# Confidence intervals on weighted proportions

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?

• What if businesses which generate large amount of waste are more likely to recycle than those with small amount of waste? Maybe small amount of waste businesses and/or businesses which dispose of waste are less likely to respond to the survey. Etc. – Mark L. Stone May 6 '16 at 22:11
• How do you know the total waste in the industry? Don't you need the other $15$ businesses (ie how to calculate the denominator for the 50% number)? – probabilityislogic May 7 '16 at 0:07
• @MarkL.Stone as you noted, it may very well may be the case that the size of the business would affect how it chooses to manage its waste. These are important attributes that need to be qualified in my presentation of the results. However, I am posing this question to the forum to ask how I could quantify confidence intervals given the fact that the sizes (not the number) of the businesses that respond to the survey would determine the percent of total waste that is accounted for in the survey. – Danny May 7 '16 at 20:54
• @probabilityislogic I have data from a supplier to the industry on the amount of a material supplied to the industry which ultimately ends up as waste. The supplier however, does not know how that material is disposed. – Danny May 7 '16 at 20:58