# 12.3.5 - Confidence Interval for Slope

12.3.5 - 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 $\beta_1$

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.

# 12.3.5.1 - Video Example: Exam.MTW

12.3.5.1 - Video Example: Exam.MTW

This example uses the EXAM.MTW 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.

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