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Chapter 12: Problem 20

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 \(\$ 1000\).

Short Answer

Expert verified
Option (c) is the correct interpretation of the slope.

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.

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Most popular questions from this chapter

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. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ Which of the following would have resulted in a violation of the conditions for inference? (a) If the entire sample was selected from one neighborhood (b) If the sample size was cut in half (c) If the scatterplot of \(x=\) appraised value and \(y=\) selling price did not show a perfect linear relationship (d) If the histogram of selling prices had an outlier (e) If the standard deviation of appraised values was different from the standard deviation of selling prices

Beavers and beetles Do beavers benefit beetles? Researchers laid out 23 circular plots, each 4 meters in diameter, at random in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth, so that beetles prefer them. If so, more stumps should produce more beetle larvae. \({ }^{8}\) Minitab output for a regression analysis on these data is shown below. Construct and interpret a \(99 \%\) confidence interval for the slope of the population regression line. Assume that the conditions for performing inference are met. $$ \begin{aligned} &\text { Regression Analysis: Beetle larvae versus Stumps }\\\ &\begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & -1.286 & 2.853 & -0.45 & 0.657 \\ \text { Stumps } & 11.894 & 1.136 & 10.47 & 0.000 \end{array}\\\ &\begin{array}{ll} S=6.41939 & R-S q=83.9 \% & R-S q(a d j)=83.1 \% \end{array} \end{aligned} $$

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. \({ }^{13}\) $$ \begin{array}{lllll} \text { Predictor } & \text { Coef } & \text { SE Coef } & \text { T } & \text { P } \\ \text { Constant } & 127.27 & 79.49 & 1.60 & 0.132 \\ \text { Appraisal } & 1.0466 & 0.1126 & 9.29 & 0.000 \\ \mathrm{~S}=69.7299 & \mathrm{R}-\mathrm{Sq}=86.1 \% & \mathrm{R}-\mathrm{Sq}(\mathrm{adj}) & =85.1 \% \end{array} $$ Which of the following is the best interpretation for the value 0.1126 in the computer output? (a) For each increase of \(\$ 1000\) in appraised value, the average selling price increases by about 0.1126 . (b) When using this model to predict selling price, the predictions will typically be off by about 0.1126 . (c) \(11.26 \%\) of the variation in selling price is accounted for by the linear relationship between selling price and appraised value. (d) There is a weak, positive linear relationship between selling price and appraised value. (e) In repeated samples of size 16 , the sample slope will typically vary from the population slope by about \(0.1126 .\)

The professor swims Here are data on the time (in minutes) Professor Moore takes to swim 2000 yards and his pulse rate (beats per minute) after swimming on a random sample of 23 days: $$ \begin{array}{lrrrrrr} \hline \text { Time: } & 34.12 & 35.72 & 34.72 & 34.05 & 34.13 & 35.72 \\ \text { Pulse: } & 152 & 124 & 140 & 152 & 146 & 128 \\ \text { Time: } & 36.17 & 35.57 & 35.37 & 35.57 & 35.43 & 36.05 \\ \text { Pulse: } & 136 & 144 & 148 & 144 & 136 & 124 \\ \text { Time: } & 34.85 & 34.70 & 34.75 & 33.93 & 34.60 & 34.00 \\ \text { Pulse: } & 148 & 144 & 140 & 156 & 136 & 148 \\ \text { Time: } & 34.35 & 35.62 & 35.68 & 35.28 & 35.97 & \\ \text { Pulse: } & 148 & 132 & 124 & 132 & 139 & \\ \hline \end{array} $$ Is there statistically significant evidence of a negative linear relationship between Professor Moore's swim time and his pulse rate in the population of days on which he swims 2000 yards? Carry out an appropriate significance test at the \(\alpha=0.05\) level.

The swinging pendulum Mrs. Hanrahan's precalculus class collected data on the length (in centimeters) of a pendulum and the time (in seconds) the pendulum took to complete one back-and forth swing (called its period). Here are their data: $$ \begin{array}{cc} \hline \text { Length (cm) } & \text { Period (s) } \\ 16.5 & 0.777 \\ 17.5 & 0.839 \\ 19.5 & 0.912 \\ 22.5 & 0.878 \\ 28.5 & 1.004 \\ 31.5 & 1.087 \\ 34.5 & 1.129 \\ 37.5 & 1.111 \\ 43.5 & 1.290 \\ 46.5 & 1.371 \\ 106.5 & 2.115 \\ \hline \end{array} $$ (a) Make a reasonably accurate scatterplot of the data by hand, using length as the explanatory variable. Describe what you see. (b) The theoretical relationship between a pendulum's length and its period is $$ \text { period }=\frac{2 \pi}{\sqrt{g}} \sqrt{\text { length }} $$ where \(g\) is a constant representing the acceleration due to gravity (in this case, \(g=980 \mathrm{~cm} / \mathrm{s}^{2}\) ). Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a). (c) Use the following graph to identify the transformation that was used to linearize the curved pattern in part (a).
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