I am conducting a survey that will identify and quantify the extent to which businesses in a certain industry reuse, recycle and dispose of their waste. I am trying to figure out how I would be able to express the confidence of my results where the responses would be weighted.
For example, let’s assume there are 20 businesses in a particular industry and I get survey responses from 5 businesses that account for 50% (Bus A), 5% (Bus B), 2% (Bus C), 2% (Bus D), and 1% (Bus E) of the total waste generated. I therefore have data from 25% of the population (5 out of 20 businesses) which accounts for 60% of the total waste generated.
Businesses A and B recycle their waste, Businesses C and D landfill their waste, and Business E disposes its waste.
I could simply scale to results to show that 92% of the waste is recycled ([50+5]/60), 7% is landfilled ([2+2]/60) and 1% reuses its waste (1/60).
However, given the low sample size I know there would be a wide confidence interval. How would I construct these confidence intervals with weighted proportions?