# 9.2.4 - Inferences about the Population Slope

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.

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