I know statistical test usually requires "enough" samples to estimate the uncertainties. But what if I have only one sample in each group, and also I "know" which distribution this one sample comes from, is it reasonable to construct a statistical test?
To be more specific, Let's say we have a case-control study and we want to know if there is some significant improvement in "case" than in "control".
The data we have is only one sample, $x$ for the case group, and $y$ for the control group, so $x$ and $y$ are both scalars.
The other information we also have is that $X\sim exp(\lambda)$ and $Y\sim exp(\mu)$, where $\lambda$ and $\mu$ are both some known parameters.
My question is to test my hypothesis that "case is greater than control", can I just construct a test statistic $T = X-Y$? And I will be able to find the distribution of $T$ by this post. Does this make sense?
My confusion mainly comes from how to write up $H_0$ and $H_1$. It seems like I am testing $H_0:x=y$ but obviously this looks very weird... Because usually we will test $H_0: \lambda = \mu$...