Let $y_i$ be a set of data points and $\hat y_i(a,b,c)$ be a set of estimation. Suppose that the chi squared statics was minimized to a real number $A$.(For simplicity, suppose that $a,b,c$ were uncorrelated in the test statics.)
$$\chi^2(a_{min},b_{min},c_{min})=\sum_i \frac{(y_i -\hat y_i)^2}{\hat y_i}=A$$
Then, I was told that $[a_{min}-\beta ,a_{min}+\alpha]$ such that $\chi^2(a_{min}-\beta,b_{min},c_{min})=\chi^2 (a_{min}+\alpha,b_{min},c_{min})=A+1$ corresponding to the 1 sigma error of the estimated parameter $a$ at $a_{min}$.
Could you provide some explanation and proof over why this was?