9.2.4 - Inferences about the Population Slope

In this section, we will present the hypothesis test and the confidence interval for the population slope. A similar test for the population intercept, \(\beta_0\), is not discussed in this class because it is not typically of interest.

Hypothesis Test for the Population Slope


Research Question

Is there a linear relationship?

Is there a positive linear relationship?

Is there a negative linear relationship?

Null Hypothesis




Alternative Hypothesis




Type of Test

Two-tailed, non-directional

Right-tailed, directional

Left-tailed, directional

The test statistic for the test of population slope is:


where \(\hat{SE}(\hat{\beta}_1)\) is the estimated standard error of the sample slope (found in Minitab output). Under the null hypothesis and with the assumptions shown in the previous section, \(t^*\) follows a \(t\)-distribution with \(n-2\) degrees of freedom.

Note! In this class, we will have Minitab perform the calculations for this test. Minitab's output gives the result for two-tailed tests for \(\beta_1\) and \(\beta_0\). If you wish to perform a one-sided test, you would have to adjust the p-value Minitab provides.

Confidence Interval for the Population Slope
\( (1-\alpha)100\)% Confidence Interval for the Population Slope

The \( (1-\alpha)100\)% confidence interval for \(\beta_1\) is:

\(\hat{\beta}_1\pm t_{\alpha/2}\left(\hat{SE}(\hat{\beta}_1)\right)\)

where \(t\) has \(n-2\) degrees of freedom.

Note! The degrees of freedom of t depends on the number of independent variables. The degrees of freedom is \(n - 2\) when there is only one independent variable.