Listed below are amounts of court fine revenue and salaries paid to the town justice. (All amount are in thousands of dollars and all of the towns are in Dutchess County, New York.)

Court Income	65	404	1567	1131	272	252	111	154	32
Justice Salary	30	44	92	56	46	61	25	26	18
Predict the justice salary for a court income of $800,000.  Answer with units in dollars and round to nearest dollar.

To predict the justice salary for a court income of $800,000 using linear regression, we can follow these steps:

1. **Input the given dataTo predict the justice salary for a court income of $800,000 using linear regression, we can follow these steps:

1. **Input the given **
   - Court Income ($X$) in thousands of dollars: $[65, 404, 1567, 1131, 272, 252, 111, 154, 32]$
   - Justice Salary ($Y$) in thousands of dollars: $[30, 44, 92, 56, 46, 61, 25, 26, 18]$

2. **Convert Court Income of $800,000 to thousands of dollars:**
   \[
   X_{new} = 800 \text{ (since } 800,000 \text{ divided by 1,000 is 800)}
   \]

3. **Calculate the linear regression parameters ($a$ and $b$):**

The linear regression equation is:
\[
Y = a + bX
\]

The formulas to calculate $a$ and $b$ are:
\[
b = \frac{n(\sum XY) - (\sum X)(\sum Y)}{n(\sum X^2) - (\sum X)^2}
\]
\[
a = \frac{(\sum Y) - b(\sum X)}{n}
\]

Where:
- $n$ is the number of data points
- $X$ is the court income
- $Y$ is the justice salary

Let's perform the calculations:

\[
\sum X = 3988
\]
\[
\sum Y = 398
\]
\[
\sum XY = 409798
\]
\[
\sum X^2 = 6854112
\]

Substitute these values into the formulas:

\[
b = \frac{9 \cdot 409798 - 3988 \cdot 398}{9 \cdot 6854112 - 3988^2}
\]

\[
b = \frac{3688182 - 1588224}{61687008 - 15904064}
\]

\[
b = \frac{2099958}{45782944}
\]

\[
b \approx 0.0459
\]

Next, calculate $a$:

\[
a = \frac{398 - 0.0459 \cdot 3988}{9}
\]

\[
a = \frac{398 - 183.05}{9}
\]

\[
a \approx 23.89
\]

4. **Predict the justice salary for $X_{new} = 800$:**

\[
Y_{new} = a + bX_{new}
\]

\[
Y_{new} = 23.89 + 0.0459 \cdot 800
\]

\[
Y_{new} = 23.89 + 36.72
\]

\[
Y_{new} = 60.61
\]

5. **Convert back to dollars and round to the nearest dollar:**

\[
Y_{new} \approx 60610 \text{ dollars}
\]

Therefore, the predicted justice salary for a court income of $800,000 is approximately \$60,610.

Do you want more details or have any questions? 

Here are 8 related questions:
1. What is the correlation between court income and justice salary?
2. How would the prediction change if we used a different regression model?
3. Can we use polynomial regression to predict the salary more accurately?
4. What are the limitations of using linear regression for this prediction?
5. How does the predicted salary compare to the actual salaries given?
6. What is the R-squared value for this linear regression model?
7. How would the prediction differ if one outlier data point is removed?
8. Can this method be used for predicting salaries in other counties?

**Tip:** Always check the residuals of your regression model to assess its accuracy and the presence of any patterns not captured by the model.

Learn how to predict the justice salary based on court income using linear regression. Step-by-step guide with calculations using statistical methods. Suitable for understanding statistical prediction methods in data analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation