1. Regression Analysis
Regression analysis is a statistical technique that is very useful for exploring the relationships between two or more variables (one independent variable and the one dependent variable)

2. Simple Linear Regression Model

4. Probabilistic Linear Regression Model

5. The Least Square Method
LSM is based on the concept of minimizing L

6. The Least Square Method

7. Example 11.1

9. See the Excel Solution

10. Estimation of Variance
Where SSE = Error sum of squares

12. Solution 11.1

13. Problem 11.11

14. Problem 11.11
Solve using Excel

15. Standard Error of the Estimates

16. HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
Objective:
Assessing the adequacy of a linear regression model by testing statistical hypotheses about the model parameters and constructing certain confidence intervals.
Assumption:
the errors are normally and independently distributed with mean zero and variance σ2

17. HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION
Suppose we wish to test the hypothesis that the slope equals a constant

18. HYPOTHESIS TESTS IN SIMPLE LINEAR REGRESSION

19. A very important special case of the hypotheses about the slope:
Either x is of little value in explaining the variation in Y and that the best estimator of Y for any x is Y or that the true relationship between x and Y is not linear
Rejecting H0: Either that the straight-line model is adequate or that, although there is a linear effect of x, better results could be obtained with the addition of higher order polynomial terms in x

20. Example11.2

21. Analysis of Variance Approach to Test Significance of Regression
The total corrected sum of squares
The error sum of squares
The regression sum of squares

22. Analysis of Variance Approach to Test Significance of Regression
The above test statistic:

23. Example 11.3
See the Excel solution

24. Confidence Intervals on the Slope and Intercept

25. Confidence Intervals on the Slope and Intercept

26. Confidence Interval on the Mean Response

27. Example 11.5

28. Example 11.5

29. Residual Analysis
Analysis of the residuals is frequently helpful in checking the assumption that the errors are approximately normally distributed with constant variance
As an approximate check of normality, the experimenter can construct a frequency histogram of the residuals or a normal probability plot of residuals.
The analysis can also be done by ploting the residuals against the independent variable x.

30. Residual Analysis

32. Coefficient of Determination(R2)
Coefficient of determination is used to judge the adequacy of a regression model.
R2 is the square of the correlation coefficient between X and Y.