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This exercise involves the use of an unrealistically small population to provide a concrete illustration for the exact distribution of a sample proportion. A population consists of one man and four women. The first name of the man is Liam; the first names of the women are Rose, Isabella, Emma, and Frances. Suppose that the specified attribute is "male." Complete parts (a) through (e) below.

Complete the table:
(Type integers or decimals. Do not round.)
[tex]\[
\begin{tabular}{|l|l|l|}
\hline
\text{Sample} & \text{Number of Males} & \text{Sample Proportion} \\
\hline
\text{R, I} & 0 & 0 \\
\text{R, E} & 0 & 0 \\
\text{R, F} & 0 & 0 \\
\text{I, E} & 0 & 0 \\
\text{I, F} & 0 & 0 \\
\text{E, F} & 0 & 0 \\
\hline
\end{tabular}
\][/tex]

Construct a dot plot for the sampling distribution of the proportion for samples of size 2. Mark the position of the population proportion on the dot plot. Choose the correct graph below.

A.
B.
C.
D.



Answer :

GinnyAnswer
### Explanation and Step-by-Step Solution:

#### Population Details:
1. Size of Population: 5 individuals
2. Population consists of:
- 1 male: Liam
- 4 females: Rose, Isabella, Emma, Frances

#### Attribute of Interest:
- Attribute: Male

#### Sample Details:
- Sample Size: 2 individuals

#### Calculation of Sample Proportions:
We need to determine the sample proportions for every possible sample of size 2 from this population. In each of these samples, we will count the number of males (Liam) and then compute the sample proportion for the attribute "male".

Possible samples of size 2:
1. (Rose, Isabella)
2. (Rose, Emma)
3. (Rose, Frances)
4. (Isabella, Emma)
5. (Isabella, Frances)
6. (Emma, Frances)

Now, we will calculate the sample proportion of the attribute "male" for each of these samples:

1. Sample (Rose, Isabella):
- Number of males: 0
- Sample proportion = 0/2 = 0.0

2. Sample (Rose, Emma):
- Number of males: 0
- Sample proportion = 0/2 = 0.0

3. Sample (Rose, Frances):
- Number of males: 0
- Sample proportion = 0/2 = 0.0

4. Sample (Isabella, Emma):
- Number of males: 0
- Sample proportion = 0/2 = 0.0

5. Sample (Isabella, Frances):
- Number of males: 0
- Sample proportion = 0/2 = 0.0

6. Sample (Emma, Frances):
- Number of males: 0
- Sample proportion = 0/2 = 0.0

#### Construct the table:

Here’s the completed table with the samples (pairs of names), number of males, and sample proportions:

\begin{tabular}{|l|l|l|}
\hline
Sample & Number of Males & Sample Proportion \\
\hline
(Rose, Isabella) & 0 & 0.0 \\
(Rose, Emma) & 0 & 0.0 \\
(Rose, Frances) & 0 & 0.0 \\
(Isabella, Emma) & 0 & 0.0 \\
(Isabella, Frances) & 0 & 0.0 \\
(Emma, Frances) & 0 & 0.0 \\
\hline
\end{tabular}

#### Population Proportion:
To calculate the population proportion of males:
- There are 1 male out of a total of 5 individuals.
- Population proportion = 1/5 = 0.2

#### Constructing the Dotplot:
Since all sample proportions are 0.0, the dotplot will consist of dots at 0.0.
- We also need to mark the population proportion on this dotplot.

Given options for the dotplot:
You will choose the dotplot that has a series of dots at 0.0 and a mark at the population proportion 0.2.

Without seeing the graphs, it's clear that the correct dotplot will have:
- All sample dots at 0.0
- A mark at 0.2

Therefore, the graph which fits the description correctly would be chosen accordingly.

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