12.3.5  Confidence Interval for Slope
12.3.5  Confidence Interval for SlopeWe can use the slope that was computed from our sample to construct a confidence interval for the population slope (\(\beta_1\)). This confidence interval follows the same general form that we have been using:
 General Form of a Confidence Interval
 \(sample statistic\pm(multiplier)\ (standard\ error)\)
 Confidence Interval of \(\beta_1\)
 \(b_1 \pm t^\ast (SE_{b_1})\)

\(b_1\) = sample slope
\(t^\ast\) = value from the \(t\) distribution with \(df=n2\)
\(SE_{b_1}\) = standard error of \(b_1\)
Example: Confidence Interval of \(\beta_1\)
Below is the Minitab Express output for a regression model using Test 3 scores to predict Test 4 scores. Let's construct a 95% confidence interval for the slope.
Term  Coef  SE Coef  TValue  PValue  VIF 

Constant  16.37  12.40  1.32  0.1993  
Test 3  0.8034  0.1360  5.91  <0.0001  1.00 
From the Minitab Express output, we can see that \(b_1=0.8034\) and \(SE(b_1)=0.1360\)
We must construct a \(t\) distribution to look up the appropriate multiplier. There are \(n2\) degrees of freedom.
\(df=262=24\)
\(t_{24,\;.05/2}=2.064\)
\(b_1 \pm t \times SE(b_1)\)
\(0.8034 \pm 2.064 (0.1360) = 0.8034 \pm 0.2807 = [0.523,\;1.084]\)
We are 95% confident that \(0.523 \leq \beta_1 \leq 1.084 \)
In other words, we are 95% confident that in the population the slope is between 0.523 and 1.084. For every one point increase in Test 3 the predicted value of Test 4 increases between 0.523 and 1.084 points.
12.3.5.1  Video Example: Exam.MTW
12.3.5.1  Video Example: Exam.MTWThis example uses the
dataset. Students' quiz averages in a course are used to predict their final exam scores in the course. In the video below, a 95% confidence interval is constructed for the slope.