# 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 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 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 scores

Data 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.

  Link ↥ Has Tooltip/Popover Toggleable Visibility