91Ó°ÊÓ

Multiple choice: Select the best answer for Exercises, which are based on the following information. To determine property taxes, Florida reappraises real estate every year, and the county appraiser's Web site lists the current "fair market value" of each piece of property. Property usually sells for somewhat more than the appraised market value. We collected data on the appraised market values \(x\) and actual selling prices \(y\) (in thousands of dollars) of a random sample of 16 condominium units in Florida. We checked that the conditions for inference about the slope of the population regression line are met. Here is part of the Minitab output from a least-squares regression analysis using these data. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ The equation of the least-squares regression line for predicting selling price from appraised value is (a) price \(=79.49+0.1126\) (appraised value). (b) price \(=0.1126+1.0466\) (appraised value). (c) price \(=127.27+1.0466\) (appraised value). (d) price \(=1.0466+127.27\) (appraised value). (e) price \(=1.0466+69.7299\) (appraised value).

Step by step solution

01

Identify the Regression Equation

The regression equation for predicting the selling price from the appraised market value is found using the coefficients given in the output. The general form of a regression equation is \( y = b_0 + b_1 x \), where \( b_0 \) is the y-intercept (Constant) and \( b_1 \) is the slope (the coefficient of Appraisal).

02

Use the Given Coefficients

Based on the output, the y-intercept \( b_0 \) is given as 127.27, and the slope \( b_1 \) is given as 1.0466. Therefore, the correct regression equation is \( y = 127.27 + 1.0466x \).

03

Match with the Options

Compare the derived regression equation with the provided options. Option (c) states "price \(=127.27+1.0466\) (appraised value)", which matches our derived equation \( y = 127.27 + 1.0466x \).

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Key Concepts

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

Property Tax Analysis

Property tax analysis in Florida involves understanding how property taxes are assessed and applied based on the appraised market value of real estate properties. Every year, Florida conducts a reappraisal of real estate, determining the "fair market value" of each property. This value is essential because property taxes are derived from it. These taxes are critical as they fund various public services, like schools, roads, and local government operations.
However, the appraised value usually differs from the selling price. Properties tend to sell for more than the appraised value, as seen in our data collection of 16 condominium units in Florida. In practice, property tax analysis helps in identifying how expected market values align with actual selling prices, which can influence decision-making for homeowners, buyers, and appraisers alike. It also serves as a basis for making predictions using statistical tools like regression analysis.

Regression Equation

A regression equation is a mathematical formula that describes the relationship between two variables. In our study, we are trying to predict the selling price of a property (dependent variable) based on its appraised value (independent variable). The general form of a regression equation is given by \( y = b_0 + b_1x \), where \( b_0 \) is the y-intercept and \( b_1 \) is the slope of the line.
From the exercise, the regression equation was determined as \( y = 127.27 + 1.0466x \). Here, 127.27 represents the y-intercept, which is the estimated selling price when the appraised value is zero. The slope, 1.0466, indicates how much the selling price is expected to increase for a one-unit increase in the appraised value. This regression line provides a best-fit line through our data, minimizing the sum of squared differences between the observed and predicted values.

Statistical Inference

Statistical inference allows us to make predictions about population parameters based on sample data. It's a significant concept in regression analysis as it helps us understand the reliability and accuracy of our regression model. For instance, in our exercise, statistical inference helps validate that the conditions are met for inference about the slope of the population regression line.
The results from the Minitab output provide a \( p \)-value of 0.000 for the slope coefficient, suggesting the relationship between appraised value and selling price is statistically significant. This low \( p \)-value indicates that there is a less than 0.1% probability that the observed relationship is due to random chance alone. Therefore, we are confident that the appraised value is a good predictor of the selling price.

Regression Coefficients

Regression coefficients are values that quantify the relationship between the predictor variable and the response variable in a regression model. In our example, they provide insight into how changes in the appraised value affect the selling price.
We have two key coefficients: the y-intercept (127.27) and the slope (1.0466). The y-intercept represents the expected selling price of a property when the appraised value is zero. While theoretically, a property with a zero appraised value might not make practical sense, the intercept can be useful in understanding the baseline influence of other factors not included in the model.
The slope coefficient of 1.0466 is immensely important as it quantifies the increase in the selling price with every increment of 1 unit in appraised value. This shows that for every $1,000 increase in the appraised value, the selling price is expected to increase by approximately $1,046.60. Understanding these coefficients helps in making informed predictions and decisions based on the regression model.