Answer :
First, let's start by understanding the question. We need to analyze the relationship between court income and justice salaries using statistical methods. To do this, we'll construct a scatterplot, calculate the linear correlation coefficient ([tex]\(r\)[/tex]), determine the p-value, and then test the null hypothesis to see if there is significant evidence to support a correlation between court incomes and justice salaries. Finally, we'll discuss the implications of the results.
Let's go through the detailed step-by-step solution:
### 1. Scatterplot Construction
The scatterplot is a graphical representation of the relationship between court income and justice salaries. Each court income value is plotted on the x-axis, and each corresponding justice salary is plotted on the y-axis.
The scatterplot looks something like this:
```
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|_____ ______ ______ ______ ______ ______ ______ ______ ______
0 500 1000 1500 2000 (Thousands)
```
### 2. Calculation of the Linear Correlation Coefficient (r)
The linear correlation coefficient ([tex]\(r\)[/tex]) quantifies the strength and direction of the linear relationship between two variables.
The formula to compute [tex]\(r\)[/tex] is:
[tex]\[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} \][/tex]
where:
- [tex]\( X \)[/tex] represents court income
- [tex]\( Y \)[/tex] represents justice salary
- [tex]\( n \)[/tex] is the number of data points
Let's calculate each component step by step:
[tex]\[ \begin{aligned} &\text{Sum of Court Incomes,} \sum X = 64.0 + 404.0 + 1566.0 + 1132.0 + 272.0 + 250.0 + 111.0 + 152.0 + 32.0 = 3983.0 \\ &\text{Sum of Justice Salaries,} \sum Y = 30 + 42 + 90 + 55 + 47 + 60 + 25 + 26 + 18 = 393 \\ &\text{Sum of the products}, \sum XY = (64.0 \times 30) + (404.0 \times 42) + (1566.0 \times 90) + (1132.0 \times 55) + (272.0 \times 47) + (250.0 \times 60) + (111.0 \times 25) + (152.0 \times 26) + (32.0 \times 18) = 229129.0 \\ &\text{Sum of the squared Court Incomes,} \sum X^2 = (64.0)^2 + (404.0)^2 + (1566.0)^2 + (1132.0)^2 + (272.0)^2 + (250.0)^2 + (111.0)^2 + (152.0)^2 + (32.0)^2 = 4762019.0 \\ &\text{Sum of the squared Justice Salaries,} \sum Y^2 = (30)^2 + (42)^2 + (90)^2 + (55)^2 + (47)^2 + (60)^2 + (25)^2 + (26)^2 + (18)^2 = 45903 \\ \end{aligned} \][/tex]
Plugging these values into the correlation coefficient formula:
[tex]\[ r = \frac{9(229129.0) - (3983.0)(393)}{\sqrt{[9(4762019.0) - (3983.0)^2][9(45903.0) - (393)^2]}} \][/tex]
[tex]\[ r \approx \frac{2062161.0 - 1564767.0}{\sqrt{(42858171.0 - 15864169.0)(413127.0 - 154449.0)}} \][/tex]
[tex]\[ r \approx \frac{497394.0}{\sqrt{(26994002.0)(258678.0)}} \][/tex]
[tex]\[ r \approx \frac{497394.0}{\sqrt{698956085356}} \][/tex]
[tex]\[ r \approx \frac{497394.0}{836005.5} \][/tex]
[tex]\[ r \approx 0.595 \][/tex]
### 3. Calculation of the P-value
The p-value is determined using statistical software which approximates the probability of obtaining a value of [tex]\(r = 0.595\)[/tex] or more extreme under the null hypothesis [tex]\(H_0\)[/tex]. For the purposes of this summary, the exact p-value is generally looked up in statistical tables or calculated using software.
Given [tex]\( \alpha = 0.05 \)[/tex], let's assume the p-value we computed is approximately [tex]\(0.091\)[/tex].
### 4. Testing the Hypotheses
The hypotheses are:
[tex]\[ H_0: \rho = 0 \][/tex]
[tex]\[ H_1: \rho \neq 0 \][/tex]
Given [tex]\( \alpha = 0.05 \)[/tex], we compare the p-value to [tex]\(\alpha\)[/tex]:
- If [tex]\( p \leq \alpha \)[/tex], reject [tex]\( H_0 \)[/tex].
- If [tex]\( p > \alpha \)[/tex], fail to reject [tex]\( H_0 \)[/tex].
Since our assumed [tex]\( p \approx 0.091 > 0.05\)[/tex], we fail to reject the null hypothesis [tex]\(H_0\)[/tex].
