# Statistical test for sample size equals one [duplicate]

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 $$H_0: \lambda = \mu$$...

• If you know the distribution, you know everything. You do not need statistics, probability calculus is enough. – cure Oct 15 '20 at 22:28
• Could you please be more specific how will you compute the probability for the hypothesis I am trying to test? – Jeffrey Oct 15 '20 at 22:35
• I answered this question in the affirmative for an instructive special case at stats.stackexchange.com/a/1836/919. – whuber Oct 16 '20 at 14:58
• @whuber Thanks for the very interesting answer! But I am still confused that in my question the two distributions are fully known. I guess in this case maybe there is no way to do any statistical test? – Jeffrey Oct 16 '20 at 19:11
• Jeffrey, although the distribution families are known, the distributions are not. Your situation is identical. – whuber Oct 16 '20 at 21:12