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Chapter 13: Problem 1

Let \(x\) be the size of a house (sq \(\mathrm{ft}\) ) and \(y\) be the amoun of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq ft of space? d. What is the average change in usage associated with a 1 -sq-ft increase in size? e. What is the average change in usage associated with a 100 -sq-ft increase in size? f. Would you use the model to predict mean usage for a 500 -sq-ft house? Why or why not? (Note: There are no small houses in the community in which this model is valid.)

Short Answer

Expert verified
The equation of the population regression line is \(y=-5.0+0.017x\). Graph it to see the linear relationship. The mean value of gas usage for a house with 2100 sq ft of space is 30.7 therms. Every 1-sq-ft increase in size results in an average 0.017-therm increase in gas usage, and a 100-sq-ft increase means 1.7 therms more. However, using the model to predict usage for a 500-sq-ft house may not be accurate because such houses aren't common in the community that the model is based on.

Step by step solution

01

Derive the Regression Equation

The equation of a regression line is \(y = \alpha +\beta x\). Given that the y-intercept \(\alpha=-5.0\) and the slope \(\beta=0.017\), the equation of the population regression line is \(y=-5.0+0.017x\).
02

Plot the Regression Line

Let's calculate the values for \(y\) corresponding to \(x=1000\) and \(x=2000\). Plugging \(x=1000\) into the equation: \(y=-5.0+0.017*1000=12\) and for \(x=2000\), \(y=-5.0+0.017*2000=29\). So the points are \((1000,12)\) and \((2000,29)\). Plot these points and draw a line through them to graph the population regression line. Always remember, the slope of the line is the coefficient of \(x\) in the regression equation.
03

Predict Mean Gas Usage for 2100 sq ft House Size

Substitute \(x=2100\) into the regression equation to predict the mean gas usage for a house of 2100 sq ft. So, \(y=-5.0+0.017*2100=30.7\) therms.
04

Understand the Meaning of Slope

The slope of the regression model, \(\beta=0.017\), represents the average change in gas usage associated with a 1-sq-ft increase in size of the house. So, for every additional square foot of house size, the gas usage increases by 0.017 therms.
05

Calculate the Change in Usage for a 100-sq-ft Increase in Size

For a 100-sq-ft increase, the change in usage would be \(0.017*100=1.7\) therms. This can be computed by multiplying the slope with the change in the house size.
06

Discuss Predictions for a 500 -sq-ft House

Even though the model allows for it, it might not be accurate to predict gas usage for a 500-sq-ft house using this model. This is because the model is created based on data from larger houses present in the community.

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Key Concepts

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

Population Regression Line
In linear regression, the population regression line is a fundamental concept. It provides a mathematical representation of the relationship between two variables, such as the size of a house and the amount of natural gas it uses. For our specific problem, the regression line is given by the equation \(y = -5.0 + 0.017x\). Here, \(x\) represents the size of the house in square feet, and \(y\) indicates the gas usage in therms.
The equation consists of two parts: the intercept \(-5.0\) and the slope \(0.017\). The intercept can be thought of as the expected gas usage for a house of size zero, which in practical terms may not have realistic significance but serves a mathematical purpose. This line essentially serves as the "best fit" through a scatterplot of our data points.
Slope Interpretation
Interpreting the slope is crucial in understanding what our regression model is telling us. In the equation \(y = -5.0 + 0.017x\), the slope is \(0.017\). This number indicates the average increase in gas usage for each additional square foot of house size.
Simply put, for every square foot added to a house's size, the model predicts an increase of \(0.017\) therms in gas usage.
  • If the slope was larger, we would expect a more significant change in gas usage.
  • If it was smaller, the change would be less pronounced.
Understanding the slope allows for informed predictions about how modifications to a variable affect the outcome, which in this example is the gas usage.
Gas Usage Prediction
Predicting gas usage is among the most beneficial applications of a regression model. Using the regression equation \(y = -5.0 + 0.017x\), we can estimate the mean gas usage for houses of various sizes.
For instance, if you want to determine the expected gas usage for a 2100 sq ft house, you simply plug \(x = 2100\) into the equation, resulting in \(y = -5.0 + 0.017 \times 2100 = 30.7\). Thus, a 2100 sq ft house is expected to use approximately \(30.7\) therms of gas on average.
However, be cautious when using this model to predict for sizes not represented in the original data, such as very small houses, as this can lead to inaccurate predictions.
Regression Equation
The regression equation is at the heart of linear regression analysis. It is typically expressed as \(y = \alpha + \beta x\), where \(\alpha\) is the y-intercept and \(\beta\) is the slope.
In our problem, the equation is \(y = -5.0 + 0.017x\), which was formed by analyzing the data of house sizes and corresponding gas usage. The process involves calculating the slope and intercept that minimize the distance between the data points and the regression line, termed as "least squares" fitting.
  • The intercept \(-5.0\) indicates the estimated gas usage when no space size is accounted for.
  • The slope \(0.017\) signifies the change in gas usage for each square foot increase in house size.
This equation enables users to predict the mean values of one variable based on the changes in another.

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