One of the biggest factors in determining the value of a...

One of the biggest factors in determining the value of a home is the square footage. The following data represent the square footage and asking price (in thousands of dollars) for a random sample of homes for sale in Naples, Florida, in December $2010 .$ (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 1092 square feet is listed at $\$ 189,900 .$ Is this home's price above or below average for a home of this size? What might be some reasons for this price? (TABLE CAN'T COPY)

One of the biggest factors in determining the value of a home is the square footage. The following data represent the square footage and asking price (in thousands of dollars) for a random sample of homes for sale in Naples, Florida, in December $2010 .$
(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 1092 square feet is listed at $\$ 189,900 .$ Is this home's price above or below average for a home of this size? What might be some reasons for this price?
(TABLE CAN'T COPY)

Key Concepts

Explanatory Variable

In regression analysis, the explanatory variable (also known as the independent variable) is the variable that is presumed to affect or predict the outcome. It is the input or cause used in building a model, while the response variable is the output or effect. Understanding this distinction is essential for modeling and interpreting relationships between variables.

Scatter Diagram

A scatter diagram, or scatter plot, is a graphical representation that displays values for two quantitative variables. By plotting one variable along the x-axis and the other along the y-axis, the scatter diagram helps reveal signs of association, trends, and potential outliers. This visual tool is fundamental in preliminary data analysis before applying statistical models.

Linear Correlation Coefficient

The linear correlation coefficient is a numerical measure that quantifies the strength and direction of a linear relationship between two variables. Typically denoted as 'r', it ranges from -1 to 1, where values close to 1 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 suggest little or no linear association. This statistic is key in evaluating how well one variable predicts another.

Least-Squares Regression Line

The least-squares regression line is a statistical method used to determine the best fitting straight line through a set of data points. This line minimizes the sum of the squared differences (residuals) between the observed values and the predicted values on the line. It is a foundational tool in predictive modeling and helps to elucidate the quantifiable relationship between variables.

Slope Interpretation

The slope in the context of a regression line represents the rate at which the response variable changes for a one-unit increase in the explanatory variable. It quantifies the relationship between the two variables, indicating both the direction (positive or negative) and the magnitude of the change. A proper interpretation of the slope is vital for understanding the practical impact of changes in the explanatory variable.

Y-intercept Interpretation

The y-intercept in a linear regression model is the predicted value of the response variable when the explanatory variable is zero. While it is an important component of the regression equation, its practical interpretability depends on whether a value of zero for the explanatory variable is meaningful within the context of the data. In some cases, the intercept may have little practical significance if zero lies outside the range of observed values.

Prediction and Residual Analysis

Prediction involves using the regression line to estimate the response variable for given values of the explanatory variable, while residual analysis examines the differences between observed and predicted values. This analysis is crucial to assess the accuracy and appropriateness of the model. Understanding whether a particular observation falls above or below the regression line helps in identifying anomalies and understanding additional factors influencing the data.

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