1. Making predictions using linear regression
The shortest distance is the one that crosses at 90° the vector u

2. 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 y1
The shortest distance is the one that crosses at 90° the vector u x1

3. Example
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4. Example

5. Prediction
Using the regression equation, it is possible to make some predictions
Ex. 1 If x = 7.5, = ?

6. Prediction In a similar fashion, we can predict x from y
Ex. 2 If y = 9.65, = ?

7. Prediction Finally! Ex. if x = 3, = ?
However, (x,y) => (3,2). Therefore, we made an error.

8. Standard error of prediction
The difference between the predicted scores and the observed scores is the standard error of prediction from x.
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9. Note
For large samples

10. Prediction from a new score
Confidence interval
Prediction from a new score
The standard error of regression is an estimate of the total error. However, it is not a good estimate for the prediction of a given x. In fact, the error estimation will be smaller when x is near the mean and larger when it is far from the mean.

11. Prediction from a new score
Confidence interval
Prediction from a new score
Example xnew= 7.5 for a CI of 95%
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12. Prediction from a new score
Confidence interval
Prediction from a new score
Example xnew= 7.5 for a CI of 95%
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