Confidence intervals of linear multiple regression

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Mingyang Sun
Mingyang Sun 2021-4-2
编辑： Mingyang Sun 2021-4-2
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 ?

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