I have the linear model $y = \beta x + \alpha + \epsilon$ with $\epsilon$ i.i.d normally distributed with variance $\sigma^2$.
I fit the linear regression using OLS and compute a prediction interval et confidence $\gamma$ for a new observation $x_0$ using the formula:
$\widehat{\alpha}+\widehat{\beta} x_0 \pm t_{n-2 ; 1-\gamma / 2} \cdot \widehat{\sigma} \sqrt{1+\frac{1}{n}+\frac{\left(x_0-\bar{x}\right)^2}{S S_X}}$
with $S S_X=\sum_{i=1}^n\left(x_i-\bar{x}\right)^2$
I'm now wondering if I can solve the two bounds for $x_0$ to obtain a range of possible values for $x_0$ at confidence level $\gamma$.
Usually, when testing multiple points, one has to use a Bonferroni correction to keep the significance level. Since I'm trying to solve for $x_0$ and get an interval(infinitely many values), I'm wondering if I should also have any sort of correction on the $t$ statistic.