/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Let \(x\) be the size of a house... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 1

Let \(x\) be the size of a house (in square feet) and \(y\) be the amount 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 \(=-5.0\) Houses in this community range in size from 1000 to 3000 square feet. 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. \(\mathrm{ft}\). of space? d. What is the average change in usage associated with a 1 sq. \(\mathrm{ft}\). increase in size? e. What is the average change in usage associated with a 100 sq. \(\mathrm{ft}\). increase in size? f. Would you use the model to predict mean usage for a 500 sq. \(\mathrm{ft}\). house? Why or why not?

Short Answer

Expert verified
a. The regression equation is \(y = -5 + 0.017x\). b. The points on the regression line are (1000, 12) and (2000, 29). c. The mean value of gas usage for houses with 2100 sq. ft. of space is 30.7 therms. d. The average change in usage with a 1 sq. ft. increase in size is 0.017 therms. e. The average change in usage with a 100 sq. ft. increase in size is 1.7 therms. f. The model should not be used to predict mean usage for a 500 sq. ft. house because it is outside the range of data used to create the model.

Step by step solution

01

Construction of the Regression Equation

Inject the given values of \(\alpha\) and \( \beta \) into the regression equation \( y = \alpha + \beta x \) to get \( y = -5 + 0.017x \).
02

Calculate y for given x

To find the points on the line, replace \(x\) with the provided values (1000 and 2000) in the equation \( y = -5 + 0.017x \). For \(x = 1000\), we find \(y = -5 + 0.017*1000 = 12\), and for \(x = 2000\), we have \(y = -5 + 0.017*2000 = 29\). The required points are thus (1000, 12) and (2000, 29).
03

Find Mean Gas Usage for House of Size 2100 sq. ft.

To find the mean gas usage for a house with 2100 sq. ft. of space, replace \(x\) with 2100 in the equation \( y = -5 + 0.017x \). Thus, \(y = -5 + 0.017*2100 = 30.7 \) therms.
04

Find Change in Usage per sq. ft.

The slope of the regression line (0.017) indicates the average change in gas usage per one unit change in house size, which in this case is sq. ft. Hence, the average gas usage increases by 0.017 therms for every sq. ft. increase in house size.
05

Find Change in Usage for 100 sq. ft

To find the change in usage for 100 sq. ft., multiply the slope of the equation (0.017) by 100. This gives 1.7 therms.
06

Model Applicability for a 500 sq. ft. House

The given model can't be used to predict the mean usage for a 500 sq.ft. house because the data used to create the model does not include houses of this size. Using it for this case would be an extrapolation which can lead to unreliable results.

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Key Concepts

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

Population Regression Line
The concept of a population regression line is central in understanding simple linear regression. In our example, the population regression line expresses the relationship between the size of a house and the amount of natural gas it uses. This relationship is expressed using the equation:
\[y = \alpha + \beta x\]where:
  • \(y\) is the dependent variable (natural gas usage).
  • \(x\) is the independent variable (size of the house).
  • \(\alpha\) is the intercept, showing the expected usage when house size is zero.
  • \(\beta\) is the slope, indicating how much gas usage changes with each unit change in house size.
In our scenario, the equation becomes \(y = -5 + 0.017x\). This line gives a general idea about gas usage across the population. It assumes a straight-line relationship throughout the data set.
Slope and Intercept
Understanding both the slope and intercept is crucial to interpreting a regression line. The intercept \(\alpha = -5.0\) shows the expected gas usage when the house size is 0 sq. ft. While it might not be practical to have a 0 sq. ft. house, it helps in constructing the line.
The slope, represented by \(\beta = 0.017\), tells us how much the gas usage changes for each additional square foot of house size. In simpler terms, if you increase the size of your house by 1 sq. ft., gas usage is expected to increase by 0.017 therms.
For larger increments, like 100 sq. ft., this becomes more significant. The usage would increase by 1.7 therms with an increase of 100 sq. ft., signifying a proportionate relationship maintained by the slope.
Thus, slope and intercept together help in forming a predictive model by showing both the baseline consumption and variation with size change.
Predictive Modeling
Predictive modeling with simple linear regression involves using past data to make predictions about future or unseen data. The regression line \(y = -5 + 0.017x\) serves as a model to predict future gas usage based on known house sizes.
For instance, if you want to predict usage for a house sized 2100 sq. ft., you substitute this value into the equation:
\[y = -5 + 0.017 \times 2100\]giving a predicted usage of 30.7 therms.
This prediction assumes that the relationship observed in past data holds true in the future, which makes the model practical for sizes between 1000 to 3000 sq. ft. However, applying it outside this range without additional data inputs introduces risk, emphasizing the need for the model to stay within its validated range.
Extrapolation
Extrapolation happens when we use a regression model to predict values outside the range of the observed data. This can be risky and may lead to inaccuracies since the model is not validated beyond its original data set.
In our example, the houses in the community range from 1000 to 3000 sq. ft. Trying to predict the gas usage for a 500 sq. ft. house would involve extrapolation.
Because the data does not include houses smaller than 1000 sq. ft., the model lacks the basis to ensure accurate predictions for a 500 sq. ft. house. Such predictions can lead to unreliable and potentially misleading conclusions.
Thus, it is crucial to use regression models within the range they are designed for to maintain reliability and accuracy. Always be cautious of predictions that fall outside the verified data range.

