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. 