In regards to confidence interval, what do we do if the variance is zero and our sample size is 15?
The confidence interval will always be the mean, no matter the confidence level.
For any confidence interval involving means, typically the formula is mean plus and minus the critical value (z or t) times the standard deviation divided by square root of sample size. If stand. deviation is 0, that makes the division part 0, and that would in the end result in the mean plus or minus 0. Now if you have a standard deviation of 0, then that means that all your data points are the same number so a confidence interval isn't really necessary.