Let consider 2 hypotheses $$\begin{cases} \mathcal{H}_{0}: & \mathcal{N}\left(\mu_{0},\sigma_{0}^{2}\right)\\ \mathcal{H}_{1}: & \mathcal{N}\left(\mu_{1},\sigma_{1}^{2}\right) \end{cases}, $$ where $\mu_{0}$, $\sigma_{0}^{2}$, $\mu_{1}$, $\sigma_{1}^{2}$ are known, and $\mu_{0}\neq \mu_1$, $\sigma^2_0 \neq \sigma^2_1$ . Can we use the fundamental lemma by Neyman and Pearson to test these hypotheses? If not, do you have any suggestion?
Thanks you in advance for your answer.
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I edit my question as commented by WarreG and BruceET.
I have a set of i.i.d. samples. I know that these samples follow either $\mathcal{N}(\alpha_0,\sigma^2_0)$ or $\mathcal{N}(\alpha_1,\sigma^2_1)$ (i.e., only one of them). I would like to use statistical hypothesis testing to detect the probability distribution of this samples set. Actually, when {$\mu_{0}\neq \mu_1$ and $\sigma^2_0 = \sigma^2_1$} or when {$\mu_{0}= \mu_1$ and $\sigma^2_0 \neq \sigma^2_1$}, I know how to do the test for the detection. But it strange when {$\mu_{0}\neq \mu_1$ and $\sigma^2_0 \neq \sigma^2_1$}.