Why Slope Tests and Confidence Intervals Agree

One Calculation, Two Ways to Report Evidence

A two-sided test asks whether the data are inconsistent with a specified population regression slope. A confidence interval gives a range of slope values compatible with the data under the regression model. When the test and interval use matching levels, their decisions agree: the test rejects a hypothesized slope exactly when that slope lies outside the interval. This is not a coincidence between two procedures; both are built from the same estimated slope, standard error, and \(t\) critical value.

The matching is precise. A test at significance level \(\alpha\) corresponds to a two-sided confidence interval with confidence level \(1-\alpha\). Thus, a 5% two-sided test matches a 95% confidence interval, and a 10% two-sided test matches a 90% confidence interval. Comparing results at different levels can produce apparent disagreement simply because the decision thresholds differ.

The Test and Interval Formulas

Let \(\beta_1\) be the population slope and \(b_1\) the slope estimated from a sample of \(n\) paired observations. The estimated standard error of the slope is \(SE(b_1)\). To test a specified value \(\beta_{1,0}\), the two-sided hypotheses are \(H_0:\beta_1=\beta_{1,0}\) and \(H_a:\beta_1\ne\beta_{1,0}\).

Definition: Under the standard linear regression model for slope inference, the test statistic for \(H_0:\beta_1=\beta_{1,0}\) is \(T=(b_1-\beta_{1,0})/SE(b_1)\), with \(df=n-2\). The two-sided test at level \(\alpha\) rejects when \(|T|>t^*_{1-\alpha/2,n-2}\), where \(t^*_{1-\alpha/2,n-2}\) is the \(1-\alpha/2\) quantile of the \(t\) distribution with \(n-2\) degrees of freedom. The corresponding \(100(1-\alpha)\%\) confidence interval for \(\beta_1\) is \(b_1\pm t^*_{1-\alpha/2,n-2}SE(b_1)\).

These formulas rely on the regression conditions required for valid slope inference, as well as a positive estimated standard error. The test and interval must use the same model, standard error, and degrees of freedom. If those inputs differ, the algebraic agreement need not hold. The regression conditions are not checked by the equivalence itself; they are needed to justify the inferential procedures.

Why the Decisions Agree

Write \(c=t^*_{1-\alpha/2,n-2}\), a positive critical value. The confidence interval has lower endpoint \(b_1-cSE(b_1)\) and upper endpoint \(b_1+cSE(b_1)\). A hypothesized slope is inside this interval when it is no smaller than the lower endpoint and no larger than the upper endpoint.

b_1-cSE(b_1)\leq \beta_{1,0}\leq b_1+cSE(b_1)

Because \(SE(b_1)>0\), subtracting \(b_1\) and dividing by the standard error preserves the inequality directions. The result is

-c\leq \frac{\beta_{1,0}-b_1}{SE(b_1)}\leq c \quad\Longleftrightarrow\quad -c\leq \frac{b_1-\beta_{1,0}}{SE(b_1)}\leq c.

The middle expression on the right is the test statistic \(T\). Therefore, the hypothesized slope is inside the closed confidence interval exactly when \(-c\leq T\leq c\), or equivalently \(|T|\leq c\). It is outside the interval exactly when \(|T|>c\), which is the two-sided test's rejection rule.

Theorem: For a two-sided slope test at level \(\alpha\) and the corresponding \(100(1-\alpha)\%\) confidence interval, using the same regression model and standard error, \(H_0:\beta_1=\beta_{1,0}\) is rejected exactly when \(\beta_{1,0}\) is outside the interval. If \(H_0\) is not rejected, its hypothesized slope is inside the interval, including at an endpoint.

This formulation uses the strict critical-value rule \(|T|>c\). At the exact boundary \(|T|=c\), the hypothesized value is an endpoint of the closed interval and the test does not reject under that rule. Some descriptions express a rejection rule using \(p\leq\alpha\); at exact equality this convention creates a boundary difference from a closed interval. In ordinary continuous-model applications exact equality has probability zero, but keeping the boundary convention explicit makes the equivalence mathematically precise.

Worked Examples

Worked Example: A Slope Excluded by a 95% Interval

Suppose a hypothetical regression relates daily study time \(x\), measured in hours, to quiz score \(y\), measured in points. A sample of \(n=20\) students gives \(b_1=2.40\) points per additional study hour and \(SE(b_1)=0.800\) points per hour. Test \(H_0:\beta_1=0\) against \(H_a:\beta_1\ne0\) at \(\alpha=0.05\), and compare the test with a 95% confidence interval. Assume the regression conditions for slope inference are satisfied; in particular, the standard error and \(t\) reference distribution are appropriate for these hypothetical data.

The degrees of freedom are \(df=n-2=20-2=18\). The two-sided critical value is \(t^*_{0.975,18}\approx2.101\). The test statistic is

T=\frac{b_1-\beta_{1,0}}{SE(b_1)} =\frac{2.40-0}{0.800}=3.00.

Since \(|T|=3.00>2.101\), reject \(H_0\) at the 0.05 level. The two-sided \(p\)-value is approximately \(0.007685\), which is below 0.05. For the matching 95% interval, the margin of error is \(2.1009(0.800)=1.6807\) points per hour, so

2.40\pm 1.6807=[0.7193,\ 4.0807]\text{ points per hour}.

