Slope of the Regression Model

26. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-26 Hypothesis Test: Airline Cost Example (Part 1)

27. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-27 Hypothesis Test: Airline Cost Example (Part 2)

28. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-28 Testing the Overall Model (Part 1)

29. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.13-29 Testing the Overall Model (Part 2) ANOVA dfSSMSFSignificance F Regression12.79803 89.0921792.7E-06 Residual100.314060.03141 Total113.11209 F = 89.09 > 4.96, reject H 0

30. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-30 Point Estimation for the Airline Cost Example

31. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-31 Confidence Interval to Estimate  Y : Airline Cost Example

32. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-32 Confidence Interval to Estimate the Average Value of Y for some Values of X: Airline Cost Example

33. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-33 Prediction Interval to Estimate Y for a given value of X

34. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-34 Confidence Intervals for Estimation 60 70 80 90100 4 5 6 Number of Passengers C o s t Regression 95% CI 95% PI Regression Plot

35. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-35 MINITAB Regression Analysis of the Airline Cost Example

36. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-36 Pearson Product-Moment Correlation Coefficient

37. Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 13-37 Three Degrees of Correlation r < 0r > 0 r = 0

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