Whenever we perform simple linear regression, we end up with the following estimated regression equation:

**ŷ = b _{0} + b_{1}x**

We typically want to know if the slope coefficient, b_{1}, is statistically significant.

To determine if b_{1} is statistically significant, we can perform a t-test with the following test statistic:

**t = b _{1} / se(b_{1})**

where:

**se(b**represents the standard error of b_{1})_{1}.

We can then calculate the p-value that corresponds to this test statistic with n-2 degrees of freedom.

If the p-value is less than some threshold (e.g. α = .05) then we can conclude that the slope coefficient is different than zero.

In other words, there is a statistically significant relationship between the predictor variable and the response variable in the model.

The following example shows how to perform a t-test for the slope of a regression line in R.

**Example: Performing a t-Test for Slope of Regression Line in R**

Suppose we have the following data frame in R that contains information about the hours studied and final exam score received by 12 students in some class:

#create data frame df <- data.frame(hours=c(1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 6, 8), score=c(65, 67, 78, 75, 73, 84, 80, 76, 89, 91, 83, 82)) #view data frame df hours score 1 1 65 2 1 67 3 2 78 4 2 75 5 3 73 6 4 84 7 5 80 8 5 76 9 5 89 10 6 91 11 6 83 12 8 82

Suppose we would like to fit a simple linear regression model to determine if there is a statistically significant relationship between hours studied and exam score.

We can use the lm() function in R to fit this regression model:

#fit simple 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 -7.398 -3.926 -1.139 4.972 7.713 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 67.7685 3.3757 20.075 2.07e-09 *** hours 2.7037 0.7456 3.626 0.00464 ** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 5.479 on 10 degrees of freedom Multiple R-squared: 0.568, Adjusted R-squared: 0.5248 F-statistic: 13.15 on 1 and 10 DF, p-value: 0.004641

From the model output, we can see that the estimated regression equation is:

Exam score = 67.7685 + 2.7037(hours)

To test if the slope coefficient is statistically significant, we can calculate the t-test statistic as:

- t = b
_{1}/ se(b_{1}) - t = 2.7037 / 0.7456
- t =
**3.626**

The p-value that corresponds to this t-test statistic is shown in the column called **Pr(> |t|)** in the output.

The p-value turns out to be **0.00464**.

Since this p-value is less than 0.05, we conclude that the slope coefficient is statistically significant.

In other words, there is a statistically significant relationship between the number of hours studied and the final score that a student receives on the exam.

**Additional Resources**

The following tutorials explain how to perform other common tasks in R:

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Interpret Regression Output in R