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Chapter 4: Getting in Line with Simple Linear Regression 69

Take your estimate, y, which you get by plugging your given x value into the
regression line, and then add and subtract the margin of error for y. The
formula for a 1 – α confidence interval for the mean of y for a given value of
x (call it x*) is equal to , where y is the value of the equation of the
line when you plug in x* for x. The standard error for y is equal to

. Luckily Minitab does

these calculations for you and reports a confidence interval for the mean of y
for a given x*.

To find a confidence interval for the mean value of y using Minitab, you ask for
a regression analysis (see instructions in the earlier section “Finding the best-
fitting line to model your data”) and click on Options. You see a box called
Prediction Intervals for New Observations; enter the value of x* that you want,
and just below that, put in your confidence level (the default is 95 percent).
Even though this box is labeled Prediction Intervals, it finds both a confidence
interval and a prediction interval for you as well. (Prediction intervals are dif-
ferent from confidence intervals, and I discuss them in the next section.) On
the computer output, the confidence interval is labeled 95% CI.

Returning to the textbook-weight example, the computer output for finding a
95 percent confidence interval for the average textbook weight for 100-pound
children is shown in Figure 4-4. The result is (14.015, 16.048) pounds. I’m 95
percent confident that the average textbook weight for the group of children
averaging 100 pounds is between 14.015 and 16.048 pounds. (Get out those
rolling backpacks, kids!)

You should only make predictions for the average value of y for x values
that are within the range of where the data was collected. Failure to do so
will result in the statistical no-no called extrapolation (see the later section
“Knowing the Limitations of Your Regression Analysis”).

Making the band with prediction intervals

Suppose instead of the mean value of y, you want to take a guess at what y
would be for some future value of x. Because you’re looking into the future,
you have to make a prediction, and to do that, you need a range of likely
values of y for a given x*. This is what statisticians call a prediction interval.

The formula for a 1 – α level prediction interval for y at a given value x* is

. Again, Minitab easily

makes these calculations for you.

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