Answer to Question #350346 in Statistics and Probability for princess

Question

possible sample of size 2 which can be drawn without replacement

Answer 1

We have population population size N and sample size n=2.

Select a random sample of size 2 without replacement. We have a sample distribution of sample mean.

The number of possible samples which can be drawn without replacement is 

$^{N}C_n= ^{N}C_2=\dfrac{N!}{2!(N-2)!}=\dfrac{N(N-1)}{2}.$

Mean of sampling distribution 

$\mu_{\bar{X}}=\mu$

The variance of sampling distribution 

$Var(\bar{X})=\sigma^2_{\bar{X}}=\dfrac{\sigma^2}{n}(\dfrac{N-n}{N-1})$ $=\dfrac{\sigma^2}{2}(\dfrac{N-2}{N-1})$