待在 are acceptable ways of writing
dai1 zai4 meaning "stay at/in". For instance, a couple sentences from iciba:
我们因天气寒冷呆在家里。 Because of the cold weather, we stayed at home.
他们整天和朋友们待在外边。 They stay out(side) all day with their friends.
So, which one of these came first? Where did the alternative come from? Was it originally a 错别字 which gained acceptance?
[I would like to emphasize that I'm not asking which you think is correct. They're both widely used; they're both pronounced the same; and there's no stable difference in usage (subjective impressions != usage).]