# 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

$$\beta_1=0$$

$$\beta_1=0$$

$$\beta_1=0$$

Alternative Hypothesis

$$\beta_1\ne0$$

$$\beta_1>0$$

$$\beta_1<0$$

Type of Test

Two-tailed, non-directional

Right-tailed, directional

Left-tailed, directional

The test statistic for the test of population slope is:

$$t^*=\dfrac{\hat{\beta}_1}{\hat{SE}(\hat{\beta}_1)}$$

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.