In Exercises 7.3-7.10, we have given population data for a variable. For each exercise, do the following tasks.
a. Find the mean, $\mu$, of the variable.
b. For each of the possible sample sizes, construct a table similar to Table 7.2 on page 293 and draw a dotplot for the sampling distribution of the sample mean similar to Fig. 7.1 on page 293.
c. Construct a graph similar to Fig. 7.3 and interpret your results.
d. For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.
e. For each of the possible sample sizes, find the probability that the sampling error made in estimating the population mean by the sample mean will be $0.5$or less (in magnitude), that is, that the absolute value of the difference between the sample mean and the population mean is at most $0.5$.
7.4 Population data: $2,5,8$.

To find the mean, $\mu$ , of the variable

Let, the mean $\mu$is calculated as follows:
$\mu =\frac{\sum x_1}{N}$
$=\frac{2+5+8}{3}$
$=\frac{15}{3}$
$=5$
As a result, the mean, $\mu$, of the variable is $5$.

To construct a table similar and draw a dotplot for the sampling distribution of the sample mean.

Let, the population data is $2,5,$and $8$.

Create a table for each of the various sample sizes as follows:
If $n=1$, is the sample size,

If $n=2$, is the sample size,

If $n=3$, is the sample size,

As a result, as shown below, construct the dot plot for the sampling distribution of the sample mean.

To construct a graph similar to Fig. 7.3 and interpret the results.

Let, the the population data is $2,5,$and $8$.

Construct the dot plot for the sampling distribution of the sample mean as:

To find the probability that the sample mean will equal the population mean.

Let, the population data is $2,5,$and $8$.
Find the probability that the sample mean will equal the population mean for each feasible sample size. Note that there is one dot equivalent to $\mu =5$ in the dot lot.
As a result, the chance of the sample mean equaling the population mean for $n=1$is $\frac{1}{3}$.
For $n=2$, the probability that the sample mean equals the population mean is the same$=\frac{1}{3}$
There is no dot in the dot lot that corresponds to $\mu =5$.
As a result, the probability that sample mean will be equal to population mean for $n=3$is $0$.

To find the probability that the sampling error made in estimating the population mean by the sample mean will be $0.5$ or less.

Let, the population data is $2,5,$and $8$.
Determine the probability that the sampling error in estimating the population mean by the sample mean will be $0.5$or less; that is, the probability that $\overline{X}$will be within $0.5$of $\mu$.
For $n=1$and $n=2$, the number of dots within $0.5$of $\mu =5$ is one out of three.
Because there are no dots in the sample for $n=3$, the mean $\overline{X}$will be within $0.5$of $\mu$.
As a result, the probability that $\overline{X}$will be within $0.5$or less of $\mu$for

$n=1$is $\frac{1}{3}$.

Correspondingly, the probability that $\overline{X}$will be within $0.5$less of $\mu$for $n=2$is $\frac{1}{3}$

As a result, the probability that $\overline{X}$will be within $0.5$or less of $\mu$for $n=3$is $0$.