# For bivariate linear model, if the residuals ARE normal, but the data is NOT normal, can I make inferences about the slope coefficient? [duplicate]

For convenience and context, I'm looking at the formula for standard error of the slope coefficient from here: https://www.statology.org/standard-error-of-regression-slope/ If X is not normally distributed but the residuals are, is the estimated slope coefficient's distribution characterized by the T-distribution?

• Residuals are never independently Normally distributed, so you likely are trying to ask about the errors in the model.
– whuber
May 31 at 21:00
• @whuber, yes your comment makes sense. Jun 1 at 1:55
• In Kutner's Applied Linear Statistical Models Ch2 Page 42, it explains that the sampling distribution of $b_1$ is normal because $b_1$ is a linear combination of $Y_i$ and if we assume $Y_i$ is normally distributed, then a linear combination of $Y_i$ is also normally distributed. I think this is the answer I was looking for, if I'm not misunderstanding the passage. Would love feedback on this.. Jun 19 at 0:58
• It contradicts what you write. Because the estimated slope is Normal, it is not characterized by a Student t distribution.
– whuber
Jun 19 at 17:01