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
$$e^{-1/2\left(\frac{\sum_{i=1}^n y_i x_i}{\sum_{i=1}^n x_i^2}\right)^2\sum_{i=1}^n x_i^2}$$
After setting $\Lambda < k$ and then solving the inequality
$\sqrt{-2\text{log}(k)}<\sum x_iy_i/\sqrt{\sum_{i=1}^n x_i^2} <-\sqrt{-2\text{log}(k)}$
(Someone told me that my last step is incorrect)
The algebra looks correct to me except you have the signs flipped on the last line. It should be $-\sqrt{-2\text{log}(k)}<\sum x_iy_i/\sqrt{\sum_{i=1}^n x_i^2} < \sqrt{-2\text{log}(k)}$