Let ${Y_1,...,Y_n}$ be independent random variables and $Y_i$~$N(\beta x_i, 1)$ where $x_1,...,x_n$ are fixed known constants, and $\beta$ is an unknown parameter.
I'm looking to find the p-value or rejection region for the test
$$H_0: \beta=0 \quad \text{vs} \quad H_1:\beta\ne0$$
The Likelihood Ratio Test statistic $\Lambda$ is
$$e^{-1/2\left(\frac{\sum_{i=1}^n y_i x_i}{\sum_{i=1}^n x_i^2}\right)^2\sum_{i=1}^n x_i^2}$$
I have already asked a similar question with $H_1: \beta>0$ here: Uniformly Most Powerful test for normal distribution.
After setting $\Lambda < k$ I'm left with $\sum x_iy_i$ after removing constants to the right, which is the same as my previous question. How will finding the p-value or rejection region for this test differ?