Question

Refer to Exercise 7.4 on page 295.

a. Use your answers from Exercise 7.4(b) to determine the mean, ${\mu }_s$, of the variable $\overline{x}$for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_s$, of the variable $\overline{x}$, using only your answer from Exercise 7.4(a).

Short answer

Part a. The variable $\overline{x}$has a mean value of ${\mu }_{\overline{x}}=5$for each of the possible sample sizes.

Part b. The population mean is $\mu =5$.

Step-by-step solution

Part (a) Step 1. Given Information

It is given that the population data is 2,5,8.

We need to determine the mean, ${\mu }_s$, of the variable $\overline{x}$for each of the possible sample sizes.

Part (a) Step 2. When the sample size is 1

For the population data: 2,5,8.

The sample and sample mean for a sample of size $n=1$are shown in the table below.

Sample	$\overline{x}$
2	2
5	5
8	8

The variable $\overline{x}$has the following mean

$$\begin{gathered} {\mu }_{\overline{x}}=\frac{2+5+8}{3} \\ {\mu }_{\overline{x}}=\frac{15}{3} \\ {\mu }_{\overline{x}}=5 \end{gathered}$$

So when the sample size is 1, the variable $\overline{x}$has a mean ${\mu }_{\overline{x}}=5$.

Part (a) Step 3. When the sample size is 2

For the population data: 2,5,8.

The sample and sample mean for a sample of size $n=2$are shown in the table below.

Sample	$\overline{x}$
2,5	$\frac{2+5}{2}=3.5$
2,8	$\frac{2+8}{2}=5$
5,8	$\frac{5+8}{2}=6.5$

The variable $\overline{x}$has the following mean

$$\begin{gathered} {\mu }_{\overline{x}}=\frac{3.5+5+6.5}{3} \\ {\mu }_{\overline{x}}=\frac{15}{3} \\ {\mu }_{\overline{x}}=5 \end{gathered}$$

So when the sample size is 2, the variable $\overline{x}$has a mean ${\mu }_{\overline{x}}=5$.

Part (a) Step 4. When the sample size is 3

For the population data: 2.5.8.

The sample and sample mean for a sample of size $n=3$are shown in the table below.

Sample	$\overline{x}$
2,5,8	$\frac{2+5+8}{3}=5$

So when the sample size is 3, the variable $\overline{x}$has a mean ${\mu }_{\overline{x}}=5$.

Thus it can be seen that the mean of all potential sample means is the same.

Part (b) Step 1. Find the population mean

For the given population data: 2,5,8, the population mean can be given as

$$\begin{gathered} \mu =\frac{2+5+8}{3} \\ \mu =\frac{15}{3} \\ \mu =5 \end{gathered}$$

So from the results, it can be observed that the population mean is equal to the mean of all potential sample means that is ${\mu }_{\overline{x}}=\mu$.