I came here to say this, and am glad someone else said it first.

That on top of the unwarranted distribution assumption is very dangerous. How dangerous? If the true distribution has heavier tails than assumed, then the true variance can be many times what is assumed. If the true variance is, say, 4x what is assumed then an average standard deviation variation will show up as passing a 95% confidence interval in a random direction. Getting to 99% confidence by accident becomes quite easy, with bad results.