I'm trying to catch up in my statistics class before finals and I am posed with this "try at home" question in our notes. It asks me to find the best critical region for testing a null hypothesis vs. an alternative hypothesis at an unspecified/arbitrary significance $\alpha$ using the Neyman-Pearson lemma.
The question starts familiarly: "Let $X_1,...,X_n$ denote a random iid sample of size n from $N(μ,σ^2)$ for $μ ∈ \mathbb{R},σ^2 > 0$. Find the best yadda yadda yadda" -- but I am thrown off by the given hypotheses:
$H_0:μ = 0,σ^2 = 2$
$H_1:μ = 1,σ^2 = 4$
Maybe my understanding is further behind than I thought, but I am unsure of what to do with both $\mu \text{ and } \sigma^2$ denoted in the hypotheses.
I attempted a solution starting by finding the likelihood function $L$ for the normal distribution. Then I constructed $L_0 \text{ and }L_1$ by plugging in the values set by $H_0 \text{ and } H_1$, respectively, and set up $\frac{L_1}{L_0} > k$, but (if that's even the right thing to do here) I'm freezing up from there.
Do I just keep plugging away at the algebra until I get to something like $\sum\limits_{i = 1}^n x_i > \text{"some function"}$ and denote the critical set that way? If not, I'm hoping a little hint can guide me in the right direction...
Related -- are there additional considerations to make when determining the power of this test?