Why do we use T distribution in linear regression?

The thing I know is- We use the T-distribution when estimating the mean of a normally distributed population in situations where the sample size is small and the population standard deviation is unknown.

In the case of linear regression, I understand that sometimes population standard deviation is unknown so it's understandable to use a T-test. However, when we talk about the distribution of the population to be normal, as far as I know, in linear regression only the errors are normally distributed and not the parameters. So, I am confused regarding are we using T-test in linear regression because the errors are normally distributed? and if yes, how can the errors being normal could be translated as the whole population being normal which then lead us to use the T distribution.

Edit- According to the book "Econometric By Examples" written by Damodar Gujrati-

"It is very important to note that the use of the t and F tests is explicitly based on the assumption that the error term, is normally distributed. If this assumption is not enabled, the t and F testing procedure is invalid in small samples, although they can still be used if the sample is sufficiently large (technically infinite)".

Can someone please tell me why this assumption of normality is required in order to perform the T test?

  • $\begingroup$ the parameters are a linear function of the residuals so also normally distributed see eg en.wikipedia.org/w/…, consistency and asymptotic normality $\endgroup$
    – seanv507
    Commented Oct 13, 2021 at 9:21

1 Answer 1


The $t$-distribution is used for performing hypothesis tests of the parameters of the model (using a $t$-test), since we do not know the true standard deviations (errors) of these parameters. So we estimate these SE's from the data and hence use a $t$-test (instead of say a $z$-test).

Your first statement about using a $t$-test for testing the mean for normally distributed data is... not true, maybe. Normally distributed data is not a really a requirement for a $t$-test (although there are others which might coincide).

  • $\begingroup$ I want to add on that that the reasoning behind using the t-distribution here is to incorporate the increased unceratainty that comes with having to estimate the standard errors here. The t-distribution encapsulates this because it has longer tails than the normal distribution. The normal assumption is not needed when the CLT kicks in. $\endgroup$
    – BalaGizeh
    Commented Oct 13, 2021 at 5:52

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