Q. 7.4 - Elementary Statistics

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. 7.1 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: $2,5,8$.

Step-by-Step Solution

Answer

(a) The mean,$\mu$, of the variable is $5$.

(b) The dot plot for the sampling distribution of the sample mean as:

(c) Constructed the dot plot for the sampling distribution of the sample mean as:

(d) The probability that sample mean will be equal to population mean for $n=3$ is $0$.

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

Part (a) Step 1: Given information

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

Part (a) Step 1: Explanation

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 $5$.

Part (b) Step 1: Given information

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

Part (b) Step 2: Explanation

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,

Sample	$\overline{X}$
2	2.0
5	5.0
8	8.0

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

Sample	$\overline{X}$
2.5	3.5
2.8	5.0
5.8	6.5

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

Sample	$\overline{X}$
2,5,8	3

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

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

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

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 $2,5,$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 $0$.

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 $2,5,$ 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 $0$.