Regression Slope: Confidence Interval
This lesson describes how to construct a
confidence interval around the
slope
of a
regression
line. We focus on the equation for simple linear regression, which is:
ŷ = b_{0} + b_{1}x
where b_{0} is a constant,
b_{1} is the slope (also called the regression coefficient),
x is the value of the independent variable, and ŷ is the
predicted value of the dependent variable.
Estimation Requirements
The approach described in this lesson is valid whenever the
standard requirements for simple linear regression are met.
- For any given value of X,
Previously, we described
how to verify that regression requirements are met.
The Variability of the Slope Estimate
To construct a
confidence interval for the slope of the regression line,
we need to know the
standard error
of the
sampling distribution of the slope.
Many statistical software packages and some graphing calculators
provide the standard error of the slope as a regression analysis
output. The table below shows hypothetical output for the following
regression equation: y = 76 + 35x .
Predictor |
Coef |
SE Coef |
T |
P |
Constant |
76 |
30 |
2.53 |
0.01 |
X |
35 |
20 |
1.75 |
0.04 |
In the output above, the standard error of the slope (shaded in gray)
is equal to 20. In this example, the standard error is referred to
as "SE Coeff". However, other software packages might use a
different label for the standard error. It might be "StDev",
"SE", "Std Dev", or something else.
If you need to calculate the standard error of the slope
(SE)
by hand, use the following formula:
SE = s_{b1} =
sqrt [ Σ(y_{i} - ŷ_{i})^{2}
/ (n - 2) ]
/ sqrt [ Σ(x_{i} -
x)^{2} ]
where y_{i} is the value of the dependent variable for
observation i,
ŷ_{i} is estimated value of the dependent variable
for observation i,
x_{i} is the observed value of the independent variable for
observation i,
x is the mean of the independent variable,
and n is the number of observations.
How to Find the Confidence Interval for the Slope of a
Regression Line
Previously, we described
how to construct confidence intervals. The confidence
interval for the slope of a simple linear regression equation uses the same general approach. Note,
however, that the critical value is based on a
t score
with n - 2
degrees of freedom.
- Identify a sample statistic. The sample statistic is the
regression slope
b_{1} calculated from sample data. In the table
above, the regression slope is 35.
- Select a confidence level. The confidence level describes the
uncertainty of a sampling
method. Often, researchers choose 90%, 95%, or 99% confidence
levels; but any percentage can be used.
- Find the margin of error. Previously, we showed
how to compute the margin of error, based on the
critical value and standard error. When calculating
the margin of error for a regression slope, use a
t score
for the critical value, with
degrees of freedom (DF) equal to
n - 2.
- Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic +
margin of error. And the uncertainty is denoted
by the confidence level.
In the next section, we work through a problem that shows how to
use this approach to construct a confidence interval for the
slope of a regression line. Note that this approach is used for
simple linear regression (one independent variable and one dependent variable).
Test Your Understanding
Problem 1
The local utility company surveys 101 randomly selected
customers. For each survey participant, the company collects
the following: annual electric bill (in dollars) and home size
(in square feet). Output from a regression analysis
appears below.
Regression equation:
Annual bill = 0.55 * Home size + 15 |
Predictor |
Coef |
SE Coef |
T |
P |
Constant |
15 |
3 |
5.0 |
0.00 |
Home size |
0.55 |
0.24 |
2.29 |
0.01 |
What is the 99% confidence interval for the slope of the regression
line?
(A) 0.25 to 0.85
(B) 0.02 to 1.08
(C) -0.08 to 1.18
(D) 0.20 to 1.30
(E) 0.30 to 1.40
Solution
The correct answer is (C). Use the following
four-step approach to construct a confidence interval.
- Identify a sample statistic. Since we are trying to estimate
the slope of the true regression line, we use the
regression coefficient for home size (i.e., the sample estimate of
slope) as the sample statistic. From the regression output, we
see that the slope coefficient is 0.55.
- Select a confidence level. In this analysis, the confidence level
is defined for us in the problem. We are working with a 99%
confidence level.
- Find the margin of error. Elsewhere on this site, we show
how to compute the margin of error. The key steps applied
to this problem are shown below.
- Specify the confidence interval. The range of the confidence
interval is defined by the sample statistic +
margin of error. And the uncertainty is denoted
by the confidence level.
Therefore, the 99% confidence interval for this sample is 0.55 + 0.63, which is -0.08 to 1.18
If we replicated the same
study multiple times with different random samples and computed a confidence interval for each sample, we would expect
99% of the confidence intervals to contain the true slope of the regression line.