That isn't a random sample, so they had to massage the numbers.

Indeed, that is stated in the article. Do you have any specific concerns with their methodology that would cause the results to have an even larger error range than they account for?

Sampling uncertainties aside, this is super sensitive to mis-estimating the false positive rate. Which they seem fairly likely to have done, using an estimated test specificity of 100% (with error bars) in one branch of their analysis after getting 30/30 negatives against pre-COVID serum samples. Too small a sample to get that confident in an atypically high specificity, especially if that collection of samples was disproportionately low antibody (e.g. summer blood rather than post cold/flu season blood).

That's to my mind the most consequential misestimation of the test characteristics, but Balaji Srinivasan details more:

https://medium.com/@balajis/peer-review-of-covid-19-antibody...