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.
In the exercises this week we show 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 can be used to construct a confidence interval for the estimates.