Refer to Exercise 7.3 on page 295 .

a. Use your answers from Exercise 7.3(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, ${\mu }_i$, of the variable $\tilde{x}$, using only your answer from Exercise 7.3(a).

Data on the population: $1,2,3$

For the variable $x$, we have population data, which is $1,2,3$.

The mean ${\mu }_{\overline{x}}$of the variable role="math" localid="1650963041682" $\overline{x}$for each of the samples is calculated as follows:

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

The following is the mean ${\mu }_{\overline{x}}$for the variable $\overline{x}$:

$=\frac{6}{3}$

$=2$

As a result, the variable $\overline{x}$ has a mean $\overline{x}=2$.

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

The following is the mean ${\mu }_{\overline{x}}$for the variable $\overline{x}$:

${\mu }_{\overline{x}}=\frac{1.5+2+2.5}{3}$

$=\frac{6}{3}$

$=2$

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

The following is the mean ${\mu }_{\overline{x}}$for the variable $\overline{x}$:

${\mu }_{\overline{x}}=\frac{2.0}{1}$

$=2$

All the sample means are equal.

Data on the population :$1,2,3$.

The following is a definition of the population mean:

$\mu =\frac{\sum x_i}{N}$

$=\frac{1+2+3}{3}$

$=2$

We can see that ${\mu }_{\overline{x}}=\mu =2$by combining the results of parts (a) and (b).