14. Correlation and Regression
At a fixed (that is important to remember) the confidence interval for is
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 :
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.