I grew up in Hawaii: I don't think we have wooly bear caterpillars, and I know we don't generally have snow, so I'd never heard of this myth until I looked it up on Net just now.
The Wooly Bear caterpillar is a moth larva which is common throughout North America. There's a folktale which says that the brown band around their middle is wide, it'll be a mild winter, and that if it's narrow, it'll be a severe winter. According to The Weather Notebook and knowwhat.com, studies have been done by the American Museum of Natural History which show that this is untrue: there is no connection between wooly bears and the severity of winter.
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