Confidence intervals are used in statistics and represent a type of estimate computed from the observed data. This gives a range of values for an unknown parameter such as the parameters \( \boldsymbol{\beta} \) from linear regression.
With the OLS expressions for the parameters \( \boldsymbol{\beta} \) we found \( \mathbb{E}(\boldsymbol{\beta}) = \boldsymbol{\beta} \), which means that the estimator of the regression parameters is unbiased.
We found also that the variance of the estimate of the \( j \)-th regression coefficient is \( \boldsymbol{\sigma}^2 (\boldsymbol{\beta}_j ) = \boldsymbol{\sigma}^2 [(\mathbf{X}^{T} \mathbf{X})^{-1}]_{jj} \).
This quantity will be used to construct a confidence interval for the estimates.