9.2.4  Inferences about the Population Slope
9.2.4  Inferences about the Population SlopeIn 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 
Twotailed, nondirectional 
Righttailed, directional 
Lefttailed, 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 \(n2\) degrees of freedom.
Note! In this class, we will have Minitab perform the calculations for this test. Minitab's output gives the result for twotailed tests for \(\beta_1\) and \(\beta_0\). If you wish to perform a onesided test, you would have to adjust the pvalue 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 \(n2\) degrees of freedom.