An application of random plane slicing to counting $\mathbb{F}_q$-points on hypersurfaces

Let $X$ be an absolutely irreducible hypersurface of degree $d$ in $\mathbb{A}^n$, defined over a finite field $\mathbb{F}_q$. The Lang-Weil bound gives an interval that contains #X(\mathbb{F}_q). We exhibit explicit intervals, which do not contain #X(\mathbb{F}_q), and which overlap with the Lang-Weil interval. In particular, we sharpen the best known lower and upper bounds for #X(\mathbb{F}_q). The proof uses a combinatorial probabilistic technique.