1. Ch11 Curve Fitting II

2. Outline Checking the adequacy of the model Correlation
Multiple linear regression (Matrix notation)

3. 11.5 Checking the adequacy of the model
For the multiple regression, we can use the fitted equation to make inferences.
Residuals:
A plot of the residual versus the predicted values is a major diagnostic tool

4. 11.6 Correlation
Correlation analysis: it is assumed that the data points are values a of pair of random variables whose joint density is given by f(x, y).
The best interpretation of the sample correlation is in terms of the standardized observation.

5. The sample correlation coefficient r
r is the sum of products of the standardized variables divided by n-1.
r=+1 if all pairs lie exactly on a straight line having a positive slope.
r>0, if the pattern in the scattergram runs from lower left to upper right.
R<0, if the pattern in the scattergram runs from upper left to lower right.
R=-1 if all pair lie exactly on a straight line having a negative slope.

6. EX
PP 376~377. Students solve it and plot the graph.

7. Correlation and regression
Alternatively,
Proof.

8. Inference about the correlation coefficient
Covariance: the measure of association between X and Y, is called population correlation coefficient.
When (rho) = 1, or -1, we say that there is a perfect linear correlation between the two random variables.
When it is 0, there is no correlation between the two random variables.

9. Test about rho
Assume that the joint distribution of X and Y is the bivariate normal distribution.
Fisher Z transformation:

10. Test Statistic for inferences about rho
is a random variable approximately the standard normal distribution

11. 11.7 Multiple Linear regression
In Matrix Notation
Hence

12. Proof.

13. EX
P365, P388.

14. EX
Use the matrix relations to fit a straight line to the data x y