This might not sound significant, until you compare it with the control group: an equal number of study participants who were not forced to freeze their feet, and presumably just sat in the lobby reading a People magazine from 1989 until the time came to collect their fee. Of these, only 9 percent got sick. That's what scientists officially call "statistically fucking significant."
It turns out you and your mom were both sort of right: Colds are viral, but research shows that cold temperatures can substantially worsen any largely dormant viruses you might already have, to the point that it becomes noticeable and you're considered "sick."
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See, you technically were sick all those times you called into work so you could binge-watch Netflix.
Right now, you're actually a lot sicker than you realize, but your immune system is doing a damn good job arresting the various viruses, and you're doing a really shitty job giving it credit. Among those who reported developing a cold in the days after the ice experiment, it's likely that most of them already harbored the cold virus, but their white blood cells were keeping that all in check -- until they worsened the situation by dunking their feet in a glacier.
Freezing temperatures constrict the blood vessels, cool the blood, and prevent the immune system from doing its job properly. As a result, the inhibited immune response allows the cold virus to get just enough of an upper hand to produce symptoms. The correct path of action is clear: Put on the sweater first, then tell your mother to suck the science.