Question

Consider all possible samples of size 2 that can be drawn with replacement

from the population consists of numbers 1,2, and 3. Find the following:

a. mean of the sampling distribution

b. variance of the sampling distribution

c. standard deviation of the sampling distribution

Answer 1

There are 9 samples of size two which can be drawn with replacement:

The corresponding sample mean are:

(a). And the mean of sampling distribution is:

$\mu _{\overline{x}}=\frac{sum\:of\:all\:sample\:means}{9}=\frac{18}{9}=2$

(b). Variance of sampling distribution is:

The variance $\sigma ^2_{\overline{x}}$ of the sampling distribution means is obtained by subtracting the mean 2 from each number, squaring the result, adding all 9 numbers obtained, and dividing by 9. The final result is:

$\sigma ^2_{\overline{x}}=\frac{3}{9}=0.3333$

(c). Standard deviation of sampling distribution is:

Standard deviation is the square root of variance obtained in b

$\sigma _{\overline{x}}=\sqrt{\sigma ^2_{\overline{x}}}$

\sigma \:_{\overline{x}}=\sqrt{0.3333}=0.5774 .