91Ó°ÊÓ

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

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

The mean ${\mathrm{μ}}_{\stackrel{¯}{x}}$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 ${\mathrm{μ}}_{\stackrel{¯}{x}}$for the variable $\overline{x}$:

${\mathrm{μ}}_{\stackrel{¯}{x}}=\frac{1+2+3+4}{4}$

$=\frac{10}{4}$

$=2.5$

$2.5$ is the mean ${\mathrm{μ}}_{\stackrel{¯}{x}}$ 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 ${\mathrm{μ}}_{\stackrel{¯}{x}}$for the variable $\overline{x}$:

${\mathrm{μ}}_{\stackrel{¯}{x}}=\frac{1.5+2.0+2.5+2.5+3+3.5}{6}$

$=\frac{15}{6}$

$=2.5$

As a result, the variable $\overline{x}$ has a mean ${\mathrm{μ}}_{\stackrel{¯}{x}}$ 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 ${\mathrm{μ}}_{\stackrel{¯}{x}}$:

${\mathrm{μ}}_{\stackrel{¯}{x}}=\frac{2.0+2.3+2.7+3.0}{4}$

$=\frac{10}{4}$

$=2.5$

The variable $\overline{x}$has a mean value of $2.5$(${\mathrm{μ}}_{\stackrel{¯}{x}}$).

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

The following is the mean ${\mathrm{μ}}_{\stackrel{¯}{x}}$for the variable $\overline{x}$:

${\mathrm{μ}}_{\stackrel{¯}{x}}=\frac{2.5}{1}$

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

As a result, the variable $\overline{x}$'s mean ${\mathrm{μ}}_{\stackrel{¯}{x}}$ 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:

$\mathrm{μ}=\frac{∑x_i}{N}$

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

$=2.5$

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

We can see that $\mathrm{μ}=2.5\$$ from the findings of parts (a) and (b).

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