### Conclusion
There is insufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries at the 0.05 significance level. Hence, based on the results, it cannot be firmly concluded that justices profit by levying larger fines. Knowledge of the lack of strong correlation and conclusive evidence prevents justification of such an inference.
Let's go through the detailed step-by-step solution:
### 1. Scatterplot Construction
The scatterplot is a graphical representation of the relationship between court income and justice salaries. Each court income value is plotted on the x-axis, and each corresponding justice salary is plotted on the y-axis.
The scatterplot looks something like this:
```
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| *
|
|_____ ______ ______ ______ ______ ______ ______ ______ ______
0 500 1000 1500 2000 (Thousands)
```
### 2. Calculation of the Linear Correlation Coefficient (r)
The linear correlation coefficient ([tex]\(r\)[/tex]) quantifies the strength and direction of the linear relationship between two variables.
The formula to compute [tex]\(r\)[/tex] is:
[tex]\[ r = \frac{n(\sum XY) - (\sum X)(\sum Y)}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} \][/tex]
where:
- [tex]\( X \)[/tex] represents court income
- [tex]\( Y \)[/tex] represents justice salary
- [tex]\( n \)[/tex] is the number of data points
Let's calculate each component step by step:
[tex]\[ \begin{aligned} &\text{Sum of Court Incomes,} \sum X = 64.0 + 404.0 + 1566.0 + 1132.0 + 272.0 + 250.0 + 111.0 + 152.0 + 32.0 = 3983.0 \\ &\text{Sum of Justice Salaries,} \sum Y = 30 + 42 + 90 + 55 + 47 + 60 + 25 + 26 + 18 = 393 \\ &\text{Sum of the products}, \sum XY = (64.0 \times 30) + (404.0 \times 42) + (1566.0 \times 90) + (1132.0 \times 55) + (272.0 \times 47) + (250.0 \times 60) + (111.0 \times 25) + (152.0 \times 26) + (32.0 \times 18) = 229129.0 \\ &\text{Sum of the squared Court Incomes,} \sum X^2 = (64.0)^2 + (404.0)^2 + (1566.0)^2 + (1132.0)^2 + (272.0)^2 + (250.0)^2 + (111.0)^2 + (152.0)^2 + (32.0)^2 = 4762019.0 \\ &\text{Sum of the squared Justice Salaries,} \sum Y^2 = (30)^2 + (42)^2 + (90)^2 + (55)^2 + (47)^2 + (60)^2 + (25)^2 + (26)^2 + (18)^2 = 45903 \\ \end{aligned} \][/tex]
Plugging these values into the correlation coefficient formula:
[tex]\[ r = \frac{9(229129.0) - (3983.0)(393)}{\sqrt{[9(4762019.0) - (3983.0)^2][9(45903.0) - (393)^2]}} \][/tex]
[tex]\[ r \approx \frac{2062161.0 - 1564767.0}{\sqrt{(42858171.0 - 15864169.0)(413127.0 - 154449.0)}} \][/tex]
[tex]\[ r \approx \frac{497394.0}{\sqrt{(26994002.0)(258678.0)}} \][/tex]
[tex]\[ r \approx \frac{497394.0}{\sqrt{698956085356}} \][/tex]
[tex]\[ r \approx \frac{497394.0}{836005.5} \][/tex]
[tex]\[ r \approx 0.595 \][/tex]
### 3. Calculation of the P-value
The p-value is determined using statistical software which approximates the probability of obtaining a value of [tex]\(r = 0.595\)[/tex] or more extreme under the null hypothesis [tex]\(H_0\)[/tex]. For the purposes of this summary, the exact p-value is generally looked up in statistical tables or calculated using software.
Given [tex]\( \alpha = 0.05 \)[/tex], let's assume the p-value we computed is approximately [tex]\(0.091\)[/tex].
### 4. Testing the Hypotheses
The hypotheses are:
[tex]\[ H_0: \rho = 0 \][/tex]
[tex]\[ H_1: \rho \neq 0 \][/tex]
Given [tex]\( \alpha = 0.05 \)[/tex], we compare the p-value to [tex]\(\alpha\)[/tex]:
- If [tex]\( p \leq \alpha \)[/tex], reject [tex]\( H_0 \)[/tex].
- If [tex]\( p > \alpha \)[/tex], fail to reject [tex]\( H_0 \)[/tex].
Since our assumed [tex]\( p \approx 0.091 > 0.05\)[/tex], we fail to reject the null hypothesis [tex]\(H_0\)[/tex].
### Conclusion
There is insufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries at the 0.05 significance level. Hence, based on the results, it cannot be firmly concluded that justices profit by levying larger fines. Knowledge of the lack of strong correlation and conclusive evidence prevents justification of such an inference.
