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 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 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  Example: Quiz and exam scores
12.3.5.1  Example: Quiz and exam scoresData from a sample of 50 students were used to build a regression model using quiz averages to predict final exam scores. Construct a 95% confidence interval for the slope.
This example uses the Minitab file: Exam.mpx
We can use the coefficients table that we produced in the previous regression example using the exam data.
Coefficients
Term  Coef  SE Coef  TValue  PValue  VIF 

Constant  12.1  11.9  1.01  0.315  
Quiz_Average  0.751  0.141  5.31  0.000  1.00 
The general form of a confidence interval is sample statistic \(\pm\) multiplier(standard error).
We have the following:
 \(b_1\) (sample slope) is 0.751
 t multiplier for degrees of freedom of (502) = 48 is 2.01
 The standard error of the slope (\(SE_{b_1}\) is 0.141 from our table
The confidence interval is...
\begin{align} \text{sample statistic} &\pm \text{multiplier*standard error}\\ 0.751 &\pm 2.01 (0.141)\\ 0.751&\pm 0.283 \\ [0.468 &, 1.034] \end{align}
Interpret
I am 95% confident that the slope for this model is between 0.468 and 1.034 in the population.