What you are interested in is the visual image given by the residual plot, and it doesn’t matter if the
residuals are plotted against the x -variable or something else, like “FITS2”—the scatter of the points
above and below 0 stays the same. All that changes are the horizontal distances between points. This is
the way it must be done in multiple regression, since there is more than one independent variable and,
as you can see, it can be done in simple linear regression.If we are trying to predict a value of y from a value of x , it is called interpolation if we are
predicting from an x -value within the range of x -values. It is called extrapolation if we are predicting
from a value of x outside of the x -values.
example: Using the age/height data from the previous example, we are interpolatingif we attempt to predict height from an age between 18 and 29 months. It is interpolation if we try to
predict the height of a 20.5-month-old baby. We are extrapolating if we try to predict the height o f a child
less than 18 months old or more than 29 months old.
If a line has been shown to be a good model for the data and if it fits the line well (i.e., we have a
strong r and a more or less random distribution of residuals), we can have confidence in interpolated
predictions. We can rarely have confidence in extrapolated values. In the example above, we might be
willing to go slightly beyond the ages given because of the high correlation and the good linear model, but
it’s good practice not to extrapolate beyond the data given. If we were to extrapolate the data in the
example to a child of 12 years of age (144 months), we would predict the child to be 156.2 inches, or
more than 13 feet tall!
Coefficient of Determination
In the absence of a better way to predict y -values from x -values, our best guess for any given x might
well be , the mean value of y .
example: Suppose you had access to the heights and weights of each of the students in your
statistics class. You compute the average weight of all the students. You write the heights of
each student on a slip of paper, put the slips in a hat, and then draw out one slip. You are asked