1. 1-1 Regression Models  Population Deterministic Regression Model Y i =  0 +  1 X i u Y i only depends on the value of X i and no other factor can affect Y i.  Population Probabilistic Regression Model Y i =  0 +  1 X i +  i   i  n.  E(Y |X i )=  0 +  1 X i, That is,  Y ij = E(Y |X i ) +  ij   0 +  1 X ij +  ij   i  n; j = 1, 2,..., N.   0 and  1 are population parameters   0 and  1 are estimated by sample statistics b 0 and b 1 u Sample Model:

2. 1-2 Assumptions Underlying Linear Regression– for Y For each value of X, there is a group of Y values, and these Y values are normally distributed. The means of these normal distributions of Y values all lie on the straight line of regression. The error variances of these normal distributions are equal (Homoscedasticity). If the error variances are not constant ( called heteroscedasticity). The Y values are statistically independent. This means that in the selection of a sample, the Y values chosen for a particular X value do not depend on the Y values for any other X values.

3. 1-3 Equation of the Simple Regression Line

4. 1-4 Ordinary Least Squares (OLS) Analysis

5. 1-5

6. 1-6 Least Squares Analysis

7. 1-7 Standard Error of the Estimate Sum of Squares Error Standard Error of the Estimate

8. 1-8 Proof: Standard Error of the Estimate Sum of Squares Error Standard Error of the Estimate

9. Coefficient of Determination The Coefficient of Determination, r 2 - the proportion of the total variation in the dependent variable Y that is explained or accounted for by the variation in the independent variable X. –The coefficient of determination is the square of the coefficient of correlation, and ranges from 0 to 1. 12-10

10. 1-10 Analysis of Variance (ANOVA)

11. 1-11 Figure: Measures of variation in regression

12. 1-12 Expectation of b 1

13. 1-13 Variance of b 1

14. 1-14 Expectation of b 0

15. 1-15 Variance of b 0

16. 1-16 Covariance of b 0 and b 1

17. 1-17

18. Confidence Interval—predict The confidence interval for the mean value of Y for a given value of X is given by: 12-20 p.483

19. 1-19 Prediction of Y 0

20. Prediction Interval of an individual value of Y 0 The prediction interval for an individual value of Y for a given value of X is given by: 12-21 p.484

21. 1-21 Figure: Confidence Intervals for Estimation Y X=6.5 Confidence Intervals for Y X Confidence Intervals for E(Y X )

22. 1-22 The Coefficient of Correlation, r The Coefficient of Correlation (r) is a measure of the strength of the relationship between two variables. –It requires interval or ratio-scaled data (variables). –It can range from -1.00 to 1.00. –Values of -1.00 or 1.00 indicate perfect and strong correlation. –Values close to 0.0 indicate weak correlation. –Negative values indicate an inverse relationship and positive values indicate a direct relationship.

23. 1-23 (Pearson Product-Moment ) Correlation Coefficient For sampleFor population p.489

24. 1-24 Covariance p. 493

25. 1-25 Coefficient of regression and correlation

26. 1-26 F and t statistics

27. 1-27 The Simple Regression Model-Matrix Denote

28. 1-28

29. 1-29

30. 1-30

31. 1-31

32. 1-32

33. 1-33