/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 The following data on sale price... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 23

The following data on sale price, size, and land-to-building ratio for 10 large industrial properties appeared in the paper "Using Multiple Regression Analysis in Real Estate Appraisal" (Appraisal Journal \([2002]: 424-430):\) \begin{tabular}{rrrr} & Sale Price (millions of dollars) & Size (thousands of sq. ft.) & Land- toBuilding \\ \hline 1 & \(10.6\) & 2166 & \(2.0\) \\ 2 & \(2.6\) & 751 & \(3.5\) \\ 3 & \(30.5\) & 2422 & \(3.6\) \\ 4 & \(1.8\) & 224 & \(4.7\) \\ 5 & \(20.0\) & 3917 & \(1.7\) \\ 6 & \(8.0\) & 2866 & \(2.3\) \\ 7 & \(10.0\) & 1698 & \(3.1\) \\ 8 & \(6.7\) & 1046 & \(4.8\) \\ 9 & \(5.8\) & 1108 & \(7.6\) \\ 10 & \(4.5\) & 405 & \(17.2\) \\ \hline \end{tabular} a. Calculate and interpret the value of the correlation coefficient between sale price and size. b. Calculate and interpret the value of the correlation coefficient between sale price and land-to-building ratio. c. If you wanted to predict sale price and you could use either size or land- to-building ratio as the basis for making predictions, which would you use? Explain. d. Based on your choice in Part (c), find the equation of the least-squares regression line you would use for predicting \(y=\) sale price.

Short Answer

Expert verified
The correlation coefficients between Sale Price and Size and Sale Price and Land-to-Building ratio would need to be computed and compared for their absolute values. The variable corresponding to the higher absolute value correlation coefficient would be a better predictor for Sale Price. The equation for the least-squares regression line is then obtained using the selected predictor.

Step by step solution

01

Calculate Correlation Coefficient between Sale Price and Size

The correlation coefficient, denoted as 'r', can be calculated using the formula:\[ r = \frac{n(\sum{xy})-\sum{x}\sum{y}}{\sqrt{[(n\sum{x^2}- (\sum{x})^2)(n\sum{y^2}- (\sum{y})^2)]}} \]where, 'x' is the Size and 'y' is the Sale Price. Plug in the values to get the value of 'r'.
02

Interpret the Correlation Coefficient between Sale Price and Size

The Correlation Coefficient ranges from -1 to 1. A value close to 1 indicates a strong positive correlation. The interpretation of this correlation coefficient value indicates how the Sale Price is related to the Size.
03

Calculate Correlation Coefficient between Sale Price and Land-to-Building Ratio

Similarly, the correlation coefficient between 'Sale Price' and 'Land-to-Building Ratio' can be calculated using the formula from Step 1. Here, 'x' is the Land-to-Building Ratio and 'y' is the Sale Price.
04

Interpret the Correlation Coefficient between Sale Price and Land-to-Building Ratio

The interpretation of this correlation coefficient value will indicate how the Sale Price is related to the Land-to-Building Ratio.
05

Determine the Better Predictor

Comparing the magnitude of the correlation coefficients calculated in step 1 and step 3, we can identify which one is better for predicting Sale Price. Whichever has the higher absolute value will be a better predictor.
06

Calculate the Least-Squares Regression Line

Using the better predictor determined in step 5, the least-squares regression line equation can be found using the formula: \(y = b0 + b1*x\), where 'b0' is the y-intercept, 'b1' is the slope, 'x' is the predictor, and 'y' is the dependent variable (Sale Price in this case). The values of 'b0' and 'b1' can be calculated using formulas for regression coefficients.

