A LOT of people will test positive for lead. With all my paint scraping days, I'm sure I would. Testing positive doesn't mean you are "lead poisoned" however. Nor does it mean they got it in your rental, or that your tenant is telling the truth.
In pre-1987 residences (or 1978, I always forget which), HUD requires LBP disclosures by the LL and that new tenants be given a HUD pamplet on LBP. You can get the disclosure and pamplet from hud.gov. I think most, if not all states follow this, as well. Check your states law. Not doing this when you rent to a new tenant will leave you liable and could cost you in court.
I had a Section 8 inspection that found flaking paint on a window sill and it sparked a lead inspection. I, too, had recently painted everything in the apartment. On unpainted pressure treated steps to the front door, he found too much lead. I asked how he could find lead on a nearly new (about 5 years old) unpainted wood steps. His response was to ask me if I remodeled the apartment, scraped paint, removed carpet, etc., which I had. He said that's how the lead got there. They are testing "parts per billion." He said in these older neighborhoods, he could test any sidewalk and find lead. I had to get a state approved lead based paint abatement contractor just to paint the steps.
In a more extreme case, a LL in the town where we lived hired a homeless person to scrape the paint on the outside of a house. The whole house. Gave him tools, ladders, everything exept masks and other safety equipment. Within weeks, he was in the hospital and died from lead poisoning. Big news. The LL was later found guilty of manslaughter or some "gross disregard for life" type of charge and got a hefty jail sentance.
And those $5-$10 lead testing kits are not reliable. There has been too much press on false positives or failing to detect.
I'd evict and not worry about it, but that's me. Do a little research for yourself, and asking a lawyer would be best.