1. Suppose that by field methods of different intensities it is possible to make the "response" stratum consist of 60,80,90 or 95 % of the whole population. For a percentage that is to be estimated, the true "response" stratum means are:60% stratum,40.7;80 % stratum,43.5;90% stratum,44.8;95% stratum,45.4; last 5%,59.0 (a) For a method that samples only the 60% stratum, show that the root mean square error of the estimated percentage for the whole population is

√(2414/n+28.94)

where n is the number of completed questionnaires obtained. 

(b) show that a root means a square error of 5% cannot be achieved by a method with a 60% response but can be obtained with slightly over 100 completed questionnaires for the methods that have a response of 80% or better. 

(c) if a root means a square error of 2% is prescribed, what methods can achieve it and what sample sizes are needed?

2. In 1. (c) suppose that it costs $5 per completed questionnaire for the field method that has a 90% response. To obtain a completed questionnaire from the next 5% of the population costs $20. For a root mean square error of 2%, is it cheaper to use the method with a 90% response rate or that with a 95% response rate?