I want to run an experiment to test the outcomes of variations A and B. I'm designing the experiment but I want to know what's the minimum sample size to obtain a result of 10% difference between treatment and control with 90% confidence and a power of at least 80%.
If the standard deviation of variations A and B are different, the formula for the sample size is
$$N = \left ( \frac{t_{0.05/2}+t_{1-0.8}}{\delta} \right)^2 \left ( \frac{\sigma_0^2}{\frac{\sigma_0}{\sigma_0 + \sigma_1}} + \frac{\sigma_1^2}{\frac{\sigma_1}{\sigma_0 + \sigma_1}} \right )$$
Where delta is the expected difference in the outcome, or 10% as mentioned above.
What we don't know is the standard deviations. We know $$\delta = p_1 - p_0 = 10\%$$ but we don't know before the experiment the value for each term. If we knew them we could calculate $$\sigma_0 = p_0(1-p_0)$$ $$\sigma_1 = p_1(1-p_1)$$
Is there a way to obtain an optimal sample size before running a test? Assume there is no literature available.