We study the exceptional set estimate for projections in $\mathbb{ F}_q^n$. For each $V\in G(k,\mathbb{ F}^n_q)$, let $$ \pi_V: \mathbb{ F}_q^n\rightarrow V $$ be the projection map. We prove the following result: If $A\subset \mathbb{ F}_q^n$ with $\#A=q^a$ ($n-1\le a\le n$) and $0< s<\frac{ a+n-2}{ 2}$, then $$ \# \{ V\in G(n-1,\mathbb{ F}^n_q): \#\pi_V(A)< q^s \}\lessapprox q^{ n-2}.$$ This improves the previous range $0<s<\frac{ n-1}{ n}a$. Also, our range of $s$ is sharp in the sense that if $s>\frac{ a+n-2}{ 2}$, then the right hand side above should be at least $q^t$ for some $t>n-2$.
Categories: math.CA