Stratified Sampling Practical In R

Consider the following population with 196 elements stratified into 6 strata.

i)      Allocate the sample size 56 to different strata using the proportional and neyman  allocation .

ii)      Which sampling strategy would be suggested?  Justify using the values of variance of population mean for these sampling designs.

Theory-

Sample size allocation to different strata in proportional allocation is given by-

n_h = N_h.(n/N)

Sample size allocation to different strata in neyman allocation is given by-

Variance of population mean in proportional allocation is given by-

Variance of population mean in neyman allocation is given by-

Where,

n_h = sample size allocation in h^th stratum

N_h = total number of elements in h^th stratum

n = total sample size

N = population size

S_h = variability in h^th stratum.

L = number of strata

The allocation method having minimum variance will be considered as better allocation method.

R Code-

#command to remove previous objects
rm(list=ls())

#loading and viewing data set
Population <- read_excel("Practical STA304/prob 10.xlsx")
View(Population)

#checking the class of the dataset
data.class(Population)

#loading dataset on environment
attach(Population)

#extracting the variables
Nh = `No of elements in different strata` 
Sdh = `sd`

#calculating some variables
n=56 #given
N=sum(Nh)
f=n/N
Wh=as.numeric(format(Nh/N,digits=4))
Sh=as.numeric(format(sqrt((N/(Nh-1))*Sdh^2),digits=4))
L=length(NUMBER)
Stratum=NUMBER

#defining some vector
n_h_prop = c()
n_h_neyman = c()

#loop to calculate sample allocation in prop. and neyman allocation
for (h in 1:L ){
  n_h_prop[h] = round(Nh[h]*n/N)
  n_h_neyman[h] = round((n*Nh[h]*Sh[h])/sum(Nh*Sh))
}

#calculating the variance of estimate of population mean (i.e. y bar st.)
y_bar_st_prop=((1-f)/n)*sum(Wh*Sh^2)  #for proportional allocation
y_bar_st_Neyman=(((1/n)*(sum(Wh*Sh))^2) - ((1/N)*sum(Wh*Sh^2)) ) #for neyman

#filling data in 7th row of dataframe for different variable
Stratum[7]="Total"
Nh[7]=N
Wh[7]=sum(Wh[1:6])
Sdh[7]="-"
Sh[7]="-"
n_h_prop[7]=sum(n_h_prop[1:6])
n_h_neyman[7]=sum(n_h_neyman[1:6])

#priting the table of different variable along with sample allocation
View(data.frame(Stratum,Nh,Wh,Sdh,Sh,n_h_prop,n_h_neyman))

#printing the variance of estimate of population mean (i.e. y bar st.)
cat("variance of estimate of population mean in Proportional allocation is: ",y_bar_st_prop,"\n")
cat("variance of estimate of population mean in Neyman allocation is: ",y_bar_st_Neyman,"\n")

Some Insights From The R Console-

Imported Dataset On R Console-

Output Table With Proportional And Neyman Allocation-

And Obtained Variances are-

> #printing the variance of estimate of population mean (i.e. y bar st.)
> cat("variance of estimate of population mean in Proportional allocation is: ",y_bar_st_prop,"\n")
variance of estimate of population mean in Proportional allocation is:  3070.006 
> cat("variance of estimate of population mean in Neyman allocation is: ",y_bar_st_Neyman,"\n")
variance of estimate of population mean in Neyman allocation is:  1278.315 

Conclusion-

From the above output on R console we conclude that in proportional allocation, the sample allocation to different strata serial wise are- 4,5,7,12,21,7

And in neyman allocation, the sample allocation to different strata serial wise are- 13,9,10,6,8,10 The variance of the estimate of population mean in neyman allocation is 1278.315 which is far less than variance of the estimate of population mean in proportional allocation which is 3070.006. hence we conclude that neyman allocation is more efficient than proportional allocation.