the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries? Based on the results, does it appear that justices might profit by levying larger fines? Court Income Justice Salary 65.0 405.0 1567.0|1132.0| 270.0| 254.0| 111.0| 151.0| 32.0 17 29 44 94 54 46 60 24 25 Court Income Court Income Court Income Court Income
the linear correlation coefficient r, and find the P-value using a = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries? Based on the results, does it appear that justices might profit by levying larger fines? Court Income Justice Salary 65.0 405.0 1567.0|1132.0| 270.0| 254.0| 111.0| 151.0| 32.0 17 29 44 94 54 46 60 24 25 Court Income Court Income Court Income Court Income
MATLAB: An Introduction with Applications
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ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Correlation
Correlation defines a relationship between two independent variables. It tells the degree to which variables move in relation to each other. When two sets of data are related to each other, there is a correlation between them.
Linear Correlation
A correlation is used to determine the relationships between numerical and categorical variables. In other words, it is an indicator of how things are connected to one another. The correlation analysis is the study of how variables are related.
Regression Analysis
Regression analysis is a statistical method in which it estimates the relationship between a dependent variable and one or more independent variable. In simple terms dependent variable is called as outcome variable and independent variable is called as predictors. Regression analysis is one of the methods to find the trends in data. The independent variable used in Regression analysis is named Predictor variable. It offers data of an associated dependent variable regarding a particular outcome.
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Identify the linear
![**Correlation Analysis of Court Incomes and Justice Salaries**
In this exercise, you are prompted to calculate the linear correlation coefficient \( r \) and determine the P-value using \( \alpha = 0.05 \). The goal is to assess if there's sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries. Furthermore, based on the results, you should consider whether it appears that justices might profit by levying larger fines.
Here's the data presented:
\[
\begin{array}{c|ccccccc}
\text{ } & \text{Court Income} & 65.0 & 405.0 & 1567.0 & 1132.0 & 270.0 & 254.0 & 111.0 & 151.0 & 32.0 \\
\hline
\text{Justice Salary} & 29 & 44 & 94 & 54 & 46 & 60 & 24 & 25 & 17 \\
\end{array}
\]
Below the table, you'll input the calculated linear correlation coefficient \( r \).
**Instructions for Calculation:**
1. Use the given values to compute the linear correlation coefficient \( r \).
2. Decision rule: Compare the calculated P-value to \( \alpha = 0.05 \) to determine if the correlation is statistically significant.
3. Conclude if there is a potential for profit by justices based on larger fines.
**The linear correlation coefficient \( r \) is:** [Input field for result]
*Note: Round to three decimal places as needed.*
Enter your answer in the answer box and then click "Check Answer".
This instructional content aligns with learning outcomes in correlation analysis, emphasizing the relationship interpretation and hypothesis testing.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F242a0dac-d7d5-4e5c-b656-00cdaf0b077c%2Fad26007b-5f94-4bf8-bc81-086ce1dbb620%2Ftt8ky8b.png&w=3840&q=75)
Transcribed Image Text:**Correlation Analysis of Court Incomes and Justice Salaries**
In this exercise, you are prompted to calculate the linear correlation coefficient \( r \) and determine the P-value using \( \alpha = 0.05 \). The goal is to assess if there's sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries. Furthermore, based on the results, you should consider whether it appears that justices might profit by levying larger fines.
Here's the data presented:
\[
\begin{array}{c|ccccccc}
\text{ } & \text{Court Income} & 65.0 & 405.0 & 1567.0 & 1132.0 & 270.0 & 254.0 & 111.0 & 151.0 & 32.0 \\
\hline
\text{Justice Salary} & 29 & 44 & 94 & 54 & 46 & 60 & 24 & 25 & 17 \\
\end{array}
\]
Below the table, you'll input the calculated linear correlation coefficient \( r \).
**Instructions for Calculation:**
1. Use the given values to compute the linear correlation coefficient \( r \).
2. Decision rule: Compare the calculated P-value to \( \alpha = 0.05 \) to determine if the correlation is statistically significant.
