https://www.cnn.com/2020/05/26/health/antibody-tests-cdc-coronavirus-wrong/index.html
[In most of the country, including areas that have been heavily impacted, the prevalence of SARS-CoV-2 antibody is expected to be low, ranging from less than 5% to 25%, so that testing at this point might result in relatively more false positive results and fewer false-negative results,” the CDC said. The higher the sensitivity, the fewer false negatives a test will give. The higher the specificity, the fewer false positives. Across populations, tests give more accurate results if the disease being tested for is common in the population. If an infection has only affected a small percentage of people being tested, even a very small margin of error in a test will be magnified.]