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.
|
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.
- \( (1-\alpha)100\)% 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.