7.35 Refer to Exercise 7.5 on page 295 .

a. Use your answers from Exercise 7.5(b) to determine the mean, ${\mu }_{\mathrm{i}}$. of the variable $\overline{x}$ for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu }_5$, of the variable $\overline{x}$, using only your answer from Exercise 7.5(a).

Data on the population: $1,2,3,4$.

The population data for the variable $x$is as follows: $1,2,3,4$.

The mean ${\mu }_{\overline{x}}$of the variable $\overline{x}$for each of the samples is calculated as follows:

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

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

${\mu }_{\overline{x}}=\frac{1+2+3+4}{4}$

$=\frac{10}{4}$

$=2.5$

$2.5$ is the mean ${\mu }_{\overline{x}}$ of the variable $\overline{x}$.

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

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

${\mu }_{\overline{x}}=\frac{1.5+2.0+2.5+2.5+3+3.5}{6}$

$=\frac{15}{6}$

$=2.5$

As a result, the variable $\overline{x}$ has a mean ${\mu }_{\overline{x}}$ of $2.5$.

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

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

${\mu }_{\overline{x}}=\frac{2.0+2.3+2.7+3.0}{4}$

$=\frac{10}{4}$

$=2.5$

The variable $\overline{x}$has a mean value of $2.5$(${\mu }_{\overline{x}}$).

The sample and sample means for a sample of size $n=4$are shown in the table below.

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

${\mu }_{\overline{x}}=\frac{2.5}{1}$

role="math" localid="1650974100795" $=2.5$

As a result, the variable $\overline{x}$'s mean ${\mu }_{\overline{x}}$ is $2.5$.

Interpretation: We can see from the preceding conclusion that the mean of all conceivable sample means is the same.

Data on the population:$1,2,3,4$.

The following is a definition of the population mean:

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

$=\frac{1+2+3+4}{4}$

$=2.5$

As a result, the population average is $2.5$.

We can see that $\mu =2.5\$$ from the findings of parts (a) and (b).

The population mean is equal to the average of all conceivable sample means.