14. Correlation and Regression

# 14.7 Confidence Interval for y′ at a Given x

At a fixed (that is important to remember) the confidence interval for is

where

where, as usual, comes from the t Distribution Table with .

Example 14.5 : Continuing from Example 14.4 (so you can see how an exam will go), say we want to predict the grade () in terms of a 95 confidence interval for the number of absences () equal to 10.

First, find the value predicted from the regression line, which we previously found to be :

at . The result is

Furthermore, from the last example, we found

and, from the completed data table (Example 14.3)

We still need and . Using our sums:

and from t Distribution Table for the 95 confidence interval, we get

Now we compute :

So

This is the 95 confidence interval for predicting the mark of a person who was absent for 10 days.

Important: is independent of but is not. So confidence intervals look like :

The reason for this variance of the width of the confidence interval comes from the uncertainty in the slope . You can make plots like the one above in SPSS.