Question

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.l on page 293.
c. Construct a graph similar to Fig.7.3 and interpret your results.
d. For each of the possible sample sises. 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 $0.5$.
7.3 Population data: $1,2,3$.

Short answer

(a) The mean, $\mu$, of the variable is $2$.

(b) A table is created, and the dot plot is created.

(c) The dot plot for the sample mean's sampling distribution as follows:

(d) The probability that sample mean will be equal to population mean for $n=3$is $1$.

(e) The probability that $\overline{X}$will be within $0.5$or less of $\mu$for $n=3$is $0$.

Step-by-step solution

Part (a) Step 1: Given information

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

Part (b) Step 1: Given information

To construct a table and draw a dotplot for the sampling distribution of the sample mean for each of the possible sample sizes.

Part (b) Step 2: Explanation

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

Sample	$\overline{X}$
1	1.0
2	2.0
3	3.0

If $n=2$is the sample size,

Sample	$\overline{X}$
1.2	1.5
1.3	2.0
2.3	2.5

If $n=3$is the sample size,

Sample	$\overline{X}$
1,2,3	2

Part (b) Step 3: Explanation

As a result, the dot plot for the sampling distribution of the sample mean is constructed as shown below:

Hence, a table is created, and the dot plot is created.

Part (c) Step 1: Given information

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

Part (c) Step 2: Explanation

Let, the population data is $1,2,$and $3$.

Construct the dot plot for the sample mean's sampling distribution as follows:

Hence, the graph is constructed.

Part (d) Step 1: Given information

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

Part (d) Step 2:Explanation

Let, the population data is $1,2,$and $3$.
Find the probability that the sample mean will equal the population mean for each feasible sample size. Note that there is one dot corresponding to $\mu =2$ in the dot lot.
The probability that sample mean will be equal to population mean$=\frac{1}{3}$
Correspondingly, the probability that the sample mean equals the population mean for $n=2$,
$=\frac{1}{3}$
As a result, the probability that sample mean will be equal to population mean for $n=3$is $1$.

Part (e) Step 1: Given information

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

Part (e) Step 2: Explanation

Let, the population data is $1,2,$and $3$.
Find the probability that the sampling error in calculating the population mean by the sample mean is $0.5$or less; that is, the probability that $\overline{X}$is $0.5$or less than $\mu$.
For $n=1$and $n=2$, the number of dots within $0.5$or less of $\mu =2$ 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 $\mu$.
Since, the probability that $\overline{X}$will be within $0.5$or less of $\mu$is 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 $0$.