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 \(df=n-2\)
\(SE_{b_1}\) = 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 | 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 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.
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 | T-Value | P-Value | 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 (50-2) = 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.