If you did want to create a confidence interval for the difference of two proportions, there are better procedures than the one you are using: you might consider making use of the Wilson procedure. The [Wikipedia article on binomial confidence intervals][1] mentions this and several other possibilities, but does not show the method applied to *differences* in proportions: for that, consult a reference such as  Newcombe, Robert G., "Interval Estimation for the Difference Between Independent Proportions: Comparison of Eleven Methods," *Statistics in Medicine*, **17**, 873-890 (1998).

However it appears that you don't want to create a confidence interval, but rather perform a hypothesis test. The formula you would want to use is a rearranged version of the given one. Let me write $p_a$ and $p_b$ for the proportions in groups A and B, and their sample sizes as $m$ and $n$ respectively. Then your test statistic is:

$$z = \frac{p_a - p_b}{\sqrt{\frac{p_a(1-p_a)}{m}+\frac{p_b(1-p_b)}{n}}}$$

Your result will be significant if this exceeds the upper critical value $z_\text{crit} = \Phi^{-1}(1 - \alpha/2)$ where $\alpha$ is your significance level (if you were interested in a 90% confidence interval, note that this is equivalent to setting $\alpha$ as 10%, not as 90%!) or if it is below the lower critical value, which by symmetry of the normal distribution is $-z_\text{crit}$. These two sides of significance correspond to whether the proportions differ because $p_a$ exceeds $p_b$ (positive $z$) or vice versa (negative $z$). If you are interested in the critical values for $p_a$ which make this just significant, then you need to solve this equation as equal to $\pm z_\text{crit}$. I take it you are interested in both possibilities.

If you feel daunted by the task of algebraic rearrangement, one option is to use a [computer algebra system][2] to do the work for you. One freely available, open source product is [Sage][3] (which is actually rather more powerful than just a CAS). Rearranging to make one variable the subject, is essentially the same as solving the equation for that variable *in terms of the other variables*. A brief [tutorial on how to solve equations symbolically in Sage][4] is here. This would then give you a formula you can set up in Excel.

A paid-for product is Mathematica, but many basic features of Mathematica are freely available online at [Wolfram Alpha][5]. Go there and type:

    solve z=(a-b)/sqrt((a(1-a))/m + (b(1-b))/n) for a

The output will be:

![Rearranged formula for binomial difference of proportions test][6]

Here I have written $z_\text{crit}$ as $z$, $p_a$ as $a$ and $p_b$ as $b$ but I hope the meaning is still clear. Simply by changing $m$, $n$, $z$ and $b$ into appropriate cell references you can easily implement this formula in Excel. If cell `A1` contains your level of signficance, $\alpha$, then then the cell you use for the critical $z$-score should contain the formula `=NORM.S.INV(1-A1/2)` so you should get about `1.96` if you set $\alpha$ at the 5% level.

Note that we actually find two solutions arise, corresponding to the two critical values for $a$, without having to check `solve -z=(a-b)/sqrt((a(1-a))/m + (b(1-b))/n) for a` for the case with negative $z$. It is clear that the first line of the rearrangement is must be:

$$z_\text{crit}^2 = \frac{(p_a - p_b)^2}{\frac{p_a(1-p_a)}{m}+\frac{p_b(1-p_b)}{n}}$$

Beyond this point it no longer matters whether we used the positive or negative value for $z_\text{crit}$. It's not so hard to see where Mathematica derives it solution from. Multiply by the denominator and we obtain:

$$z_\text{crit}^2 \left(\frac{p_a(1-p_a)}{m}+\frac{p_b(1-p_b)}{n}\right)= (p_a - p_b)^2$$

Then multiply by $mn$:

$$z_\text{crit}^2 \left(np_a(1-p_a) + mp_b(1-p_b)\right)= mn(p_a - p_b)^2$$

Once the brackets are multiplied out and terms are collected together, this will be a quadratic in $p_a$. The form of Mathematica's solutions were just the two roots to the [quadratic formula][7] but it's easier to let it deal with the simplification!


  [1]: http://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval
  [2]: http://en.wikipedia.org/wiki/Computer_algebra_system
  [3]: http://en.wikipedia.org/wiki/Sage_%28mathematics_software%29
  [4]: http://www.sagemath.org/doc/tutorial/tour_algebra.html
  [5]: http://www.wolframalpha.com/
  [6]: https://i.sstatic.net/0D9lD.png
  [7]: http://en.wikipedia.org/wiki/Quadratic_formula