Confidence Interval for the Mean in Linear Regression

Definition

Confidence Interval for the Mean in Simple Linear Regression

Consider a Simple Linear Regression model $Y_i={\beta }_0+{\beta }_1{X}_i+{ϵ}_i$ The $100(1-\alpha )\%$ confidence region for the mean response $E(Y)$, when $x$ is given, is defined as $({\hat{\beta }}_0+{\hat{\beta }}_{1}x)\pm t_{\alpha /2}(n-2)\times s\sqrt{\frac{1}{n}+\frac{(x-\bar{X}{)}^2}{S_{XX}}}$ where $s^2=\frac{{\sum }_{i=1}^n(Y_i-\hat{Y}{)}^2}{n-2}$

Confidence Interval for the Mean in Multiple Linear Regression

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ The $100(1-\alpha )\%$ confidence region for the mean response $E(Y)$, when ${\mathbf{x}}_0$ is given, is defined as ${\mathbf{x}}_0^{⊺}\mathbf{\beta }\pm t_{\alpha /2}(n-p)\times s\sqrt{{\mathbf{x}}_0^{⊺}({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{\mathbf{x}}_0}$ where $s^2=\frac{{\sum }_{i=1}^n(Y_i-\hat{Y}{)}^2}{n-p}$ and $p-1$ is the number of explanatory variables