Short Academic case study Proposal - Sampling Technique

We Short Academic case study Proposal in Sampling Technique - Two Stage Cluster analysis.

the case is we have 33 regions, each region has a number of hospitals as below

Region # of hospitals
Region 01 24
Region 02 20
Region 03 29
Region 04 17
Region 05 22
Region 06 26
Region 07 19
Region 08 15
Region 09 19
Region 10 11
Region 11 29
Region 12 15
Region 13 31
Region 14 20
Region 15 20
Region 16 6
Region 17 17
Region 18 4
Region 19 21
Region 20 32
Region 21 20
Region 22 15
Region 23 7
Region 24 18
Region 25 11
Region 26 21
Region 27 11
Region 28 8
Region 29 37
Region 30 6
Region 31 22
Region 32 18
Region 33 16

-we don't know the intercorrelation or DEEF from any previous study. and we need the paper to answer;

- what is the optimal number of clusters should be selected in first stage [with equations reference].
- what is the optimal number of hospital should be selected from each selected cluster in second stage [with equations reference].
- what is the total sample size.
- How to select the clusters, how to select the hospitals.

assuming MOE is 5% and the tested percentage is 50%