**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.

We can use the following formula to calculate a confidence interval for the value of β_{1}, the value of the slope for the overall population:

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

where:

**b**= Slope coefficient shown in the regression table_{1}**t**= The t critical value for confidence level 1-∝ with n-2 degrees of freedom where_{1-∝/2, n-2}*n*is the total number of observations in our dataset**se(b**= The standard error of b_{1})_{1}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 intercept:

- 95% C.I. for β
_{1}: b_{1}± t_{1-α/2, n-2}* se(b_{1}) - 95% C.I. for β
_{1}: 1.982 ± t_{.975, 15-2}* .248 - 95% C.I. for β
_{1}: 1.982 ± 2.1604 * .248 - 95% C.I. for β
_{1}: [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.

**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