Question

Consider a population consisting of 1,2,3,4 and 5. Supposed samples of size 2 are drawn from this population. describe the sampling distribution of the sample means.

• What is the mean and variance of the sampling distribution of the sample means?
• Compare these values to the mean and variance of the population.
• Draw the histogram of the sampling distribution of the population mean.

Answer 1

Here population size : N = 5

and which are 1,2,3,4 and 5

And we have to draw a sample of size 2

So, there are $^{N}C_n$ possible samples

that is, $^{5}C_2= 10$

Mean of population $(\mu)$ = $\dfrac{1+2+3+4+5}{5}=3$

Variance of population $(\sigma^2)=\dfrac{\Sigma(x_i-\bar{x})^2}{n}=\dfrac{4+1+0+1+4}{5}=2$

So all the possible samples are:

(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5) and (4,5)

Mean of sampling distribution $(\mu_{\bar{x}})=\mu=3$

The variance of sampling distribution $(\sigma^2_{\bar{x}})= \dfrac{\sigma^2}{N}=\dfrac{4}{5}=0.8$

So mean of population = mean of sample = 3

Variance of population = 2 and variance of sample = 0.8

Histogram of the mean of population:

Histogram of the mean of sampling: