Confidence intervals of linear multiple regression

Considering following linear multiple regression model:

where observations , coefficents and is a white Gaussian noise term.

By the definition of the confidence interval:

where is the critical value, and standard error

I understand that

My question is that why in post:https://stats.stackexchange.com/questions/16493/difference-between-confidence-intervals-and-prediction-intervals

and

https://stats.stackexchange.com/questions/9131/obtaining-a-formula-for-prediction-limits-in-a-linear-model-i-e-prediction-in

they use sqaure root of Variance

rather than the standard error?? Where is the ?