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I've been working on review problems, and this one has me completely stumped.

Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the likelihood ratio test, determine a 5%-level critical region test for $H_0 : \sigma^2 = 1 $ vs. $H_1 : \sigma^2 \neq 1$ (and, trivially, $\sigma^2 >0$).

It appears that in the general case, when one is testing a hypothesis about the variance, a chi-square statistic is used, which gives me something of an end-goal, but I'm not sure how to get there.

The joint pdf for the 10 r.v.s should be $\large(\frac{1}{\sqrt{2\pi\sigma^2}})^{10}\cdot e^-\frac{\sum_{i=1}^{10} (X_i - 3)^2}{2\sigma^2}$

Under the null hypothesis, this yields $\large(\frac{1}{\sqrt{2\pi}})^{10}\cdot e^-\frac{\sum_{i=1}^{10} (X_i - 3)^2}{2}$, since $\sigma^2 = 1$

Under the alternative hypothesis, we have $\large(\frac{1}{\sqrt{2\pi\hat\sigma^2}})^{10}\cdot e^-\frac{\sum_{i=1}^{10} (X_i - 3)^2}{2\hat\sigma^2}$

Setting these as numerator and denominator, respectively, I get

$\LARGE\frac{exp(^-\frac{\sum_{i=1}^{10} (X_i - 3)^2}{2})}{(\frac{1}{\sigma})^{10}\cdot exp(^-\frac{\sum_{i=1}^{10} (X_i - 3)^2}{2\hat\sigma^2})}$

From here, I believe I can take the log of this, which should yield $\LARGE\frac{\hat\sigma^2}{10ln(\frac{1}{\hat\sigma})}$.

At this point (assuming these steps are valid), I have no idea how to proceed! Am I missing something obvious here? Any help is greatly appreciated!

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