Question

An article on the cost of housing in California that appeared in the San Luis Obispo Tribune (March 30,2001 ) included the following statement: "In Northern California, people from the San Francisco Bay area pushed into the Central Valley, benefiting from home prices that dropped on average \(\$ 4000\) for every mile traveled east of the Bay area." If this statement is correct, what is the slope of the least-squares regression line, \(\hat{y}=a+b x\), where \(y=\) house price (in dollars) and \(x=\) distance east of the Bay (in miles)? Explain.

Short answer

The slope of the least-squares regression line, \(\hat{y}=a+bx\), where \(y=\) house price (in dollars) and \(x=\) distance east of the Bay (in miles), is -4000.

Step-by-step solution

Understand the Model

From the information given, it can be seen that the model \(y=a+bx\) represents the relationship between the housing price and the distance from the Bay area. Here, \(y=\) house price (in dollars) and \(x=\) distance east of the Bay (in miles). The slope of the regression line, represented by \(b\), shows the change in the y-variable (house price) for each one unit increase in the x-variable (distance).

Identify the Slope

According to the article, home prices drop an average of $4000 for every mile travelled east of the Bay area. This change is associated with the slope. Hence, the slope can be identified as the change in the housing price for each increase in the distance from the Bay area. That is, \(b = -4000\)

Explain the Slope

The slope of -4000 means that for each mile increase in the distance east of the Bay area, the price of the house decreases by $4000. This negative relationship indicates that the further you go from the Bay area, the less expensive the houses become.

Key concepts

Statistical Analysis

Statistical analysis is a component of data analysis that involves collecting, reviewing, and summarizing data to discover patterns or relationships. In the context of housing price data, statistical analysis can be used to determine how various factors, such as location, affect housing prices. By examining trends and making inferences, analysts can provide valuable insights for potential buyers, sellers, and policymakers.

For example, analyzing the distance from a high-demand area like the San Francisco Bay and correlating it with housing prices can be a powerful demonstration of how location affects value. Statistical tools, including regression analysis, enable the quantification of this relationship and help in making data-driven decisions.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables by fitting a linear equation to the observed data. The general form of a linear regression model is \( y = a + bx \), where \( y \) is the dependent variable, \( x \) is the independent variable, \( a \) is the y-intercept, and \( b \) is the slope.

In housing market studies, linear regression can predict housing prices (\( y \) variable) based on the distance from a central location (\( x \) variable). This type of analysis allows for an understanding of how various factors may influence the price of a home and can be particularly useful for forecasting future price trends based on current data.

Slope Interpretation

In the context of a least-squares regression line, the slope \( b \) signifies the rate of change in the dependent variable for every one unit increase in the independent variable. It's a measure of how much \( y \) (in this case, house price) is expected to change when \( x \) (distance from Bay area) increases by one mile.

A slope of \( -4000 \) in the regression equation \( \hat{y} = a + bx \) means that, with every additional mile you move east of the San Francisco Bay area, the housing price is predicted to decrease by $4,000. This negative slope indicates an inverse relationship: as one variable increases, the other decreases. It's a crucial piece of information for anyone looking to understand real estate price trends in relation to location proximity.

Housing Price Data

Housing price data is a critical resource for various stakeholders in the real estate market, providing insight into the current and historical prices of homes. This data can be analyzed to identify patterns, trends, and anomalies—which is essential for making informed decisions, whether it be for investment, purchasing, or market analysis purposes.

Applying statistical analysis to housing price data, such as in the case of the regression problem with California houses, enables individuals and businesses to recognize the impact of location and other factors on housing prices. It also provides a basis for predictions on how prices might change as these factors vary, aiding in the financial planning and forecasting that is vital to the housing industry.