How to Calculate Confidence Interval for Regression Slope

Simple linear regression is used to quantify the relationship between a predictor variable and a response variable.

This method finds a line that best “fits” a dataset and takes on the following form:

ŷ = b₀ + b₁x

where:

ŷ: The estimated response value
b₀: The intercept of the regression line
b₁: The slope of the regression line
x: The value of the predictor variable

Often we’re interested in the value for b₁, which tells us the average change in the response variable associated with a one unit increase in the predictor variable.

We can use the following formula to calculate a confidence interval for the value of β₁, the value of the slope for the overall population:

Confidence Interval for β₁: b₁ ± t_{1-α/2, n-2} * se(b₁)

where:

b₁ = Slope coefficient shown in the regression table
t_{1-∝/2, n-2} = The t critical value for confidence level 1-∝ with n-2 degrees of freedom where n is the total number of observations in our dataset
se(b₁) = The standard error of b₁ shown in the regression table

The following example shows how to calculate a confidence interval for a regression slope in practice.

Example: Confidence Interval for Regression Slope

Suppose we’d like to fit a simple linear regression model using hours studied as a predictor variable and exam score as a response variable for 15 students in a particular class:

We can perform simple linear regression in Excel and receive the following output:

Using the coefficient estimates in the output, we can write the fitted simple linear regression model as:

Score = 65.334 + 1.982*(Hours Studied)

The value for the regression slope is 1.982.

This tells us that each additional one hour increase in studying is associated with an average increase of 1.982 in exam score.

We can use the following formula to calculate a 95% confidence interval for the slope:

95% C.I. for β₁: b₁ ± t_{1-α/2, n-2} * se(b₁)
95% C.I. for β₁: 1.982 ± t_{.975, 15-2} * .248
95% C.I. for β₁: 1.982 ± 2.1604 * .248
95% C.I. for β₁: [1.446, 2.518]

The 95% confidence interval for the regression slope is [1.446, 2.518].

Since this confidence interval doesn’t contain the value 0, we can conclude that there is a statistically significant association between hours studied and exam score.

Note: We used the Inverse t Distribution Calculator to find the t critical value that corresponds to a 95% confidence level with 13 degrees of freedom.