Equation for standard error of weighted mean

Question

What should I use when to calculate standard errors (and thus a confidence interval) for weighted means? Do I simply substitute the weighted mean for the simple arithmetic mean?

Answer 1

Your weighted mean is $\sum w_ix_i$ where the $w$'s are the weights and the $x$'s the values. Use the fact that the variance of a sum is the sum of the variances to conclude the variance of the average is $\sum w_i\sigma_i^2$