# prediction interval formula

I have a model $Y_i= \beta_0 + \beta_1X_i+\beta_2X_i^2+\beta_3X_i^3+\epsilon_i$ with $\epsilon_i\sim\mathcal{N}(0,\sigma^2)$. Is the following formula correct for calculating the width of a 95% prediction interval for a new datapoint $x_{new}$:

$2 \times t_{0.975,n-4} \sigma_p$,

where $\sigma_p^2=\sigma^2[x_{new}(X^tX)^{-1}x_{new}+1]$ and $X$ is the design matrix.

If I have only 4 points in the design matrix ($n=4$), it means I need to take a quantile of a t-distribution with 0 degrees of freedom. There must be something wrong with my formula.

It is not correct, the width of the interval is $$2\times t_{0.975, n-p}\sigma_p,$$ i.e., without the square.