I have a system with a positive, Real parameter x.
I can measure x directly, but the measurements are noisy. And probably has a small systematic bias.
My priors are that x=a or x=b, where a & b differ by 0.8%-7%.
I have 12-20 measurements of x. From the measurements, I get a t-distribution for the value of x.
To distinguish between the hypotheses x=a, and x=b, I've been looking at the (log of the) ratio of the t probability density function at x=a, and at x=b.
Usually, but not always, I get a result where one of x=a or x=b is 1 to 2 standard deviations from the sample mean, and the other is 10+ standard deviations away. So there's a fairly clear answer.
Is there a significance test of the log likelihood ratio that I should/could be using? Or should I be doing something completely different?