Answer to Question #350345 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 

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