Confidence Intervals for Non-normally distributed data

Ok, say I have a set of data. All of the data is either a -1, or a number from 0 to 5. There are no numbers smaller than -1, no numbers between -1 and 0 and no numbers greater than 5. Clearly this is not a Normal distribution.

The standard deviation is 0.9 and the mean is -0.11. There are 375 pieces of data. How can I find a 95% confidence interval for the mean?

Also, how large of a sample size do I need to be "95% confident" that the true "expecation" is within +/- 0.02 of my estimated expectation. (In other words, how much data do I need to be 95% confident that the expecation is between -0.13 and -0.09?

I can do confidence intervals if my data is "normal" using "t-values" and what not, but this one has me puzzled.

Any help would be appreciated.