91Ó°ÊÓ

Find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions. Listed below are amounts of court income and salaries paid to the town justices (based on data from the Poughkeepsie Journal). All amounts are in thousands of dollars, and all of the towns are in Dutchess County, New York. For the prediction interval, use a \(99 \%\) confidence level with a court income of \(\$ 800,000\). $$\begin{array}{l|l|r|r|r|r|r|r|r|r}\hline \text { Court Income } & 65 & 404 & 1567 & 1131 & 272 & 252 & 111 & 154 & 32 \\\\\hline \text { Justice Salary } & 30 & 44 & 92 & 56 & 46 & 61 & 25 & 26 & 18 \\\\\hline\end{array}$$

Step by step solution

01

- Calculate Regression Equation

Use the given data to calculate the regression equation. Assume the form of the regression equation is \( y = b_0 + b_1x \) where \( y \) is the justice salary and \( x \) is the court income. Calculate the values of \( b_0 \) and \( b_1 \).

02

- Find Explained Variation

The explained variation (SS\text{regression}) is computed using the formula: \[ \text{SS\text{regression}} = \sum ( \hat{y} - \bar{y})^2 \] Here, \( \hat{y} \) are the predicted values from the regression equation, and \( \bar{y} \) is the mean of the observed justice salaries.

03

- Find Unexplained Variation

The unexplained variation (SS\text{residual}) is computed using the formula: \[ \text{SS\text{residual}} = \sum ( y - \hat{y})^2 \] where \( y \) are the observed justice salaries and \( \hat{y} \) are the predicted values from the regression equation.

04

- Calculate Total Variation

The total variation (SS\text{total}) is the sum of the explained and unexplained variations: \[ \text{SS\text{total}} = \text{SS\text{regression}} + \text{SS\text{residual}} \]

05

- Find Indicated Prediction Interval

For the prediction interval, calculate the margin of error using the formula: \[ \text{Margin of Error} = t_{\text{critical}} \cdot s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - x\bar)^2}{\sum (x_i - x\bar)^2}} \] where \( t_{\text{critical}} \) is based on the given confidence level (99%), \( s \) is the standard error of the estimate, and \( x_0 = 800 \) is the court income value. The prediction interval can thus be calculated using: \[ \text{Prediction Interval} = \hat{y} \pm\text{Margin of Error} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Explained Variation

Explained variation, denoted as SS\text{regression}, tells us how much of the total variation in the dependent variable (justice salary) can be explained by the linear relationship with the independent variable (court income). It gives us a sense of how well our regression model captures the trend in the data.
In mathematical terms, we calculate explained variation using the predicted values \( \hat{y} \) from our regression equation and the mean \( \bar{y} \) of the observed justice salaries:

• \[ \text{SS\text{regression}} = \sum ( \hat{y} - \bar{y})^2 \]

This formula calculates the sum of the squared differences between each predicted value and the mean of the observed values. A high SS\text{regression} means that the model explains a lot of the variability, while a low SS\text{regression} means it does not.

Unexplained Variation

Unexplained variation (also known as residual variation or SS\text{residual}) tells us how much of the variation in the dependent variable cannot be explained by our regression model. It represents the random error or noise in our data that the model fails to capture.
This can be calculated by taking the differences between the observed values \( \ y \) and the predicted values \( \ hat{y} \), squaring them, and summing them up:

• \[ \text{SS\text{residual}} = \sum ( y - \ hat{y})^2 \]

When the SS\text{residual} is low, the model's predictions are close to the actual observed values, indicating a good fit. Conversely, a high SS\text{residual} suggests a poor fit.

Prediction Interval

A prediction interval provides a range within which we can expect a future observation to fall, given a certain level of confidence (in this case, 99%). It is wider than a confidence interval for the mean because it accounts for the variation in individual observations.
To calculate this interval, we first determine the margin of error, which takes into account both the standard error of the estimate and the critical value from the t-distribution:

• \[ \text{Margin of Error} = t_{\text{critical}} \cdot s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - x\bar)^2}{\sum (x_i - x\bar)^2}} \]

Here, \( t_{\text{critical}} \) is the value from the t-table for a 99% confidence level, \( s \) is the standard error, and \( x_0 = 800 \) (the court income).Finally, the prediction interval is given by:

• \[ \text{Prediction Interval} = \ hat{y} \pm \text{Margin of Error} \]

This interval helps us understand the range within which the justice salary is likely to fall given a certain court income.

Linear Correlation

Linear correlation measures the strength and direction of the linear relationship between two variables. If the correlation between court income and justice salary is strong, it means these two variables change together in a predictable way. The correlation coefficient (r) typically ranges from -1 to 1.

• A value close to 1 indicates a strong positive correlation, meaning as court income increases, justice salary tends to increase.
• A value close to -1 indicates a strong negative correlation, meaning as one variable increases, the other decreases.
• A value near 0 suggests no linear relationship.

In our case, the problem statement mentions there is sufficient evidence to support a claim of a linear correlation, hence we use regression to predict values confidently.

Regression Equation

The regression equation is a mathematical formula that describes the relationship between the independent variable (court income) and the dependent variable (justice salary). It has the form \( y = b_0 + b_1x \).

• \( y \) is the predicted value (justice salary)
• \( b_0 \) is the y-intercept (the value of y when x is 0)
• \( b_1 \) is the slope (the change in y for a one-unit change in x)
• \( x \) is the independent variable (court income)

To find \( b_0 \) and \( b_1 \), we use methods like least squares to minimize the sum of the squared differences between observed and predicted values.This equation allows us to make predictions and understand the linear relationship between the variables.