Dear Odundo:
Since neither Mark nor Kevin have submitted an answer to your question, let me give it a try. The simplest formula to calculate 95% confidence intervals assuming simple random sampling (or its equivalent) is:
Lower confidence interval= Mean or proportion - (1.96 x standard error)
Upper confidence interval= Mean or proportion + (1.96 x standard error)
However, as you correctly state, the confidence intervals must account for complex sampling. The design effect (DEFF) is the multiplier to determine by how much the sample size must be inflated to maintain the same position if complex sampling will be done rather than simple random sampling. But when calculating measures of precision, such as confidence intervals, we use the square root of DEFF. Therefore, the equation to calculate confidence intervals for complex sampling surveys are:
Lower confidence interval= Mean or proportion - (1.96 x standard error x square root of DEFF)
Upper confidence interval= Mean or proportion + (1.96 x standard error x square root of DEFF)
So if you have the mean or proportion, the standard error calculated assuming simple random sampling, and the DEFF, you can calculate the appropriate 95% confidence intervals for estimates derived from data obtained by complex sampling.
There are other formulas to calculate confidence intervals which may or may not be more accurate in your situation. My general recommendation would be to use a statistical analysis software package which can account for cluster sampling and automatically calculate the appropriate 95% conference intervals. Such software includes Epi Info, SAS, SPSS, STATA, and R.
I hope this is helpful.