Likelihood Ratio Test statistic to test $H_0$ vs $H_1$ 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 $\Lambda &lt; k$ and then solving the inequality

$\sqrt{-2\text{log}(k)}&lt;\sum x_iy_i/\sqrt{\sum_{i=1}^n x_i^2} &lt;-\sqrt{-2\text{log}(k)}$

(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
$-\sqrt{-2\text{log}(k)}&lt;\sum x_iy_i/\sqrt{\sum_{i=1}^n x_i^2} &lt;
\sqrt{-2\text{log}(k)}$
The algebra looks correct to me except you have the signs flipped on the last line. It should be
$-\sqrt{-2\text{log}(k)}&lt;\sum x_iy_i/\sqrt{\sum_{i=1}^n x_i^2} &lt;
\sqrt{-2\text{log}(k)}$