**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 = b _{0} + b_{1}x**

where:

**ŷ**: The estimated response value**b**: The intercept of the regression line_{0}**b**: The slope of the regression line_{1}**x**: The value of the predictor variable

Often we’re interested in the value for b_{1}, which tells us the average change in the response variable associated with a one unit increase in the predictor variable.

However, in rare circumstances we’re also interested in the value for b_{0}, which tells us the average value of the response variable when the predictor variable is equal to zero.

We can use the following formula to calculate a confidence interval for the value of β_{0}, the true population intercept:

Confidence Interval for β_{0}: b_{0}± t_{α/2, n-2}* se(b_{0})

The following example shows how to calculate a confidence interval for an intercept in practice.

**Example: Confidence Interval for Regression Intercept**

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:

The following code shows how to fit this simple linear regression model in R:

#create data frame df <- data.frame(hours=c(1, 2, 4, 5, 5, 6, 6, 7, 8, 10, 11, 11, 12, 12, 14), score=c(64, 66, 76, 73, 74, 81, 83, 82, 80, 88, 84, 82, 91, 93, 89)) #fit simple linear regression model fit <- lm(score ~ hours, data=df) #view summary of model summary(fit) Call: lm(formula = score ~ hours, data = df) Residuals: Min 1Q Median 3Q Max -5.140 -3.219 -1.193 2.816 5.772 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 65.334 2.106 31.023 1.41e-13 *** hours 1.982 0.248 7.995 2.25e-06 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 3.641 on 13 degrees of freedom Multiple R-squared: 0.831, Adjusted R-squared: 0.818 F-statistic: 63.91 on 1 and 13 DF, p-value: 2.253e-06

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 intercept value is 65.334. This tells us that the mean estimated exam score for a student who studies for zero hours is **65.334**.

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

- 95% C.I. for β
_{0}: b_{0}± t_{α/2, n-2}* se(b_{0}) - 95% C.I. for β
_{0}: 65.334 ± t_{.05/2, 15-2}* 2.106 - 95% C.I. for β
_{0}: 65.334 ± 2.1604 * 2.106 - 95% C.I. for β
_{0}: [60.78, 69.88]

We interpret this to mean that we’re 95% confident that the true population mean exam score for students who study for zero hours is between 60.78 and 69.88.

**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.

**Cautions on Calculating a Confidence Interval for a Regression Intercept**

We often don’t calculate a confidence interval for a regression intercept in practice because it usually doesn’t make sense to interpret the value of the intercept in a regression model.

For example, suppose we fit a regression model that uses height of a basketball player as a predictor variable and average points per game as a response variable.

It’s not possible for a player to be zero feet tall, so it wouldn’t make sense to interpret the intercept literally in this model.

There are countless scenarios like this where a predictor variable can’t take on a value of zero so it doesn’t make sense to interpret the intercept value of the model or create a confidence interval for the intercept.

For example, consider the following potential predictor variables in a model:

- Square footage of a house
- Length of a car
- Weight of a person

Each of these predictor variables can’t take on a value of zero, so it wouldn’t make sense to calculate a confidence interval for the intercept of a regression model in any of these circumstances.

**Additional Resources**

The following tutorials provide additional information about linear regression:

Introduction to Simple Linear Regression

Introduction to Multiple Linear Regression

How to Read and Interpret a Regression Table

How to Report Regression Results