Why the Direction of a Slope Test Matters
A regression analysis may be used to assess a directional claim: for example, whether a higher level of a predictor is associated with a greater average response, or whether it is associated with a smaller one. A one-sided test is designed for one such direction. Its rejection region lies in only one tail of the t distribution, so choosing the direction correctly is part of the test—not a detail to decide after calculating the slope.
The procedures here concern inference about the population slope in a simple linear regression. They are aligned with the SOA VEE Mathematical Statistics syllabus and use the exact t test under the normal-error model.
The least-squares slope estimate is denoted by \(\hat{\beta}_1\). Write \(SSE=\sum_{i=1}^n(Y_i-\hat{Y}_i)^2\) for the residual sum of squares. The residual variance estimate, residual standard error, and standard error of the slope are
The \(n-2\) degrees of freedom account for estimating the two regression coefficients, the intercept and slope. If \(n\leq 2\), this variance estimate is not defined; if \(S_{xx}=0\), the slope cannot be estimated from variation in the predictor. In either case, the stated t test cannot be used.
Hypotheses and Rejection Tails
Let \(\beta_{10}\) be the slope specified by the null boundary. The directional alternative determines the null hypothesis and the rejection tail. The null is written as a composite inequality, while the test statistic is evaluated at its boundary value.
| Claim being tested | Hypotheses | Rejection region at level \(\alpha\) |
|---|---|---|
| Slope is greater than the specified value | \(H_0:\beta_1\leq\beta_{10}\), \(H_1:\beta_1>\beta_{10}\) | \(t>t_{1-\alpha,n-2}\) |
| Slope is less than the specified value | \(H_0:\beta_1\geq\beta_{10}\), \(H_1:\beta_1<\beta_{10}\) | \(t<-t_{1-\alpha,n-2}\) |
Here \(t_{1-\alpha,n-2}\) denotes the \(1-\alpha\) quantile of a central t distribution with \(n-2\) degrees of freedom. Equivalently, if \(T\) has that distribution, \(P(T>t_{1-\alpha,n-2})=\alpha\). The statistic for testing the specified slope is
A large positive value supports an upper-tailed alternative; a large negative value supports a lower-tailed alternative. The two-sided test uses a different rejection region, with probability \(\alpha/2\) in each tail. It is not interchangeable with either one-sided test.
Why the Slope Statistic Has a t Distribution
The justification combines the sampling distribution of the least-squares slope with the distribution of the residual variance estimate. Under the model assumptions, conditional on the predictor values, \(\hat{\beta}_1\) is normally distributed with mean \(\beta_1\) and variance \(\sigma^2/S_{xx}\). Also, \(SSE/\sigma^2\) has a chi-square distribution with \(n-2\) degrees of freedom and is independent of \(\hat{\beta}_1\). These are the standard exact regression sampling results.
At the boundary null value \(\beta_1=\beta_{10}\), the numerator standardized by the true standard deviation is standard normal. The estimated standard error replaces \(\sigma\) with \(s\), and the independent chi-square quantity supplies its random scaling:
This is why a t critical value, rather than a standard normal critical value, is used when \(\sigma\) is unknown and estimated from the residuals. For the upper-tailed composite null, values \(\beta_1<\beta_{10}\) shift the statistic toward smaller values; they do not make an upper-tail rejection more likely than it is at the boundary. The lower-tailed test is analogous: its largest rejection probability under the null occurs at \(\beta_1=\beta_{10}\). Thus the boundary distribution gives the exact level of the test under the stated model.
A Reliable Testing Procedure
Use \(H_1:\beta_1>\beta_{10}\) for a claim of a greater slope, or \(H_1:\beta_1<\beta_{10}\) for a claim of a smaller slope. The alternative determines the tail.
For an exact t test, verify the regression assumptions, \(n>2\), and \(S_{xx}>0\).
Use \(s=\sqrt{SSE/(n-2)}\), \(SE(\hat{\beta}_1)=s/\sqrt{S_{xx}}\), and \(t=(\hat{\beta}_1-\beta_{10})/SE(\hat{\beta}_1)\).
