Question

An article on the cost of housing in California (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." 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)? Justify your answer.

Short answer

The slope of the least squares regression line is -\(\$4000). For every mile traveled eastward from the San Francisco Bay, the house prices drop by \(\$4000).

Step-by-step solution

Understand the Problem

The first step is to understand the problem here. We're looking at a direct relationship between distance and house pricing. Based on the information given, we can understand that as the distance from the Bay increases (a move to the east), house prices decrease. This relationship is linear and can be expressed using a simple linear regression line.

Identify Variables and Constants

Let's identify what we know. The dependent variable is the house price (y) and the independent variable (x) is the distance from the Bay. The drop in price per mile (\(\$4000)) is essentially the slope of the regression line, denoted by (b). The slope (b) represents the amount of change in y (price of the house) for a unit change in x (distance from the Bay). Since the prices are decreasing with distance, our slope is negative.

Determine the Slope of the Regression Line

In the question, it's stated that for each mile travelled east of the Bay, the price drops by \(\$4000). This exactly corresponds to the slope of our linear regression line, because the slope represents the rate of change of y with respect to x. So, the slope of the linear regression line, (b), is equal to -\(\$4000).

Key concepts

Slope of the Regression Line

Understanding the slope of a regression line is paramount when analyzing the relationship between two variables in statistics. In the context of linear regression, the slope is a measure that shows the average change in the dependent variable for every one-unit change in the independent variable.

To put this into a real-world context like the housing cost problem mentioned, imagine plotting a graph where every point represents a house's price based on its distance from a specific location. The slope indicates how steeply house prices are increasing or decreasing as you move further from that point. The exercise states that house prices drop by \(\$4000\) for every mile away from the Bay Area, which is interpreted as a negative slope. Mathematically, this slope (denoted as \(b\) in the regression equation \(\hat{y}=a+bx\)) is \(-\$4000\). It indicates that for every mile you travel east of the Bay Area, the average house price falls by \(\$4000\), showing a negative linear relationship between distance and price.

This concept is essential in predicting outcomes. For instance, if someone wanted to know the average price of a home 10 miles east of the Bay Area, they could use the slope to calculate this estimate.

Dependent and Independent Variables

In any scientific experiment or statistical analysis, it's critical to identify which variable is the cause (independent) and which is the effect (dependent). The independent variable is the one you change on purpose to see what effect it has. In contrast, the dependent variable is what you measure to see if it was affected.

In our housing cost scenario, the independent variable, represented by \(x\), is the distance from the Bay Area. It's what we presume is influencing the other variable. The dependent variable, denoted by \(y\), is the house price, because it's expected to change in response to the distance from the Bay. By understanding these roles, one can discern that changing the independent variable will result in a variation in the dependent variable, which is depicted graphically as a regression line where the slope reflects the direction and rate of this change.

Rate of Change

The rate of change is a concept that goes hand-in-hand with the notion of slope. It quantifies how rapidly the dependent variable changes in relation to the independent variable. In the context of linear regression, the rate of change is consistent along the regression line, which reflects the 'linear' aspect of the relationship between the two variables.

For our example with housing costs, the constant rate of change is indicated by the constant slope of \(-\$4000\) per mile. This means that no matter how far you are from the Bay Area, the price will always decrease by the same amount for each additional mile eastward. This simplicity enables us to make straightforward predictions about house pricing at varying distances. It's a powerful piece of information for potential homeowners who want to estimate housing costs based on location, or for economists studying the impact of location on housing prices.