91Ó°ÊÓ

To find the mean, $\mathrm{μ}$ , of the variable

Let, the mean $\mathrm{μ}$is calculated as follows:
$\mathrm{μ}=\frac{∑x_1}{N}$
$=\frac{2+5+8}{3}$
$=\frac{15}{3}$
$=5$
As a result, the mean, $\mathrm{μ}$, 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 $\mathrm{μ}=5$ 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 $\mathrm{μ}=5$.
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 $\mathrm{μ}$.
For $n=1$and $n=2$, the number of dots within $0.5$of $\mathrm{μ}=5$ 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 $\mathrm{μ}$.
As a result, the probability that $\stackrel{¯}{X}$will be within $0.5$or less of $\mathrm{μ}$for

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

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

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