For an upper-tailed test, reject for a sufficiently large positive statistic. For a lower-tailed test, reject for a sufficiently large negative statistic.
Describe whether the data provide sufficient evidence for the directional claim at the stated significance level. A failure to reject is not proof that the slope equals the boundary value.
Worked Tests
Worked Example: Testing for a Slope Above a Benchmark
Method: Use an upper-tailed t test. A hypothetical study examines whether greater weekly use of a training simulator is associated with an increase in an employee performance score. Assume the fixed-predictor normal-error regression model stated above. For \(n=11\) employees, the fitted regression summaries are \(\hat{\beta}_1=1.3\), \(S_{xx}=25\), and \(SSE=36\). Test \(H_0:\beta_1\leq0.3\) against \(H_1:\beta_1>0.3\) at \(\alpha=0.05\).
The degrees of freedom are \(n-2=11-2=9\). The residual variance estimate is \(s^2=36/9=4\), so \(s=\sqrt{4}=2\). The slope standard error is \(SE(\hat{\beta}_1)=2/\sqrt{25}=2/5=0.4\).
The observed statistic is \(t=(1.3-0.3)/0.4=1.0/0.4=2.5\). The upper-tail 5% critical value for 9 degrees of freedom is approximately \(1.833\). Since \(2.5>1.833\), reject \(H_0\). At the 5% significance level, the data provide evidence that the population slope exceeds \(0.3\) score points per unit of weekly simulator use.
Check: The estimated slope is above the null boundary and the statistic is positive, as expected for an upper-tailed test. The common mistake to avoid is using a two-sided critical value even though the directional alternative was specified in advance.
Worked Example: Testing for a Slope Below a Negative Benchmark
Method: Use a lower-tailed t test. In a hypothetical quality-control study, a technician relates a sensor reading to the average number of surface defects on a manufactured panel. Assume the fixed-predictor normal-error model. The summaries are \(n=12\), \(\hat{\beta}_1=-0.55\), \(S_{xx}=100\), and \(SSE=40\). Test \(H_0:\beta_1\geq-0.10\) against \(H_1:\beta_1<-0.10\), using \(\alpha=0.05\).
There are \(12-2=10\) degrees of freedom. Calculate \(s^2=40/10=4\), hence \(s=2\), and \(SE(\hat{\beta}_1)=2/\sqrt{100}=2/10=0.2\).
The statistic is \(t=(-0.55-(-0.10))/0.2=(-0.45)/0.2=-2.25\). For 10 degrees of freedom, the lower-tail 5% critical value is approximately \(-1.812\). Because \(-2.25<-1.812\), reject \(H_0\). The data provide evidence that the population slope is below \(-0.10\) defects per unit of sensor reading.
Check: Subtracting the negative benchmark gives \(-0.55+0.10=-0.45\), so the statistic must be negative. The common mistake is to use the upper-tail critical value simply because critical values are often listed as positive numbers; the lower-tail rejection region is below the negative critical value.
Worked Example: A Pre-Specified Positive-Slope Test Near the Cutoff
Method: Use an upper-tailed t test and compare with a one-sided, not a two-sided, critical value. A hypothetical analyst studies whether a higher calibration setting predicts a higher instrument response. Assume independent normal errors with common positive variance and fixed predictor values. The summaries are \(n=8\), \(\hat{\beta}_1=0.95\), \(S_{xx}=18\), and \(SSE=24\). Test \(H_0:\beta_1\leq0\) against \(H_1:\beta_1>0\) at \(\alpha=0.05\).
The degrees of freedom are \(8-2=6\). Thus \(s^2=24/6=4\) and \(s=2\). The standard error is \(2/\sqrt{18}\), which is approximately \(0.4714\). Therefore, \(t=(0.95-0)/(2/\sqrt{18})=0.95\sqrt{18}/2\), approximately \(2.0158\).
