Let $X \sim f(x, \theta)$ and suppose an estimator $S_n$ of $\theta$ is such that $S_n \to \chi^2_1$.
If we want to test at level $a$ the hypothesis $H_0 : \theta= \theta_0$ vs $H_1 : \theta \not= \theta_0$ we just check $S_n> \chi^2_a$.
What can we do if we want to test $H_0 : \theta= \theta_0$ vs $H_1 : \theta > \theta_0$?
Thanks!
For most purposes, these two tests are equivalent.
A $\chi^2$ test is a two-sided test but we are usually interested in only one of the tails of the distribution. If the statistic were too far to the left, we would be worried that it fits $too$ well, which we aren't usually concerned about.
For example, noting that if
$$Z_1,..,Z_k \sim N(0,1)$$ and letting $$Q=\sum_{i=1}^k Z_i^2$$ we have$$Q\sim\chi_k^2$$
the left tail of a chi-squared distribution would essentially be like looking at z-scores that are close to $0$. We wouldn't reject these just because they fit $too$ well, by chance.
For an example where a two-sided $\chi^2$ comes into play, Fisher used a two-tailed test in assessing whether Mendel's results were "too good to be true". Here is a link to a relavant article if you're interested.