12.3.4 - Confidence Interval for Slope

We 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 \(df=n-2\)
\(SE_{b_1}\) = standard error of \(b_1\)

Example: Confidence Interval of \(\beta_1\) Section

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.

Coefficients
Term Coef SE Coef T-Value P-Value 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 \(n-2\) degrees of freedom.

\(df=26-2=24\)

\(t_{24,\;.05/2}=2.064\)

\(b_1 \pm t \times SE(b_1)\)

\(0.8034 \pm 2.064 (0.1360) = 0.8034 \pm 0.2807 = [0.523,\;1.084]\)

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.