Page 87 - Statistics II for Dummies

Chapter 4: Getting in Line with Simple Linear Regression 71

This difference also makes sense from a statistical point. A prediction interval
has more variability than a confidence interval because it’s harder to make
a prediction about y for a single value of x* than it is to estimate the average
value of y for a given x*. (For example, individual test scores vary more than
average test scores do.) A prediction interval will be wider than a confidence
interval; it will have a larger margin of error.

A similarity between prediction intervals and confidence intervals is that
their margin of error formulas both contain x*, which means the margin of
error in either case depends on which value of x* you use. It turns out in
both cases that if you use the mean value of x as your x*, the margin of error
for each interval is at its smallest because there’s more data around the
mean of x than at any other value. As you move away from the mean of x, the
margin of error increases for each interval.

Checking the Model’s Fit (The Data,
Not the Clothes!)

After you’ve established a relationship between x and y and have come up
with an equation of a line that represents that relationship, you may think
your job is done. (Many researchers erringly stop here, so I’m depending on
you to break the cycle!) The most-important job remains to be completed:
checking to be sure that the conditions of the model are truly met and that
the model fits well in more specific ways than the scatterplot and correlation
measure (which I cover in the earlier section “Exploring Relationships with
Scatterplots and Correlations”).

This section presents methods for defining and assessing the fit of a simple
linear regression model.

Defining the conditions

Two major conditions must be met before you apply a simple linear
regression model to a data set:

✓ The y’s must have an approximately normal distribution for each value
of x.
✓ The y’s must have a constant amount of spread (standard deviation) for
each value of x.

09_466469-ch04.indd 71 7/24/09 10:20:39 AM