Q.7.28 - Elementary Statistics

Question

Does the sample size have an effect on the mean of all possible sample mean? Explain your answer.

Step-by-Step Solution

Answer

No, sample size does not have an effect on the mean of all possible sample mean .

Step 1 : Given Information

We have to explain whether  the sample size have an effect on the mean of all possible sample mean .

Step 2: Explanation

Because the mean of all conceivable sample means is always equal to the population mean, regardless of sample size.

Assume we have a population of size$4$ and the population observations are $x_1,x_2,x_3,x_{4.}$

Consider all possible size 2 samples now.

$X_1,X_2\frac{X_1+X_2}{2}$

$X_1,X_3\frac{X_1+X_3}{2}$

$X_1,X_3\frac{X_1+X_3}{2}$

$X_1,X_4\frac{X_1+X_4}{2}$

$X_2,X_3\frac{X_2+X_3}{2}$

$X_2,X_4\frac{X_2+X_4}{2}$

$X_3,X_4\frac{X_3+X_4}{2}$

The mean of all possible sample is

$=\frac{\frac{x_1+x_2}{2}+\frac{x_1+x_3}{2}+\frac{x_1+x_4}{2}+\frac{x_2+x_3}{2}+\frac{x_2+x_4}{2}+\frac{x_3+x_4}{2}}{6}$

$=\frac{3\left(X_1+X_2+X_3+X_4\right)}{2\times 6}$

$=\frac{X_1+X_2+X_3+X_4}{4}$

 $$\begin{gathered} =\mu \\ =population mean \end{gathered}$$

For sample size of $3$ 

sample mean:

$X_1,X_2,X_3\frac{X_1+X_2+X_3}{3}$

$X_1,X_3,X_4\frac{X_1+X_3+X_4}{3}$

$X_2,X_3,X_4\frac{X_2+X_3+X_4}{3}$

$X_1,X_2,X_4\frac{X_1+X_2+X_4}{3}$

The mean of all possible sample means of size $3$

$=\frac{\frac{x_1+x_2+x_3}{3}+\frac{x_1+x_3+x_4}{3}+\frac{x_2+x_3+x_4}{3}+\frac{x_1+x_2+x_4}{3}}{4}$

$=\frac{\frac{x_1+x_2+x_3+x_1+x_3+x_4+x_2+x_3+x_4+x_1+x_2+x_4}{3}}{4}$

$=\frac{3\left(X_1+X_2+X_3+X_4\right)}{3\times 4}$ 

$=\frac{X_1+X_2+X_3+X_4}{4}$

$=\mu \left(population mean\right)$

For the sample size 4

The only possible sample= population mean

therefore Sample mean = population mean

i.e., mean of sample mean = population mean

So, we observe that for every sample size mean of the sample mean is equal to the population mean and it does not depend on the sample size.

as the result, No, sample size does not have an effect on the mean of all possible sample mean .