Question

Conduct a study to determine how well the binomial distribution approxi mates the hypergeometric distribution. Consider a bag with n balls, 25% of which are red. A sample of size (0.10)n is taken. Let X be the number of red balls in the sample. Find P(X (0.02)n) for increasing values of n when | sampling is (i) with replacement and (i) without replacement. Use R

Answer 1

With replacement:

Here in the with replacement case the  
XBin(0.10n, p)
, where  
p 0.25
.

XBin(0.1n,0.25)

Without replacement:

In the without replacement case  

XHypergeometric(n, 0.25n, 0.1n)

Now we have to calculate the probability  
PIX0.02n)
   for each distribution and study if binomial distribution approximates the hypergeometric distribution for in creasing values of   .

R-codes:

m=2:5
N=10^m
p=0.25
n=0.1*N
a=p*N
b=N-a
x=0.02*N
p1=round(phyper(x,a,b,n),digits = 8)
p2=round(pbinom(x,n,p),digits = 8)
P=matrix(c(n,p1,p2),ncol = 3,byrow = F)
colnames(P)=(c("n","Hypergeometric","Binomial"))
rownames(P)=sprintf("P(X<=%s)",x)
P

Output:

               n Hypergeometric   Binomial
P(X<=2)       10     0.52166241 0.52559280
P(X<=20)     100     0.13579398 0.14883105
P(X<=200)   1000     0.00005045 0.00010898
P(X<=2000) 10000     0.00000000 0.00000000

Conclusion:

From the above table we can see the probabilities from the binomial distribution is getting closer and closer to the values of probability from the hypergeometric distribution as the    increases. So here we can say the binomial distribution approximates the hypergeometric distribution very well as for sufficiently large   .