Page "Least squares" Paragraph 55

Confidence limits can be found if the probability distribution of the parameters is known, or an asymptotic approximation is made, or assumed.

Likewise statistical tests on the residuals can be made if the probability distribution of the residuals is known or assumed.

The probability distribution of any linear combination of the dependent variables can be derived if the probability distribution of experimental errors is known or assumed.

Inference is particularly straightforward if the errors are assumed to follow a normal distribution, which implies that the parameter estimates and residuals will also be normally distributed conditional on the values of the independent variables.