# Relationship between the t-statistic and F-statistic in simple linear regression

Consider the simple linear regression model $$y_i = \beta_0 + \beta_1{x_i} + \epsilon_i$$

Suppose in an OLS, the $$t$$-statistic for the null hypothesis $$\beta_1 = 0$$ is $$1.92$$, what is the $$F$$-statistic for the overall significance of the model?

I am aware that $$F$$ statistic for testing exclusion of a single variable is equal to the square of the corresponding $$t$$ statistic: however, I can't seem to find how to link that information to this question. Am I missing information?

• Consider what the null hypothesis would be for the “significance of the entire model”.
– Dave
Commented Nov 28, 2021 at 22:45
• Oh it would just be $\beta_1$ = 0, wouldn't it? Commented Nov 28, 2021 at 23:11
• That’s what I would say, and that’s what R gives you in a “summary” of a linear model. And then the alternative hypothesis would be $\ne$. // This is a subtle point, but good for you for not writing $\hat\beta_1=0$.
– Dave
Commented Nov 28, 2021 at 23:16
• Thank you! I spent way too much time looking for an answer when it was sitting right in front me Commented Nov 28, 2021 at 23:19
• Since you pretty much solved the problem yourself, you might consider posting a self-answer. (Cross Validated does not consider it arrogant or poor etiquette to do so. In fact, I have posted several self-answers.)
– Dave
Commented Nov 28, 2021 at 23:37

In the simple linear regression model, testing the significance of the model requires testing the hypothesis: $$H_0 \colon \beta_1 = 0 \\ H_0 \colon \beta_1 \neq 0$$
Given that we are whether a single parameter is equal to 0, the $$F$$ statistic will simply equal the square of the corresponding $$t$$ statistic. This means that if the $$t$$-statistic for the null is $$1.92$$, the $$F$$-statistic will simply be $$1.92^2$$.