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Chapter 4: Getting in Line with Simple Linear Regression 65

You can find the value of t* in any t-distribution table (check your textbook for
one). For example, suppose you want to find a 95 percent confidence interval
based on sample size n = 10. The value of t* is found in Table A-1 in the appen-
dix in the row marked 10 – 2 = 8 degrees of freedom, and the column marked
0.025 (because α ÷ 2 = 0.05 ÷ 2 = 0.025). This value of t* is 2.306. (Statistics For
Dummies can tell you a lot more about the t-distribution and the t-table.)

To put together a 95 percent confidence interval for the slope using computer
output, you pull off the pieces that you need. For the textbook-weight exam-
ple, in Figure 4-2 you see that the slope is equal to 0.11337. (Recall that slope
is the coefficient of the x variable in the equation, which is why you see the
abbreviation Coef in the output.)

Because the slope changes from sample to sample, it’s a random variable
with its own distribution, its own mean, and its own standard error. (Recall
from Stats I the standard error of a statistic is likened to the standard
deviation of a random variable.) If you look just to the right of the slope in
Figure 4-2, you see SE Coef; this stands for the standard error of the slope
(which is 0.01456 in this case.)

Now all you need is the value of t* from the t-table (Table A-1 in the appen-
dix). Because n = 12, you look in the row where degrees of freedom is
12 – 2 = 10. You want a 95 percent confidence interval, so you look in the
column for (1 – 0.95) ÷ 2 = 0.25. The t* value you get is 2.228.

Putting these pieces together, a 95 percent confidence interval for the slope
of the best-fitting regression line for the textbook-weight example is
0.11337 ± 2.228 * 0.01456 which goes from 0.0809 to 0.1458. The units are in
pounds (textbook) per pounds (child weight). Note this interval is large due
to the small sample size, which increases the standard error.

A hypothesis test for slope
You may be interested in conducting a hypothesis test for the slope of a
regression line as another way to assess how well the line fits. If the slope
is zero or close to it, the regression line is basically flat, signifying that no
matter the value of x, you’ll always estimate y by using its mean. This means
that x and y aren’t related at all, so a specific value of x doesn’t help you pre-
dict a specific value for y. You can also test to see if the slope is some value
other than zero, but that’s atypical. So for all intents and purposes, I use the
hypotheses Ho: β = 0 versus Ha: β ≠ 0, where β is the slope of the true model.
To conduct a hypothesis test for the slope of a simple linear regression
line, you follow the basic steps of any hypothesis test. You take the statistic
(b) from your data, subtract the value in Ho (in this case it’s zero), and

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