# Understanding hypothesis testing-compare two series

This is rather simple, but am always confused so perhaps somebody can explain me once for all. I have two sets of observations coming from two sources that I want to compare, say $e_1$ and $e_2$ both having $N$ points. Three things could happen:

i) $e_1$ is consistently larger than $e_2$, ii) $e_1$ is consistently smaller than $e_2$, iii) there is no significant difference between $e_1$ and $e_2$.

I compute the difference in the two series, $e_1 - e_2$, create a test statistic. This could be for example the mean of $e_1 - e_2$ divided by the standard deviation of $e_1 - e_2$ (divided by square root of $N$), or it could be another test statistic. If I wanted to do the Diebold-Mariano test it would be another test statistic. But the point is I get some test statistic. I don't know how to interpret this statistic.

I know I need to compare it against say the extreme value of a $T$ distribution or Normal distribution. But know my test statistic could be >1.96, <-1.96, or between -1.96 and 1.96. Help me understand how to interpret the three cases to answer the question- which is consistently larger, $e_1$ or $e_2$.