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 slope \(\beta\) of the population regression line describes (a) the exact increase in the selling price of an individual unit when its appraised value increases by \(\$ 1000\). (b) the average increase in the appraised value in a population of units when selling price increases by \(\$ 1000\). (c) the average increase in selling price in a population of units when appraised value increases by \(\$ 1000\). (d) the average increase in the appraised value in the sample of units when selling price increases by \(\$ 1000\). (e) the average increase in selling price in the sample of units when the appraised value increases by \(\$ 1000\).

Step by step solution

01

Understanding the Question

We need to determine what the slope \( \beta \) of the population regression line indicates in this context. The options provided relate to the interpretation of the slope in terms of appraised and selling values of the properties based on the regression analysis.

02

Identifying the Regression Slope

From the Minitab output, the slope (labeled as 'Appraisal') of the regression line is given as \( 1.0466 \). This suggests that for every one unit increase in the appraised market value \( x \), the selling price \( y \) increases by 1.0466 units.

03

Interpreting the Slope

The slope represents the average change in the dependent variable \( y \) (selling price) for a one-unit change in the independent variable \( x \) (appraised value). Since the units are in thousands of dollars, for every \(1000 increase in the appraised value, the selling price increases on average by \)1046.6.

04

Matching the Interpretation with Options

Option (c) states: 'the average increase in selling price in a population of units when appraised value increases by $1000.' This directly matches our interpretation of the slope, indicating that the slope reflects the average increase in selling price for a corresponding increase in appraised value.

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Key Concepts

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

Population Regression Line

In statistics, the population regression line is a tool used to predict the relationship between two variables; it is akin to drawing the best line through a scatterplot of data points. This line seeks to capture the underlying trend within a larger dataset, representing how the dependent variable (in this case, the selling price of a property, denoted as \( y \)) changes in response to changes in an independent variable (here, the appraised market value, denoted as \( x \)).

The equation for a population regression line is often expressed in the form \( y = \beta_0 + \beta_1 x \), where:

• \( \beta_0 \) is the y-intercept, indicating the estimated selling price when the appraised value is zero.
• \( \beta_1 \) is the slope that represents the rate of change in the selling price for each unit change in the appraised value.

This line provides an approximation of the general trend found within a larger population from which our sample is drawn, giving stakeholders a reliable gauge of expected outcomes under varying conditions of the independent variable.

Slope Interpretation

The interpretation of the slope in regression analysis is crucial because it provides insights into how two variables are related. Specifically, it shows how much the dependent variable is expected to increase (or decrease) as the independent variable increases by one unit.

In the context of the property market example provided, the slope of the regression line is given as 1.0466. This means that for every \\(1000 increase in the appraised market value of a condominium, the selling price is expected to increase by \\)1046.6. This knowledge can be valuable for real estate investors and assessors to understand market trends and price expectations.

A positive slope, as seen in this case, indicates a direct correlation where increases in one variable lead to increases in the other. Conversely, a negative slope would indicate an inverse relationship, where higher values of the independent variable lead to lower values of the dependent variable.

Least-Squares Regression

The least-squares regression method is a standard approach for finding the best-fitting line through a set of points on a graph. It aims to minimize the sum of the squares of the differences (residuals) between observed and predicted values; this process enhances the predictive accuracy of the model.

The differences between each observed value and the corresponding value on the regression line are squared to eliminate negative numbers, and these are then summed up to assess how well the model fits the data.

The goal of least-squares regression is to make this cumulative squared difference as small as possible, ensuring the line is an optimal predictor of the dependent variable \( y \) for given values of the independent variable \( x \). This method provides a foundation for drawing meaningful inferences from real-world data and is widely used in disciplines like economics, biology, and engineering.