I have a situation where I want to test for a statistically significant difference in the percentage of customers who own a product among two groups. The treated group gets a marketing message, and the control group does not. However, these two groups were used for a number of different studies, and since the size of each group was large (~150,000 customers in Treated and ~50,000 customers in Control), we found out afterward that the starting populations had a statistically significant difference in percent of customers who own the product.
What kind of statistical significance test can I do to rectify this situation?
TREATED (received marketing, measured product ownership before and after test was conducted)
- N = 150,000
- % that own product before marketing = 55.0%
- % that own the product after marketing = 57.5%
CONTROL (did not receive marketing, but we measured product ownership before and after the test was conducted as well)
- N = 50,000
- % that own product before = 53.2% *(statistically different than TREATED's 55%)
- % that own the product after = 56.2%
In this scenario, the CONTROL group actually has a bigger increase in the percent of customers that own the product, but if I just did the test on the two AFTER groups, it would show a statistically significant result (150,000 customers at 57.5% is statistically different than 50,000 customers at 56.2% if you just plug those numbers into a t-test).
Is there a paired t-test for two groups or something that can account for the initial difference to give the answer to the question of whether or not the marketing had a statistically significant impact on product ownership?