Putting It Together: Housing Prices One of the biggest...

Putting It Together: Housing Prices One of the biggest factors in determining the value of a home is the square footage. The following data represent the square footage and selling price (in thousands of dollars) for a random sample of homes for sale in Naples, Florida in January $2015 .$ $$ \begin{array}{cc} \text { Square Footage, } x & \text { Selling Price }(\$ 000 \mathrm{~s}), y \\ \hline 2204 & 379.9 \\ \hline 3183 & 375 \\ \hline 1128 & 189.9 \\ \hline 1975 & 338 \\ \hline 3101 & 619.9 \\ \hline 2769 & 370 \\ \hline 4113 & 627.7 \\ \hline 2198 & 375 \\ \hline 2609 & 425 \\ \hline 1708 & 298.1 \\ \hline 1786 & 271 \\ \hline 3813 & 690.1 \\ \hline \end{array} $$ (a) Which variable is the explanatory variable? (b) Draw a scatter diagram of the data. (c) Determine the linear correlation coefficient between square footage and asking price. (d) Is there a linear relation between the square footage and asking price? (e) Find the least-squares regression line treating square footage as the explanatory variable. (f) Interpret the slope. (g) Is it reasonable to interpret the $y$ -intercept? Why? (h) One home that is 1465 square feet is sold for $\$ 285,000$. Is this home's price above or below average for a home of this size? What might be some reasons for this price?

Putting It Together: Housing Prices One of the biggest factors in determining the value of a home is the square footage. The following data represent the square footage and selling price (in thousands of dollars) for a random sample of homes for sale in Naples, Florida in January $2015 .$
$$
\begin{array}{cc}
\text { Square Footage, } x & \text { Selling Price }(\$ 000 \mathrm{~s}), y \\
\hline 2204 & 379.9 \\
\hline 3183 & 375 \\
\hline 1128 & 189.9 \\
\hline 1975 & 338 \\
\hline 3101 & 619.9 \\
\hline 2769 & 370 \\
\hline 4113 & 627.7 \\
\hline 2198 & 375 \\
\hline 2609 & 425 \\
\hline 1708 & 298.1 \\
\hline 1786 & 271 \\
\hline 3813 & 690.1 \\
\hline
\end{array}
$$
(a) Which variable is the explanatory variable?
(b) Draw a scatter diagram of the data.
(c) Determine the linear correlation coefficient between square footage and asking price.
(d) Is there a linear relation between the square footage and asking price?
(e) Find the least-squares regression line treating square footage as the explanatory variable.
(f) Interpret the slope.
(g) Is it reasonable to interpret the $y$ -intercept? Why?
(h) One home that is 1465 square feet is sold for $\$ 285,000$. Is this home's price above or below average for a home of this size? What might be some reasons for this price?

Key Concepts

Residual Analysis

Residual analysis involves examining the differences between observed values and the values predicted by the regression model. These residuals help determine how well the model fits the data and can indicate whether an individual observation is above or below the model’s average prediction, highlighting potential issues like outliers or influential points.

Interpretation of y-intercept

The y-intercept in a regression equation is the predicted value of the response variable when the explanatory variable is zero. However, this value may not always be meaningful, especially if a zero value for the explanatory variable lies outside the range of observed data or does not make practical sense in the context of the study.

Least-Squares Regression Line

The least-squares regression line is a line fitted through a set of data points that minimizes the sum of the squared differences between the observed values and the values predicted by the line. This line is used to make predictions about the response variable based on values of the explanatory variable.

Interpretation of Slope

In the context of a regression line, the slope represents the estimated change in the response variable for a one-unit increase in the explanatory variable. It provides insight into the magnitude and direction of the relationship between the variables.

Scatter Diagram

A scatter diagram is a graphical representation where each pair of numerical observations is plotted as a point on a coordinate system. This tool is used to visually assess the potential relationship, trend, and outliers between two quantitative variables, which can indicate whether a linear model might be appropriate.

Explanatory and Response Variables

In statistical modeling, the explanatory variable (also known as the independent or predictor variable) is the variable believed to influence or help predict another variable, while the response variable (or dependent variable) is the outcome being studied. Understanding which variable plays each role is essential for setting up a regression analysis correctly.

Linear Correlation Coefficient

The linear correlation coefficient, often represented by 'r', is a numerical measure that quantifies the strength and direction of the linear relationship between two quantitative variables. It ranges from -1 to 1, with values close to 1 or -1 indicating strong relationships and values near 0 suggesting little to no linear relationship.

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