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 trying to find the likelihood ratio test statistic $\Lambda$ for $$H_0: \beta=0 \quad \text{vs} \quad H_1:\beta\ne0$$
Here is what I have
$$\Lambda=\frac{L(0)}{L\left(\frac{\sum_{i=1}^ny_ix_i}{\sum_{i=1}^nx_i}\right)}$$
$$=\frac{e^{-1/2 \sum^n(y_i-0x_i)^2}}{e^{-1/2\sum^n\left(y_i-\frac{\sum^ny_ix_i}{\sum^nx_i}x_i\right)^2}}$$
The numerator simplifies easily. For the denominator I expand the squared term and distribute. Then im just left to distribute the $-1/2\sum^n$ term. I'm getting puzzled because one of the terms im distributing the $-1/2\sum^n$ term to already has a sum. After finding out how to simplify the exponent of the denominator I plan to subtract it from the exponent of the numerator. Then the result of this is the likelihood ratio test statistic.
If what I'm doing here is correct I would appreciate it if you can show me how to finish this. If not then what would be the correct/best way to find the LRT statistic?