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Key Concepts

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

Multiple Regression Analysis
Multiple Regression Analysis is a powerful statistical tool used when we have more than one independent variable potentially influencing a dependent variable. Unlike simple regression, which involves only one independent variable, multiple regression allows for a more nuanced analysis by considering multiple factors simultaneously.
  • In the provided exercise, factors like Size and Land-to-Building Ratio are considered to see their effect on Sale Price. This is a typical scenario where multiple regression would be applicable if we want to model the outcome based on more than one factor.
  • Although the exercise focuses on evaluating each variable separately, multiple regression would help in creating a more comprehensive model by examining how each factor contributes to the Sale Price when considering the joint effect.
Through multiple regression, you can determine not only which factor is most predictive but also how each factor interacts with one another. It's useful in real estate and numerous other fields to derive insights from complex datasets.
Regression Line
A Regression Line represents the relationship between the dependent and independent variables in a linear regression model. It is a straight line that best fits the data on a scatter plot, minimizing the differences between the actual data points and the predicted values.
  • The formula for a simple linear regression line is given as: \( y = b_0 + b_1 x \)
  • Here, 'y' is the dependent variable (Sale Price), 'x' is the independent variable (either Size or Land-to-Building Ratio), 'b_0' is the y-intercept, and 'b_1' is the slope of the line.
Determining this line involves calculating the slope and intercept using the least-squares method, which diminishes the total of squared deviations of the points from the line.
This regression line is crucial as it serves as a predictive model for estimating the dependent variable based on given values of 'x'. It provides a simplified representation of the correlation, enabling us to make predictions and decisions based on historical data.
Predictive Modeling
Predictive Modeling involves the use of statistical techniques and algorithms to forecast outcomes based on historical data. It is a key application of regression analysis in numerous fields, including real estate, finance, and healthcare.
  • For the given exercise, once the regression line is established, it acts as a predictive model for Sale Price based on Size or Land-to-Building Ratio.
  • This enables real estate appraisers and analysts to make informed predictions about future sale prices using past trends, improving decision-making processes.
Predictive modeling offers insights into potential future trends and risks by identifying patterns within data. While this exercise looks at using individual predictors for price estimation, combining them with other predictive analytics could give even more valuable forecasts.

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Most popular questions from this chapter

The accompanying data resulted from an experiment in which weld diameter \(x\) and shear strength \(y\) (in pounds) were determined for five different spot welds on steel. A scatterplot shows a strong linear pattern. With \(\sum(x-\bar{x})^{2}=1000\) and \(\sum(x-\bar{x})(y-\bar{y})=8577\), the least-squares line is \(\hat{y}=-936.22+8.577 x\) \(\begin{array}{cccccc}x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0\end{array}\) \(\begin{array}{llllll}y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2\end{array}\) a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new \(y=0.4536(\) old \(y)\). What is the equation of the least-squares line when \(y\) is expressed in kilograms? b. More generally, suppose that each \(y\) value in a data set consisting of \(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the units of measurement for \(y\) ). What effect does this have on the slope \(b\) (i.e., how does the new value of \(b\) compare to the value before conversion), on the intercept \(a\), and on the equation of the least-squares line? Verify your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace \(y\) with \(c y\), and see what happensand remember, this conversion will affect \(\bar{y}\).)

The relationship between hospital patient-tonurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables, indicate whether you expect the slope of the least-squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of patient quality of care.

