How to Perform t-Test for Slope of Regression Line in R

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

ŷ = b₀ + b₁x

We typically want to know if the slope coefficient, b₁, is statistically significant.

To determine if b₁ is statistically significant, we can perform a t-test with the following test statistic:

t = b₁ / se(b₁)

where:

se(b₁) represents the standard error of b₁.

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 = b₁ / se(b₁)
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.