How to determine sample size for a cost-per-conversion experiment

Hi all,

I am trying to perform a scientific experiment in online marketing which would be focused on lowering the cost per customer acquisition.

There is currently an ad going on that converts customers at X euros on average. I want to test another version of the ad hoping to lower that cost.

In order to do so, the first thing I need to know is what the minimum sample size is for the desired decrease in average cost per acquisition. And here is the problem.

Typically I have used online statistical calculators such as Evan Miller to determine the minimum sample size per variation. In this calculator you type your current Conversion Rate first, and then type your desired improvement in that Conversion Rate, and it tells you the sample size.

What happens is that I am not aiming to improve a conversion rate, but a cost per conversion.

This improvement, if achieved, might surely occur through a better conversion rate of the second ad, but it could also happen that the improvement comes via a better CPC for whatever reason while having the same conversion rate.

Therefore, the tools I have used for calculating sample size cannot help me now, since they calculate based on conversion rate and percentages, not based on a cost.

I would really appreciate if anyone could let me know any insight into how to proceed for calculating that sample size for a case like this one, or if you know any such online calculator that can be used for it when the baseline is not a conversion rate but a cost.

Thank you!!