Given the following problem:
Suppose pairs $(x_i,y_i)$ are observed independently for $i = 1,2,...,n,$ where the $x_i$’s are constants and $Y_i = g(x_i) + ε_i$ with $g(x_i) = \beta_0 + \beta_1x_i$ and the $ε_i \sim N(0, \sigma^2)$. In the classic callibration problem discussed by Scheffe, an additional measurement, $Y_0$ is made, but no corresponding $x_0$ is available. Find functions $L$ and $U$, depending on pairs $(x_i,Y_i)$ and $Y_0$ such that $P(L \leq x_0 \leq U)=0.95$.
I was able to find my pivot: $$\frac{Y_0 - \hat{g(x_0)}}{s\sqrt{1+x_0^T(X^TX)^{-1}x_0}}\sim t_{n-2}$$ and so $$0.95 = P\left(-t_{n-1,0.025} \leq \frac{Y_0 - \hat{g(x_0)}}{s\sqrt{1+x_0^T(X^TX)^{-1}x_0}}\leq t_{n-1,0.025}\right)$$
However, I am not sure how to solve this for $x_0$.