3. Conclude if there is a potential for profit by justices based on larger fines.
**The linear correlation coefficient \( r \) is:** [Input field for result]
*Note: Round to three decimal places as needed.*
Enter your answer in the answer box and then click "Check Answer".
This instructional content aligns with learning outcomes in correlation analysis, emphasizing the relationship interpretation and hypothesis testing.
![## Analysis of Court Incomes and Salaries of Town Justices
### Data Overview
Listed below are the amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars. The objective is to analyze the relationship between these two variables to determine if there is a correlation.
### Data:
| Court Income ($thousands) | 65.0 | 405.0 | 1567.0 | 1132.0 | 270.0 | 254.0 | 111.0 | 151.0 | 32.0 |
|-----------------------------|-------|-------|--------|-------|-------|--------|-------|-------|-------|
| Justice Salaries ($thousands) | 30.0 | 44.8 | 91.2 | 51.0 | 45.0 | 29.5 | 33.0 | 51.2 | 47.8 |
### Analysis Requirements
1. **Scatterplot Construction**: Plot the court incomes against justice salaries to visualize the relationship between the two variables.
2. **Linear Correlation Coefficient**: Calculate the linear correlation coefficient (r) to measure the strength and direction of the linear relationship.
3. **P-value Calculation**: Determine the P-value using a significance level of α = 0.05.
4. **Conclusion on Correlation**: Evaluate if there is sufficient evidence to conclude that a linear correlation exists between court incomes and justice salaries.
5. **Implications**: Based on the results, assess if it appears that justices might profit by levying larger fines.
### Required Calculations
1. **Scatterplot**: A graph with court income on the x-axis and justice salaries on the y-axis will help visualize any potential linear relationship.
2. **Linear Correlation Coefficient (r)**: This is computed using the formula:
\[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}} \]
3. **P-value Determination**: Based on the calculated correlation coefficient, evaluate the P-value to test the null hypothesis (\( H_0 \)): There is no linear correlation between court income and justice salaries.
### Findings](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F242a0dac-d7d5-4e5c-b656-00cdaf0b077c%2Fad26007b-5f94-4bf8-bc81-086ce1dbb620%2F6ff0mks.png&w=3840&q=75)
Transcribed Image Text:## Analysis of Court Incomes and Salaries of Town Justices
### Data Overview
Listed below are the amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars. The objective is to analyze the relationship between these two variables to determine if there is a correlation.
### Data:
| Court Income ($thousands) | 65.0 | 405.0 | 1567.0 | 1132.0 | 270.0 | 254.0 | 111.0 | 151.0 | 32.0 |
|-----------------------------|-------|-------|--------|-------|-------|--------|-------|-------|-------|
| Justice Salaries ($thousands) | 30.0 | 44.8 | 91.2 | 51.0 | 45.0 | 29.5 | 33.0 | 51.2 | 47.8 |
### Analysis Requirements
1. **Scatterplot Construction**: Plot the court incomes against justice salaries to visualize the relationship between the two variables.
2. **Linear Correlation Coefficient**: Calculate the linear correlation coefficient (r) to measure the strength and direction of the linear relationship.
3. **P-value Calculation**: Determine the P-value using a significance level of α = 0.05.
4. **Conclusion on Correlation**: Evaluate if there is sufficient evidence to conclude that a linear correlation exists between court incomes and justice salaries.
5. **Implications**: Based on the results, assess if it appears that justices might profit by levying larger fines.
### Required Calculations
1. **Scatterplot**: A graph with court income on the x-axis and justice salaries on the y-axis will help visualize any potential linear relationship.
2. **Linear Correlation Coefficient (r)**: This is computed using the formula:
\[ r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}} \]
3. **P-value Determination**: Based on the calculated correlation coefficient, evaluate the P-value to test the null hypothesis (\( H_0 \)): There is no linear correlation between court income and justice salaries.
### Findings
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