Hi Kate,
You could do what you suggest, but as you note the CIs would probably be very wide – just how wide depending on the cluster total size and cluster sample size. If the sample size is small relative to the population, you may well not have a representative sample of the cluster even. You would need to calculate the CIs as for a simple or stratified random sample.
The main reason for not doing this is the likely width of the CIs will preclude any meaningful conclusions.
Why do you want to do this? If it is to identify areas that need specific attention, then a different approach, eg LQAS (Lot Quality Assurance Sampling) may be more appropriate. This will classify areas into those that reach or do not reach a pre-determined level. Those that surpass the level can be “ignored”; those that are only just above or below can be “watched” and those below can be targetted for support. Using this approach you can combine results from smaller areas to give an answer for a larger area, but NOT vice versa.
Hope this is helpful.
Sean
