Let's say I have $X_i \sim Bi(1, \theta$) and want to test $H_0: \theta \geq \theta_0$ vs $H_1: \theta < \theta_0$.
I've found that $\lambda = \frac{\sup_{\theta \in \Theta_0}L(\theta)}{\sup_{\theta \in \Theta}L(\theta)} = \frac{\theta_0^{\sum_{i=1}^n X_i}(1-\theta_0)^{n-\sum_{i=1}^n X_i}}{\bar{X}^{\sum_{i=1}^n X_i}(1-\bar{X})^{n-\sum_{i=1}^n X_i}}$. Now we know that $U = -2\ln(\lambda) \sim \chi^2_1$.
Would the rejection region for this test be $U \geq \chi^2_{1,1-\alpha}$ or $U \leq \chi^2_{1,\alpha}$?
