Confidence Interval for division of means of two different data sets

I have two sets of data (unpaired). Set $A$ has a mean of $157$, $n=54, SD=54.0, SEM=7.3$. Set $B$ has a mean of $704$, $n=42,SD=142.4,SEM=22.0$. $A$ and $B$ are both approximately normally distributed.

I want to determine a 95% confidence interval for the ratio of the means for the mean of $A$ divided by the mean of $B$, which in this case is $157/704=.223$.

I'm not sure how to go about this. The only idea I've come up with so far is to fit $A$ and $B$ with a normal and binomial distribution, respectively, and then plug them into the propagation of uncertainty formula:

$$f=\frac{A}{B}, \space \sigma_f^2 \approx f^2 \left[\left(\frac{\sigma_A}{A}\right)^2 + \left(\frac{\sigma_B}{B}\right)^2 - 2\frac{\sigma_{AB}}{AB} \right]$$

Is this the right thing to do? Any advice would be appreciated

• You say A is binomial with $n = 54$ and variance $np(1-p) = 54^2,$ but that leaves $p(1-p) = 54,$ which makes no sense (since $0 \le p \le 1.$ // If A and B were both normal, then $F = S_A^2/S_B^2$ would have an F distribution with degrees of freedom 53, and 41. In that case most applied statistics books have a formula for the CI of $\sigma_A^2/\sigma_B^2.$ // Also what do you mean by $A/B$? Are you taking the ratio of means, of variances, or of what? // Maybe it would help if you were to give the practical context of your Question. – BruceET Mar 21 '18 at 1:12
• If the sample size is big enough, you may want to consider using the delta method en.wikipedia.org/wiki/Delta_method – user103828 Mar 22 '18 at 13:24
• Bruce, thanks for the reply. I think you are correct in that both $A$ and $B$ are normal, and I edited my question to reflect that. For the context, I am trying to analyze some neuroscience data. $A$ represents the number of quantal releases measures from all active zones in a synapse. $B$ represents number of active zones in a synapse, as determined by imaging studies. I am looking for the confidence interval on the mean probability of an active zone releasing a vesicle. So, in this case the mean probability of $A/B$ is .224, and I want to know the CI about that mean. – nmjbio Mar 22 '18 at 13:25