Q.7.35 - Elementary Statistics

Question

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).

Step-by-Step Solution

Answer

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

b). $2.5$ is the population average.

Part (a) Step 1: Given Information

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

Part (a) Step 2: Explanation

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.

Sample	$\overline{x}$
$1$	$1$
$2$	$2$
$3$	$3$
$4$

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 $\overline{x}$.

Part (a) Step 3: Explanation

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

Sample	$\overline{x}$
$1,2$	$\frac{1+2}{2}=1.5$
$1,3$	$\frac{1+3}{2}=2.0$
$1,4$	$\frac{1+4}{2}=2.5$
$2,3$	$\frac{2+3}{2}=2.5$
$2,4$	$\frac{2+4}{2}=3$
$3,4$	$\frac{3+4}{2}=3.5$

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 $2.5$.

Part (a) Step 4: Explanation

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

Sample	$\overline{x}$
$1,2,3$	$\frac{1+2+3}{3}=2.0$
$1,2,4$	$\frac{1+2+4}{3}=2.3$
$1,3,4$	$\frac{1+3+4}{3}=2.7$
$2,3,4$	$\frac{2+3+4}{3}=3.0$

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.

Sample	$\overline{x}$
$1,2,3,4$	$\frac{1+2+3+4}{4}=2.5$

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

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

$=2.5$

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.

Part (b) Step 1: Given Information

Data on the population: $1,2,3,4$.

Part (b) Step 2: Explanation

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.