# Distinguishing 2 alternative hypotheses by relative likelihood

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?

• How do you obtain a t distribution and why do you refer to "the" value of x when there are 12 - 20 such values? Exactly how do you obtain the parameters of this distribution? Your description doesn't sound much like the likelihood ratio that would be appropriate in this setting.
– whuber
Aug 21 at 20:51
• The "12-20" measurements have an arithmetic mean, which I assume is t-distributed about the population mean with 11-19 df. I have prior information that the population mean is either a or it is b. Given my measurements, I'm trying to figure out whether a or b is more likely - which is obvious, and whether, for example, a is significantly more likely than b. Aug 22 at 3:54
• That is such an unusual distributional assumption that I can't help thinking you might be confusing Normal and Student t distributions.
– whuber
Aug 22 at 14:17
• "That is such an unusual distributional assumption that I can't help thinking you might be confusing Normal and Student t distributions." No, it seems you are confused. If the measurements are normally distributed, the sample mean will be t-distributed about the population mean. Please refer to any statistics textbook. Or Wikipedia. Aug 27 at 5:25