Q.7.33 - Elementary Statistics

Question

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).

Step-by-Step Solution

Answer

a). The variable $\overline{x}$ has a mean ${\mu }_{\overline{x}}$ of $2$.

b). The average for the population is $2$.

Part (a) Step 1: Given Information

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

Part (a) Step 2: Explanation

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

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

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

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

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

$=\frac{6}{3}$

$=2$

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

Part (a) Step 3: Explanation

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

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

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

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

$=\frac{6}{3}$

$=2$

Part (a) Step 4: Explanation

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

Sample	$\overline{x}$
$1,2,3$	$\frac{1+2+3}{3}=2.0$

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.

Part (b) Step 1: Given Information

Data on the population : $1,2,3$.

Part (b) Step 2: Explanation

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).