Setup Of Regression Models To Analyze House Age And County

Setup of regression models to analyze house age and county taxes

An auditor for a county government would like to develop a model to predict the county taxes based on the age of a single-family houses. A random sample of 19 single-family houses has been selected, with the following results: Taxes and Age data are available to establish regression models.

a) Set up a scatter diagram between age and county taxes.

b) State the linear regression equation predicting county taxes based on house age.

c) State the quadratic regression equation for the same data. Determine whether there is a significant overall relationship between age and county taxes at the 0.05 level of significance.

d) Determine which of the models (linear or quadratic) is better for predicting average county taxes for a house, using a significance level of 0.025.

e) Using appropriate plots for each model, perform residual analysis to determine whether the assumptions of regression have been violated.

f) Using the quadratic regression equation, predict the average county taxes for a house that is 20 years old.

Data for this analysis includes the house ages and corresponding county taxes. The goal is to establish the most appropriate model for predicting taxes based on house age and to verify the validity of the assumptions underlying these models.

Paper For Above instruction

Predicting county taxes based on the age of single-family houses is a crucial task for local government officials seeking to optimize tax assessment and revenue collection. By examining the relationship between house age and taxes through various regression models, policymakers can gain insights into tax patterns and potential factors influencing property valuation. This paper outlines the steps for establishing and evaluating linear and quadratic regression models for this purpose, including data visualization, model fitting, significance testing, model comparison, residual analysis, and specific predictions.

Introduction

Urban and suburban tax assessments often consider property characteristics, among which the age of a house is significant. Older houses might depreciate in value or, alternatively, may benefit from historical preservation, affecting their tax assessments. Developing an accurate predictive model enables efficient and equitable tax collection, supporting local government budgets and community planning. This study involves 19 house samples where both age and county taxes are recorded, aiming to establish a statistical model explaining the variation in taxes as a function of age.

Establishing the Scatter Diagram

The initial step involves creating a scatter plot of property age versus county taxes. This visual representation helps identify the nature of the relationship—whether linear, non-linear, or more complex. For instance, if the scatter shows a downward trend, it suggests that taxes decrease as houses age, possibly reflecting depreciation. Conversely, an irregular distribution may indicate the need for more sophisticated modeling techniques. Creating this plot involves plotting house ages on the horizontal axis and corresponding taxes on the vertical axis, facilitating visual assessment of the data trend.

Linear Regression Model

The linear regression model assumes a relationship of the form:

Taxes = β₀ + β₁ * Age + ε

where β₀ is the intercept, β₁ is the slope coefficient, and ε is the error term. Using least squares estimation, the parameters β₀ and β₁ are determined to minimize the sum of squared residuals. The estimated regression equation enables prediction of taxes for any given house age within or near the sample range.

Quadratic Regression Model

The quadratic model extends the linear form by including a squared term:

Taxes = α₀ + α₁ Age + α₂ Age² + ε

This model captures potential curvature in the relationship, such as taxes decreasing at a decreasing rate or increasing after a certain age. Fitting this model involves estimating parameters α₀, α₁, and α₂ using least squares.

Significance Testing of the Regression Relationship

To determine if the relationship between house age and taxes is statistically significant, an F-test is performed at the 0.05 significance level. The null hypothesis states that none of the independent variables (age, and quadratic term if applicable) explain a significant variance in taxes. A p-value less than 0.05 indicates rejecting the null, confirming an overall significant relationship.

Model Comparison

Between the linear and quadratic models, we assess which provides a better fit for the data using criteria such as adjusted R², Akaike Information Criterion (AIC), or Bayesian Information Criterion (BIC). The significance level for model comparison is set at 0.025, meaning the model showing statistically significant improvement at this threshold is preferred. The model with higher adjusted R² and lower AIC/BIC values generally indicates a better fit.

Residual Analysis

Residual plots for each model are evaluated to check compliance with regression assumptions, including linearity, homoscedasticity, independence, and normality of residuals. Plotting residuals against fitted values helps identify patterns suggesting violations such as heteroscedasticity or non-normal residuals, which necessitate model revision or transformation.

Prediction for a 20-Year-Old House

Once the suitable model is selected, the quadratic equation is used to predict taxes for a house aged 20 years. Substituting age = 20 into the regression equation yields an estimate of the average county taxes for such a property, providing a practical application of the model.

Conclusion

Developing an accurate predictive model requires careful data visualization, model fitting, hypothesis testing, comparison, and residual diagnostics. The choice between linear and quadratic models depends on statistical evidence and the behavior of residuals. The resulting model not only aids in efficient tax assessment but also enhances understanding of how property age influences tax values. These insights can inform policy decisions and improve the fairness and accuracy of property taxation.

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