Here's an example in Stata of how to create the ratio and test a hypothesis using nlcom:
. webuse regress
. regress y x1 x2 x3
Source | SS df MS Number of obs = 148
-------------+------------------------------ F( 3, 144) = 96.12
Model | 3259.3561 3 1086.45203 Prob > F = 0.0000
Residual | 1627.56282 144 11.3025196 R-squared = 0.6670
-------------+------------------------------ Adj R-squared = 0.6600
Total | 4886.91892 147 33.2443464 Root MSE = 3.3619
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x1 | 1.457113 1.07461 1.36 0.177 -.666934 3.581161
x2 | 2.221682 .8610358 2.58 0.011 .5197797 3.923583
x3 | -.006139 .0005543 -11.08 0.000 -.0072345 -.0050435
_cons | 36.10135 4.382693 8.24 0.000 27.43863 44.76407
------------------------------------------------------------------------------
. nlcom ratio:_b[x1]/_b[x2], post
ratio: _b[x1]/_b[x2]
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
ratio | .6558606 .4221027 1.55 0.122 -.1784571 1.490178
------------------------------------------------------------------------------
. test ratio=.5
( 1) ratio = .5
F( 1, 144) = 0.14
Prob > F = 0.7125
There are formulas in the pdf manual under nlcom. A terse explanation can be found in Stata FAQ on delta method.