Question

6. Consider a population of size 10 with population mean, =140 and population Standard deviation, = 30 and all possible samples of size 5. Find the following if sampling is without replacement.

A. Number of samples.

B. Mean of the sampling distribution of the sample mean.

C. Standard deviation of the sampling distribution of the sample means.

Answer 1

Mean $\mu=140$

Variance $\sigma^2=30^2=900$

Standard deviation $\sigma=30$

We have  population size $N=10$ and sample size $n=5.$ 

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

$$\dbinom{N}{n}=\dbinom{10}{5}=\dfrac{10!}{5!(10-5)!}=252$$

B. The mean of the sampling distribution of the sample means is equal to the

the mean of the population.

$$E(\bar{X})=\mu_{\bar{X}}=\mu=140$$

C.

$$Var(\bar{X})=\sigma_{\bar{X}}^2=\dfrac{\sigma^2}{n}(\dfrac{N-n}{N-1})$$ $$=\dfrac{30^2}{5}(\dfrac{10-5}{10-1})=100$$

Standard deviation of the sampling distribution of the sample means is

$$\sigma_{\bar{X}}=\sqrt{\sigma_{\bar{X}}^2}=\sqrt{100}=10$$