5.15 The authors of the paper "Evaluating Existing Movement Hypotheses in Linear Systems Using Larval Stream Salamanders" (Canadian journal of Zoology [2009]: \(292-298\) ) investigated whether water temperature was related to how far a salamander would swim and whether it would swim upstream or downstream. Data for 14 streams with different mean water temperatures where salamander larvae were released are given (approximated from a graph that appeared in the paper). The two variables of interest are \(x=\) mean water temperature \(\left({ }^{\circ} \mathrm{C}\right)\) and \(y=\) net directionality, which was defined as the difference in the relative frequency of the released salamander larvae moving upstream and the relative frequency of released salamander larvae moving downstream. A positive value of net directionality means a higher proportion were moving upstream than downstream. A negative value of net directionality means a higher proportion were moving downstream than upstream. \begin{tabular}{cc} Mean Temperature \((x)\) & Net Directionality \((y)\) \\ \hline \(6.17\) & \(-0.08\) \\ \(8.06\) & \(0.25\) \\ \(8.62\) & \(-0.14\) \\ \(10.56\) & \(0.00\) \\ \(12.45\) & \(0.08\) \\ \(11.99\) & \(0.03\) \\ \(12.50\) & \(-0.07\) \\ \(17.98\) & \(0.29\) \\ \(18.29\) & \(0.23\) \\ \(19.89\) & \(0.24\) \\ \(20.25\) & \(0.19\) \\ \(19.07\) & \(0.14\) \\ \(17.73\) & \(0.05\) \\ \(19.62\) & \(0.07\) \\ \hline \end{tabular} a. Construct a scatterplot of the data. How would you describe the relationship between \(x\) and \(y\) ? b. Find the equation of the least-squares line describing the relationship between \(y=\) net directionality and \(x=\) mean water temperature. c. What value of net directionality would you predict for a stream that had mean water temperature of \(15^{\circ} \mathrm{C}\) ? d. The authors state that "when temperatures were warmer, more larvae were captured moving upstream, but when temperatures were cooler, more larvae were captured moving downstream." Do the scatterplot and least-squares line support this statement? e. Approximately what mean temperature would result in a prediction of the same number of salamander larvae moving upstream and downstream?

Example \(5.15\) described a study that involved substituting sunflower meal for a portion of the usual diet of farm-raised sea breams (Aquaculture [2007]: 528-534). This paper also gave data on \(y=\) feed intake (in grams per 100 grams of fish per day) and \(x=\) percentage sunflower meal in the diet (read from a graph in the paper). $$ \begin{array}{rrrrrrrrr} x & 0 & 6 & 12 & 18 & 24 & 30 & 36 \\ y & 0.86 & 0.84 & 0.82 & 0.86 & 0.87 & 1.00 & 1.09 \end{array} $$ A scatterplot of these data is curved and the pattern in the plot resembles a quadratic curve. a. Using a statistical software package or a graphing calculator, find the equation of the least-squares quadratic curve that can be used to describe the relationship between percentage sunflower meal and feed intake. b. Use the least-squares equation from Part (a) to predict feed intake for fish fed a diet that included \(20 \%\) sunflower meal.

The paper "The Shelf Life of Bird Eggs: Testing Egg Viability Using a Tropical Climate Gradient" (Ecology [2005]: 2164-2175) investigated the effect of altitude and length of exposure on the hatch rate of thrasher eggs. Data consistent with the estimated probabilities of hatching after a number of days of exposure given in the paper are shown here. Probability of Hatching Exposure (days) \(\quad\)\begin{tabular}{cccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline Proportion (lowland) & \(0.81\) & \(0.83\) & \(0.68\) & \(0.42\) & \(0.13\) & \(0.07\) & \(0.04\) & \(0.02\) \\ Proportion (mid-elevation) & \(0.49\) & \(0.24\) & \(0.14\) & \(0.037\) & \(0.040\) & \(0.024\) & \(0.030\) \\ \\\ Proportion \(0.75\) (cloud forest) & \(0.67\) & \(0.36\) & \(0.31\) & \(0.14\) & \(0.09\) & \(0.06\) & \(0.07\) \\ \hline \end{tabular} a. Plot the data for the low- and mid-elevation experimental treatments versus exposure. Are the plots generally the shape you would expect from "logistic" plots? b. Using the techniques introduced in this section, calculate \(y^{\prime}=\ln \left(\frac{p}{1-p}\right)\) for each of the exposure times in the cloud forest and fit the line \(y^{\prime}=a+b(\) Days \()\). What is the significance of a a negative slope to this line? c. Using your best-fit line from Part (b), what would you estimate the proportion of eggs that would, on average, hatch if they were exposed to cloud forest conditions for 3 days? 5 days? . d. At what point in time does the estimated proportion of hatching for cloud forest conditions seem to cross from greater than \(0.5\) to less than \(0.5\) ?
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