[cshimmin]

Basically, the result of an experiment has to be boiled down to a single numerical value, called the test-statistic. Typically the test-statistic is a (log) likelihood ratio. It is the distribution of the t.s. that must be considered when determining the significance of a measurement. Obviously the measurement itself only gives you a single value of the t.s., so you need to know the distribution to ask "does this result seem significant?". This is done by considering all the factors of random variation (statistical and systematic) that could have an effect on the t.s. Often, the distributions of these individual random factors are assumed to be Normal, but the resulting distribution considering all of their conspiring effects is very seldom normal distribution. Even in the central limit theorem, I think the distribution of the LLR ends up being something like a noncentral chi^2 distribution.