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Practical 9:

1. Please note that the sample sizes ni for Neyman Allocation is proportional to s.d. Si NOT variance Si^2.

2. You should stratify the population into 4 subpopulations and draw 1 SRS from each subpopulation. However many students 
   draw 1 SRS from the whole population and poststratify the sample into 4 subsamples. This results in a poststratified 
   random sample NOT a stratified random sample. Each subsample may not all lie in the corresponding subpopulation.  
   Accordingly, you should use the formula for a stratifed SRS NOT a post-stratified SRS in the calculation of var.

3. You should round the sample sizes ni to nearest integer before each SRS of size ni is drawn from the subpopulation Pi. 
   Otherwise the computer will always round ni DOWN so that you may not get samples with total size n=100 say in the end.

4. You should use vector in calculation to save you from writing a long statement and hence to eliminate the changes of 
   making mistakes. 

   For example, in the calculation of Neyman allocation the statements using a vector format are

   Nl=c(N1,N2,N3,N4)
   Sl=c(sd(ys1),sd(ys2),sd(ys3),sd(ys4))
   n.ney=Nl*Sl/sum(Nl*Sl)   

   instead of

   n1.ney=N1*sd(ys1)/(N1*sd(ys1)+N2*sd(ys2)+N3*sd(ys3)+N4*sd(ys4))
   n2.ney=N2*sd(ys2)/(N1*sd(ys1)+N2*sd(ys2)+N3*sd(ys3)+N4*sd(ys4))
   n3.ney=N3*sd(ys3)/(N1*sd(ys1)+N2*sd(ys2)+N3*sd(ys3)+N4*sd(ys4))
   n4.ney=N4*sd(ys4)/(N1*sd(ys1)+N2*sd(ys2)+N3*sd(ys3)+N4*sd(ys4))
   n.ney=c(n1.ney,n2.ney,n3.ney,n4.ney)