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Most popular questions from this chapter

Carbon aerosols have been identified as a contributing factor in a number of air quality problems. In a chemical analysis of diesel engine exhaust, \(x=\) mass \(\left(\mu \mathrm{g} / \mathrm{cm}^{2}\right)\) and \(y=\) elemental carbon \(\left(\mu \mathrm{g} / \mathrm{cm}^{2}\right)\) were recorded. The estimated regression line for this data set is \(\hat{y}=31+.737 x\). The accompanying table gives the observed \(x\) and \(y\) values and the corresponding standardized residuals. \(\begin{array}{lrrrrr}x & 164.2 & 156.9 & 109.8 & 111.4 & 87.0 \\\ y & 181 & 156 & 115 & 132 & 96 \\ \text { St. resid. } & 2.52 & 0.82 & 0.27 & 1.64 & 0.08 \\ x & 161.8 & 230.9 & 106.5 & 97.6 & 79.7 \\ y & 170 & 193 & 110 & 94 & 77 \\ \text { St. resid. } & 1.72 & -0.73 & 0.05 & -0.77 & -1.11 \\\ x & 118.7 & 248.8 & 102.4 & 64.2 & 89.4 \\ y & 106 & 204 & 98 & 76 & 89 \\\ \text { St. resid. } & -1.07 & -0.95 & -0.73 & -0.20 & -0.68 \\ x & 108.1 & 89.4 & 76.4 & 131.7 & 100.8 \\ y & 102 & 91 & 97 & 128 & 88 \\ \text { St. resid. } & -0.75 & -0.51 & 0.85 & 0.00 & -1.49\end{array}\) \(\begin{array}{lllll}78.9 & 387.8 & 135.0 & 82.9 & 117.9\end{array}\) a. Construct a standardized residual plot. Are there any unusually large residuals? Do you think that there are any influential observations? b. Is there any pattern in the standardized residual plot that would indicate that the simple linear regression model is not appropriate? c. Based on your plot in Part (a), do you think that it is reasonable to assume that the variance of \(y\) is the same at each \(x\) value? Explain.

A sample of \(n=61\) penguin burrows was selected, and values of both \(y=\) trail length \((\mathrm{m})\) and \(x=\) soil hardness (force required to penetrate the substrate to a depth of \(12 \mathrm{~cm}\) with a certain gauge, in \(\mathrm{kg}\) ) were determined for each one. The equation of the least- squares line was \(\hat{y}=11.607-1.4187 x\), and \(r^{2}=.386\). a. Does the relationship between soil hardness and trail length appear to be linear, with shorter trails associated with harder soil (as the article asserted)? Carry out an appropriate test of hypotheses. b. Using \(s_{e}=2.35, \bar{x}=4.5\), and \(\sum(x-\bar{x})^{2}=250\), predict trail length when soil hardness is \(6.0\) in a way that conveys information about the reliability and precision of the prediction. c. Would you use the simple linear regression model to predict trail length when hardness is \(10.0\) ? Explain your reasoning.

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