The hypothesized slope of zero is below the interval's lower endpoint, so the interval also excludes it. The interval and test therefore give the same decision.

Worked Example: A Nonzero Hypothesized Slope Inside the Interval

In a separate hypothetical setting, a researcher estimates the association between a training program's duration, measured in weeks, and a performance score, measured in points. For \(n=16\) observations, suppose \(b_1=1.20\) points per week and \(SE(b_1)=0.500\) points per week. Test \(H_0:\beta_1=1.00\) against a two-sided alternative at \(\alpha=0.05\). Assume the conditions for slope inference are met.

Here \(df=16-2=14\), and \(t^*_{0.975,14}\approx2.145\). The test statistic is

T=\frac{1.20-1.00}{0.500}=0.400.

Because \(|0.400|<2.145\), do not reject \(H_0\) at the 0.05 level. The matching 95% confidence interval has margin of error \(2.1448(0.500)=1.0724\) points per week:

1.20\pm1.0724=[0.1276,\ 2.2724]\text{ points per week}.

The hypothesized slope, \(1.00\) point per week, lies between \(0.1276\) and \(2.2724\). This agrees with the test's failure to reject. The result does not establish that the true slope equals 1.00; it says that this test does not find sufficient evidence against that value at the chosen level.

Worked Example: Matching a 10% Test With a 90% Interval

Suppose a hypothetical regression estimates the change in a laboratory measurement, in units, per one-unit increase in a predictor. Let \(n=12\), \(b_1=-0.800\) units per predictor unit, and \(SE(b_1)=0.350\) units per predictor unit. Consider \(H_0:\beta_1=-1.00\) against a two-sided alternative at \(\alpha=0.10\). Assume the regression conditions for slope inference are appropriate.

The degrees of freedom are \(12-2=10\). A 10% two-sided test matches a 90% confidence interval, with critical value \(t^*_{0.95,10}\approx1.812\). The test statistic is

T=\frac{-0.800-(-1.00)}{0.350} =\frac{0.200}{0.350}\approx0.5714.

Since \(0.5714<1.812\), do not reject the null hypothesis at \(\alpha=0.10\). The interval's margin of error is \(1.8125(0.350)=0.6344\) units per predictor unit, giving

-0.800\pm0.6344=[-1.4344,\ -0.1656]\text{ units per predictor unit}.

The value \(-1.00\) lies within this 90% interval, so the interval agrees with the test. A 95% interval would use a larger critical value and would match a 5% test instead—not the 10% test used here.

What the Agreement Does and Does Not Say

The equivalence is useful because it lets a reader check a regression analysis in two forms. If a table reports a two-sided test of a specified slope and a confidence interval based on the same standard error, degrees of freedom, and model, a rejection should correspond to exclusion of the tested value. A mismatch is a reason to check whether the confidence and significance levels match, whether the test is one-sided, or whether the calculations used different assumptions or inputs.

It does not mean that the test and interval answer identical questions in every respect. A test gives a decision relative to one specified null value and a chosen \(\alpha\). The interval presents a range of values that are not rejected by the corresponding family of two-sided tests, one for each possible slope value. Nor does either method establish that a regression relationship is causal.

The match also depends on the test being two-sided. A one-sided test allocates its rejection probability to one tail, whereas a conventional two-sided interval has two endpoints and splits \(\alpha\) between the tails. A one-sided test therefore does not generally correspond to asking whether the null value lies outside a standard two-sided \(100(1-\alpha)\%\) interval.

Common Mistakes

Comparing unmatched levels. A 5% test matches a 95% interval, not a 90% interval. A 10% test matches a 90% interval.
Checking whether zero is in every interval. The test concerns the specific null value \(\beta_{1,0}\), which may be nonzero. Check whether that hypothesized value is in the interval.
Using the wrong tail structure. The stated equivalence concerns a two-sided test and a two-sided interval. A one-sided alternative has a different rejection threshold.
Treating non-rejection as proof of equality. If the hypothesized slope lies in the matching interval, the test does not reject it; this is not evidence that the population slope is known to equal that value.
Mixing standard errors or degrees of freedom. The equivalence follows only when the test and interval use the same model-based standard error and \(df=n-2\).

A useful check is to calculate the test's critical value and the interval's critical value before comparing decisions. For matched two-sided procedures, both must be \(t^*_{1-\alpha/2,n-2}\). The test statistic's magnitude then tells whether the null value is beyond the interval's corresponding endpoint.

Check Your Understanding

Answer each question using the test-interval relationship developed above.

Which confidence level matches a two-sided test conducted at \(\alpha=0.02\)?
A matching confidence interval includes the tested slope value. What is the test decision under the strict critical-value rule, and what does that decision not establish?
Why must the test and interval use the same standard error and degrees of freedom for their decisions to agree?
A researcher tests \(H_0:\beta_1=2.5\). Which value should be checked against the confidence interval: zero or 2.5?
Explain why a one-sided slope test does not generally correspond to checking whether a value falls outside a conventional two-sided interval at the same nominal level.