Prediction interval for a New Response in Linear Regression

Definition

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

Facts

The prediction interval is always wider than the Confidence Interval.