Answer :
Sure! Let's explore the relationship between domestic savings and income in Catonia using the provided data. We will go through each question step-by-step.
### Data Table
The data provided is as follows:
| Savings (\[tex]$) | Income (\$[/tex]) |
|--------------|-------------|
| 40,000 | 100,000 |
| 60,000 | 150,000 |
| 80,000 | 200,000 |
| 100,000 | 250,000 |
### Questions and Solutions
1. Is the relationship between domestic savings and income in Catonia a linear relationship? Explain your answer.
A relationship is considered linear if the slope (rate of change) between points is constant. Here, we find the slope between each consecutive pair of data points:
[tex]\[
\text{Slope} = \frac{\text{Change in Savings}}{\text{Change in Income}}
\][/tex]
- Between \[tex]$100,000 and \$[/tex]150,000:
[tex]\(\frac{60,000 - 40,000}{150,000 - 100,000} = \frac{20,000}{50,000} = 0.4\)[/tex]
- Between \[tex]$150,000 and \$[/tex]200,000:
[tex]\(\frac{80,000 - 60,000}{200,000 - 150,000} = \frac{20,000}{50,000} = 0.4\)[/tex]
- Between \[tex]$200,000 and \$[/tex]250,000:
[tex]\(\frac{100,000 - 80,000}{250,000 - 200,000} = \frac{20,000}{50,000} = 0.4\)[/tex]
Since the slope (0.4) is the same between all points, the relationship is linear.
2. Calculate the slope of the line between an income of \[tex]$150,000 and \$[/tex]200,000.
As calculated above, the slope is:
[tex]\[
\frac{\text{Change in Savings}}{\text{Change in Income}} = \frac{80,000 - 60,000}{200,000 - 150,000} = 0.4
\][/tex]
3. Calculate the slope of the line between an income of \[tex]$200,000 and \$[/tex]250,000.
Again, the slope calculation is:
[tex]\[
\frac{\text{Change in Savings}}{\text{Change in Income}} = \frac{100,000 - 80,000}{250,000 - 200,000} = 0.4
\][/tex]
4. Does the slope of the line stay the same?
Yes, the slope remains the same (0.4) throughout, indicating the relationship is linear.
5. How is your answer to question (iv) related to the answer to question (i)?
The answer to question (iv) confirms the answer to question (i). Since the slope is constant between all income intervals, this supports the conclusion that the relationship between domestic savings and income in Catonia is linear.
If you have any more questions or need further clarification, feel free to ask!
### Data Table
The data provided is as follows:
| Savings (\[tex]$) | Income (\$[/tex]) |
|--------------|-------------|
| 40,000 | 100,000 |
| 60,000 | 150,000 |
| 80,000 | 200,000 |
| 100,000 | 250,000 |
### Questions and Solutions
1. Is the relationship between domestic savings and income in Catonia a linear relationship? Explain your answer.
A relationship is considered linear if the slope (rate of change) between points is constant. Here, we find the slope between each consecutive pair of data points:
[tex]\[
\text{Slope} = \frac{\text{Change in Savings}}{\text{Change in Income}}
\][/tex]
- Between \[tex]$100,000 and \$[/tex]150,000:
[tex]\(\frac{60,000 - 40,000}{150,000 - 100,000} = \frac{20,000}{50,000} = 0.4\)[/tex]
- Between \[tex]$150,000 and \$[/tex]200,000:
[tex]\(\frac{80,000 - 60,000}{200,000 - 150,000} = \frac{20,000}{50,000} = 0.4\)[/tex]
- Between \[tex]$200,000 and \$[/tex]250,000:
[tex]\(\frac{100,000 - 80,000}{250,000 - 200,000} = \frac{20,000}{50,000} = 0.4\)[/tex]
Since the slope (0.4) is the same between all points, the relationship is linear.
2. Calculate the slope of the line between an income of \[tex]$150,000 and \$[/tex]200,000.
As calculated above, the slope is:
[tex]\[
\frac{\text{Change in Savings}}{\text{Change in Income}} = \frac{80,000 - 60,000}{200,000 - 150,000} = 0.4
\][/tex]
3. Calculate the slope of the line between an income of \[tex]$200,000 and \$[/tex]250,000.
Again, the slope calculation is:
[tex]\[
\frac{\text{Change in Savings}}{\text{Change in Income}} = \frac{100,000 - 80,000}{250,000 - 200,000} = 0.4
\][/tex]
4. Does the slope of the line stay the same?
Yes, the slope remains the same (0.4) throughout, indicating the relationship is linear.
5. How is your answer to question (iv) related to the answer to question (i)?
The answer to question (iv) confirms the answer to question (i). Since the slope is constant between all income intervals, this supports the conclusion that the relationship between domestic savings and income in Catonia is linear.
If you have any more questions or need further clarification, feel free to ask!