The one-sided upper critical value for 6 degrees of freedom at \(\alpha=0.05\) is approximately \(1.943\). Since \(2.0158>1.943\), reject \(H_0\) for this upper-tailed test. The result is just beyond the one-sided cutoff. By contrast, the 5% two-sided critical value is approximately \(2.447\), which is larger than \(2.0158\); that comparison belongs to a different hypothesis test.
Check: The standard error is \(2/\sqrt{18}\), not \(2/18\), because the formula divides by the square root of \(S_{xx}\). The test direction must come from the stated claim, not from choosing whichever tail makes the observed statistic look more persuasive.
Common Mistakes and Interpretation
- Choosing the tail after seeing the data. If the scientific question is whether the slope is positive, specify the upper-tailed alternative before examining the estimated slope. Switching directions after observing the sign changes the testing procedure.
- Using the wrong sign in the statistic. Always subtract the hypothesized boundary: \(\hat{\beta}_1-\beta_{10}\). In a lower-tailed test, the relevant evidence is a statistic that is sufficiently negative.
- Using a two-sided cutoff. A one-sided test places all of \(\alpha\) in one tail; a two-sided test places \(\alpha/2\) in each tail. They can therefore lead to different decisions.
- Forgetting residual degrees of freedom. The slope test uses \(n-2\), not \(n-1\), because the intercept and slope have both been estimated.
- Ignoring whether the test is defined. The exact t procedure requires \(n>2\) and \(S_{xx}>0\), in addition to the model assumptions. A numerical slope alone does not establish that the test is valid.
A rejection supports the directional alternative at the chosen significance level, conditional on the regression model being appropriate. It does not establish that the slope has a particular practical size, and a failure to reject does not show that the slope is exactly the null boundary. Keep the estimated slope, the tested boundary, the direction of the alternative, and the critical tail distinct when interpreting the result.
Check Your Understanding
For each problem, apply the stated one-sided alternative and use the exact regression t test. Critical values are approximate.
- A hypothetical study tests whether a population slope is below \(-0.2\). The summaries are \(n=9\), \(\hat{\beta}_1=-0.8\), \(S_{xx}=16\), and \(SSE=28\). At \(\alpha=0.05\), calculate the statistic and decide using the lower-tail critical value for 7 degrees of freedom, approximately \(-1.895\). (Answer: \(s=2\), \(SE=0.5\), \(t=-1.2\); do not reject \(H_0\).)
- For an upper-tailed test of \(H_0:\beta_1\leq0.15\) against \(H_1:\beta_1>0.15\), a hypothetical regression has \(n=15\), \(\hat{\beta}_1=0.65\), \(S_{xx}=64\), and \(SSE=52\). Use \(\alpha=0.05\) and the upper critical value \(1.771\) for 13 degrees of freedom. (Answer: \(s=2\), \(SE=0.25\), \(t=2.0\); reject \(H_0\).)
- A lower-tailed test uses \(H_0:\beta_1\geq0.5\) against \(H_1:\beta_1<0.5\). The hypothetical summaries are \(n=10\), \(\hat{\beta}_1=0.1\), \(S_{xx}=36\), and \(SSE=32\). At \(\alpha=0.05\), use the lower critical value \(-1.860\) for 8 degrees of freedom. (Answer: \(s=2\), \(SE=1/3\), \(t=-1.2\); do not reject \(H_0\).)
- A hypothetical study tests \(H_0:\beta_1\leq0.3\) against \(H_1:\beta_1>0.3\) at the 5% level. Its summaries are \(n=7\), \(\hat{\beta}_1=0.8\), \(S_{xx}=20\), and \(SSE=10\). The upper critical value for 5 degrees of freedom is approximately \(2.015\). Calculate the standard error and make the decision. (Answer: \(s=\sqrt{2}\), \(SE=\sqrt{2/20}\approx0.3162\), \(t\approx1.5811\); do not reject \(H_0\).)
- An analyst wants to test whether a population slope is positive, but after estimating a negative slope proposes changing the alternative to \(H_1:\beta_1<0\). Explain why this change is not a valid way to conduct the originally planned test. (Answer: The direction must be specified before examining the estimate; changing it after seeing the data selects the favorable tail and does not preserve the stated test’s significance level.)