Testing whether the ratio of two sums is equal to a certain value I have the following hypothesis:
$h_0 : \frac {\sum {a_{i}}} {\sum {b_{i}}} = x \\
h_1 : \frac {\sum {a_{i}}} {\sum {b_{i}}} \neq x$
Where $a_i \propto X b_i$ and $X$ is a random variable.
How would I test this hypothesis?
Edit: Switched a and b around in the 3rd equation
 A: As I interpret your question, you have observed a certain number of values for a and b that are not matched in a meaningful way.  Your aim is to test whether the ratio of the sums is significantly different from x.  I think you have already noticed that this is remarkably different from testing whether the sum of the ratios is different from X (that would have a directly corresponding parametric hypothesis test).  Parametric hypothesis tests like that rely on the concept of distributions... but you do not have enough information here to (directly) parametrically estimate the variance of the ratio of sums.  As a caveat, I leave room for the possibility that if a and b are drawn from some known distribution that you could find an analytically solution based on estimates of those parameters.  However, sticking closer to stuff I know... it seems your data has only yielded you only a single ratio and your interest in testing the ratio versus x.  This is indeed a problem for standard parametric hypothesis tests.
I think your most viable option is to do some kind of Monte-Carlo case re-sampling (1).  In short, take NSIM (where the value of NSIM depends on how precise an estimate you want) random draws from your available values of a and b.  Sum for each of those draws and take their ratios.  In this way, you build a sampling distribution of the ratio of the sum of a and the sum of b.  You can then use this distribution like you would a Z distribution to determine the proportion of bootstrapped samples that met or exceeded the null hypothesis value for x (one-tailed).
