Listed below are amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars. Construct a scatterplot, find the value of 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 64.0 404.0 1567.01131.0 270.0 250.0 111.0 155.0 30.0 P Justice Salary 30 43 91 58 44 62 26 27 18 Court Income Court Income Court Income Court Income The linear correlation coefficient r is
Listed below are amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars. Construct a scatterplot, find the value of 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 64.0 404.0 1567.01131.0 270.0 250.0 111.0 155.0 30.0 P Justice Salary 30 43 91 58 44 62 26 27 18 Court Income Court Income Court Income Court Income The linear correlation coefficient r is
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Concept explainers
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.
Question
find linear
![### Linear Correlation Between Court Income and Justice Salaries
#### Dataset Overview
Listed below are amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars.
| Court Income | 64.0 | 404.0 | 1567.0 | 1131.0 | 270.0 | 250.0 | 111.0 | 1555.0 | 30.0 |
|--------------|-------|-------|--------|--------|-------|-------|-------|--------|-------|
| Justice Salary | 30 | 43 | 91 | 58 | 44 | 62 | 26 | 27 | 18 |
#### Instructions
1. **Construct a Scatterplot**
- Plot the *Court Income* values on the x-axis.
- Plot the *Justice Salary* values on the y-axis.
- Each point on the scatterplot corresponds to a pair of court income and justice salary values.
2. **Calculate the Linear Correlation Coefficient (r)**
- Use the formula:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}
\]
where
- \( n \) is the number of pairs,
- \( x \) represents *Court Income*,
- \( y \) represents *Justice Salary*.
3. **Find the P-value**
- Use an α level of 0.05.
- The P-value helps determine if the linear correlation is statistically significant.
4. **Analysis**
- Based on the results, determine if there is sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries.
- Discuss whether it appears that justices might profit by levying larger fines.
#### Result Interpretation
- **Scatterplot**: Visual representation of the data to look for any pattern or trend.
- **Linear Correlation Coefficient (r)**: Indicates the strength and direction of the linear relationship.
- Values close to 1 imply a strong positive correlation.
- Values close to -1 imply a strong negative correlation.
- Values around 0 imply no correlation.
- **P-value**:
- A P-value less than](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bbe1ef9-cacf-48b7-b15f-bf536b5508cd%2Fd25372a4-0853-4524-85f8-ac01143167e6%2Fo8mfqwr.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Correlation Between Court Income and Justice Salaries
#### Dataset Overview
Listed below are amounts of court income and salaries paid to the town justices. All amounts are in thousands of dollars.
| Court Income | 64.0 | 404.0 | 1567.0 | 1131.0 | 270.0 | 250.0 | 111.0 | 1555.0 | 30.0 |
|--------------|-------|-------|--------|--------|-------|-------|-------|--------|-------|
| Justice Salary | 30 | 43 | 91 | 58 | 44 | 62 | 26 | 27 | 18 |
#### Instructions
1. **Construct a Scatterplot**
- Plot the *Court Income* values on the x-axis.
- Plot the *Justice Salary* values on the y-axis.
- Each point on the scatterplot corresponds to a pair of court income and justice salary values.
2. **Calculate the Linear Correlation Coefficient (r)**
- Use the formula:
\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}
\]
where
- \( n \) is the number of pairs,
- \( x \) represents *Court Income*,
- \( y \) represents *Justice Salary*.
3. **Find the P-value**
- Use an α level of 0.05.
- The P-value helps determine if the linear correlation is statistically significant.
4. **Analysis**
- Based on the results, determine if there is sufficient evidence to conclude that there is a linear correlation between court incomes and justice salaries.
- Discuss whether it appears that justices might profit by levying larger fines.
#### Result Interpretation
- **Scatterplot**: Visual representation of the data to look for any pattern or trend.
- **Linear Correlation Coefficient (r)**: Indicates the strength and direction of the linear relationship.
- Values close to 1 imply a strong positive correlation.
- Values close to -1 imply a strong negative correlation.
- Values around 0 imply no correlation.
- **P-value**:
- A P-value less than
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