Finding standard error of variation when z-score and population are known?

Hi all, this is probably more a maths/statistics question rather than Excel, however I'm using Excel so I hope I will get the answer here.

I am currently running an A/B Test to determine which web page has the best conversion rate and the statistical significance of that test. My data and formulas are as follows:

Web Page A
Unique Visits = 6000
Conversions = 37
Conversion Rate = 37 / 6000 = 0.62%
Standard Error = SQRT((0.62% x (1-0.62%)/6000)) = 0.0010

Web Page B
Unique Visits = 2000
Conversions = 21
Conversion Rate = 21 / 2000 = 1.05%
Standard Error = SQRT((1.05% x (1-1.05%)/2000)) = 0.0023

Therefore
Z-Score = (0.62%-1.05%)/SQRT(POWER(0.0010,2)+POWER(0.0023,2)) = -1.738
P Value = NORMSDIST(-1.738) = 0.0411
Confidence Level = 1-0.0411 = 0.9589 = 95.89%

I think I am doing everything right up to this point. What I want to do now is reverse the formula, and find out the required conversions, conversion rate and standard error of web page B, when the z-score and individual sample sizes are both known, in other words:

Web Page A
Unique Visits = 6000
Conversions = 37
Conversion Rate = 37 / 6000 = 0.62%
Standard Error = SQRT((0.62% x (1-0.62%)/6000)) = 0.0010

Other 'knowns'
P-Value = 0.05
Z-Score = -1.64

Web Page B
Unique Visits = 2000
Conversions = ?Formula?
Conversion Rate = ?Formula?
Standard Error = ?Formula?

Can anyone help?

UA = page views A, UB = page views B
CA = number conversions A, CB = number conversions B
Rn = Conversion rate for any Page n = Cn / Un
EA = Standard Error A, EB = Standard Error B
En = f(Rn, Un)
Z = Z score of the two

Functionally, you've got 1 formula and 1 unknown:
En = f(Rn, Un) => Rn = f(Cn, Un) therefore En = f(Cn, Un)
So
Zboth = f(CA, UA, CB, UB)
you have the value of Zboth, and all three variables except CB.
So it's one formula, one unknown; the scenario is theoreticaly solvable.

Well, solving it algebraically would be annoying, but fortunately we're doing this numerically, so just use the SOLVER add-in, start with a guess of like UB / 100, and let the computer turn the crank until it converges on the solution.

Once you've done that and found the Conversions for page B, finding the rate and the error is straightforward.

@ben

Thanks for the reply... I hadn't used SOLVER before. What it seems to do though is just give me the result. I'm looking for the actual formula; then I can apply that formula to other situations.

If you're looking for the formula, then you need to take the function you have with the unknown you want, which is

Zboth = f(CA, UA, CB, UB)

and reorganize it to output what you don't know based on what you do:

CB = g(Zboth, CA, UA, CB)

The way to do that is,

close excel,
and take out a pad of paper and a pencil,
and do algebra to re-arrange the equation until it looks the way you want.

Well, just on inspection I think that way will be a hassle, so my suggestion was to just "cheat" and use algorythmic iteration to solve it without rearraging it. Give the computer a guess, and let it see if the result matches the answer, and then it will change the guess, and after doing that a hundred times it will converge on the answer.

But if you want the general case formula, well...

that's not an excel problem, that's an algebra problem.

@ben and @mrshorty

Thanks for your replies. I think you are both correct, it is more of an algebra problem rather than Excel. I wasn't sure if there was maybe a built in function that I was missing.

I also asked the same question on a Mathematics forum and the answer that came back was very similar to yours, MrShorty. I guess I need to start learning about quadratic equations.

Thanks for taking the time to help out guys.