In a linear regression model, a regression coefficient 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 a regression coefficient:

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

where:

**b**= Regression 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 Coefficient in R**

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 use the lm() function 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 linear regression model fit <- lm(score ~ hours, data=df) #view model summary 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)

Notice that the regression coefficient for hours 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 **confint()** function to calculate a 95% confidence interval for the regression coefficient:

#calculate confidence interval for regression coefficient for 'hours' confint(fit, 'hours', level=0.95) 2.5 % 97.5 % hours 1.446682 2.518068

The 95% confidence interval for the regression coefficient 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.

We can also confirm this is correct by calculating the 95% confidence interval for the regression coefficient by hand:

- 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 coefficient is **[1.446, 2.518]**.

**Note #1**: 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.

**Note #2**: To calculate a confidence interval with a different confidence level, simply change the value for the **level** argument in the **confint()** function.

**Additional Resources**

The following tutorials provide additional information about linear regression in R:

How to Interpret Regression Output in R

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Perform Logistic Regression in R