Weighted confidence interval for mean?

Lets say we have values
v = (0,1,2,4,8,10)
with weights (real numbers not necessarily integers)
w = (0,10,1,1,1,1)

Calculating the weighted mean is obvious: m = sum( w_i * v_i) / sum (w_i),
but how do I get weighted confidence interval (lets say with 95% confidence)?

The standard confidence interval formula does not work, since it is in the form
(m- Z95 * omega , m- Z95 * omega), but obviously the low limit should be closer to m than the upper limit (for the data presented above).

Above omega can be chosen as the weighted standard deviation:
D = sum( w_i * (v_i-m)^2) / sum (w_i),
omega = sqrt(D)
Yes, the reason is because I have never heard of it :)
I'm a programmer, not a statistician, but thank you for the advice, I will look into it.