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200 Chapter 5 Regression with a Single Regressor: Hypothesis Tests and Confidence Intervals

Key Concept Confidence Interval for b1

5.3 A 95% two-sided confidence interval for b1 is an interval that contains the true
value of b1 with a 95% probability; that is, it contains the true value of b1 in 95%
of all possible randomly drawn samples. Equivalently, it is the set of values of b1
that cannot be rejected by a 5% two-sided hypothesis test. When the sample size
is large, it is constructed as

95% confidence interval for b1 = 3bn1 - 1.96SE(bn1), bn1 + 1.96SE(bn1)4. (5.12)

bn1 { 1.96SE(bn1). This implies that the 95% confidence interval for b1 is the inter-
val 3bn1 - 1.96SE(bn1), bn1 + 1.96SE(bn1)4. This argument parallels the argument
used to develop a confidence interval for the population mean.

The construction of a confidence interval for b1 is summarized as Key
Concept 5.3.

Confidence interval for b0. A 95% confidence interval for b0 is constructed as in
Key Concept 5.3, with bn0 and SE(bn0) replacing bn1 and SE(bn1).

Application to test scores. The OLS regression of the test score against the student–
teacher ratio, reported in Equation (5.8), yielded bn1 = -2.28 and SE(bn1) = 0.52.
The 95% two-sided confidence interval for b1 is 5 -2.28 { 1.96 * 0.526, or
-3.30 … b1 … -1.26. The value b1 = 0 is not contained in this confidence interval,
so (as we knew already from Section 5.1) the hypothesis b1 = 0 can be rejected at the
5% significance level.

Confidence intervals for predicted effects of changing X. The 95% confidence
interval for b1 can be used to construct a 95% confidence interval for the pre-
dicted effect of a general change in X.

Consider changing X by a given amount, ∆x. The predicted change in Y asso-
ciated with this change in X is b1∆x. The population slope b1 is unknown, but
because we can construct a confidence interval for b1, we can construct a confi-
dence interval for the predicted effect b1∆x. Because one end of a 95% confidence
interval for b1 is bn1 - 1.96SE(bn1), the predicted effect of the change ∆x using
this estimate of b1 is 3bn1 - 1.96SE(bn1)4 * ∆x. The other end of the confidence