1. –The shortest distance is the one that crosses at 90° the vector u Statistical Inference on correlation and regression

2. –The shortest distance is the one that crosses at 90° the vector u Statistical Inference on Correlation Angle between two variablesRelationship between two variables

3. Statistical Inference on Correlation The null hypothesis is that there is no correlation between the two variables in the population. In other words, we seek to know if the two variables are linearly independent. If the hull hypothesis is rejected, then it means that the two variables are not independent and that there is a linear relationship between the two.

4. Statistical Inference on Correlation Example In this case, we cannot use the standard normal distribution (Z distribution). We will use the F ratio distribution instead. (see pdf file)

5. Statistical Inference on Correlation Example Nb of variables - 1 Nb of participants- 2

6. Statistical Inference on Correlation Because F xy > F(0.05, 1, 3) (10.3>10.13) we reject H0 and therefore accept H1. There is a linear dependency between the two variables. Example

7. –The shortest distance is the one that crosses at 90° the vector u Linear regression We want a functional relationship between 2 variables; not only a strength of association. In other words, we want to be able to predict the outcome given a predictor x1x1 y1y1 Recall: finding the slope and the constant of a line

8. Linear regression Regression: b e

9. –By substitution, we can isolate the b 1 coefficient. Linear regression Regression: The formula to obtain the regression coefficients can be deducted directly from geometry If we generalized to any situation (multiple, multivariate) (true for 2 variables only)

10. If we replace b 0 Parameters of the linear regression Equation of prediction

11. We know that: If we replace the covariance we then obrain: Note

12. Example Participant

13. Example