/*! 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 23 Data on \(x=\) size of a house (... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 23

Data on \(x=\) size of a house (in square feet) and \(y=\) amount of natural gas used (therms) during a specified period were used to fit the least squares regression line. The slope was 0.017 and the intercept was \(-5.0 .\) Houses in this data set ranged in size from 1,000 to 3,000 square feet. a. What is the equation of the least squares regression line? b. What would you predict for gas usage for a 2,100 sq. ft. house? c. What is the approximate change in gas usage associated with a 1 sq. ft. increase in size? d. Would you use the least squares regression line to predict gas usage for a 500 sq. ft. house? Why or why not?

Short Answer

Expert verified
a. The equation of the least squares regression line is \(y = 0.017x - 5\).\nb. Predicted gas usage for a 2,100 sq. ft. house can be found by substituting \(x = 2100\) in the equation.\nc. The change in gas usage for a 1 sq. ft. increase in size is 0.017 therms.\nd. The least squares regression line should not be used to predict gas usage for a 500 sq. ft. house because it's an extrapolation of the current model fitted within a range from 1,000 to 3,000 sq. ft.

Step by step solution

01

Formulate the regression line equation

The equation of the least squares regression line is of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In this case, the slope \(m\) is given as 0.017 and the intercept \(c\) as -5.0. So, the equation becomes \(y = 0.017x - 5\).
02

Predict gas usage for a specific house size

To predict the gas usage for a 2,100 sq. ft. house, substitute \(x = 2100\) into the regression line equation, we get \(y = 0.017 * 2100 - 5\). Calculating the expression on the right will give the predicted gas usage.
03

Interpret the meaning of the slope

The slope of the regression line (0.017) indicates the change in gas usage for a unit (1 sq. ft.) increase in size of the house. So, for a 1 sq. ft. increase in size, the gas usage would increase by 0.017 therms.
04

Determine the limitations of the model

No, the least squares regression line shouldn't be used to predict gas usage for a 500 sq. ft. house. The reason is that it's outside the given range of the data set (1,000 to 3,000 square feet) used to fit the regression line. Using it for predictions outside this range leads to extrapolation, which might not be accurate as the relationship might not hold outside the range of the given data.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Slope Interpretation
The slope of a linear regression line provides valuable insights into how two variables are related. In our case, the slope is 0.017. This means for every additional square foot of house size, the predicted increase in gas usage is 0.017 therms.

In more straightforward terms, if you increase the size of your house by one square foot, you can expect to use an extra 0.017 therms of natural gas. This slope tells us the rate at which gas usage changes relative to the size of the house.

Thus, the slope is a measure of sensitivity; it quantifies how the dependent variable (gas usage) reacts to changes in the independent variable (house size). Understanding this helps us make predictions and find trends between the variables.
Extrapolation Limitations
Extrapolation involves making predictions outside the range of the data used to fit the regression model. In the provided data set, the houses ranged from 1,000 to 3,000 square feet in size.

Predicting gas usage for a house smaller than 1,000 square feet, like 500 square feet, is extrapolation. This practice can be risky because the relationship derived from the data may not extend beyond these bounds.
  • The model might not accommodate unusual factors outside the data range.
  • The trend observed might alter, making predictions unreliable.
It is always safer to use regression models within the range of your data set when making predictions, ensuring the conclusions drawn are based on the observed trends.
Regression Analysis Prediction
Predicting outcomes using a regression line is a primary application of regression analysis. To estimate the gas usage for a house that is 2,100 square feet, you plug the size into the regression equation: \[ y = 0.017x - 5 \]This requires substituting \( x = 2100 \): \[ y = 0.017 \times 2100 - 5 \]Carrying out this calculation gives the predicted gas usage. This step illustrates the power of regression analysis, helping to make informed forecasts.

Moreover, regression analysis prediction enables decision-making in uncertain scenarios. By understanding how variables relate to each other, we can use the regression line for accurate predictions within the data鈥檚 range. Remember, predictions should be made within the data's known range to avoid the risks associated with extrapolation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The paper "Digit Ratio as an Indicator of Numeracy Relative to Literacy in 7-Year-Old British Schoolchildren" (British Journal of Psychology [2008]: \(75-85\) ) investigated a possible relationship between \(x=\) digit ratio (the ratio of the length of the second finger to the length of the fourth finger) and \(y=\) difference between numeracy score and literacy score on a national assessment. (The digit ratio is thought to be inversely related to the level of prenatal testosterone exposure.) The authors concluded that children with smaller digit ratios tended to have larger differences in test scores, meaning that they tended to have a higher numeracy score than literacy score. This conclusion was based on a correlation coefficient of \(r=-0.22 .\) Does the value of the correlation coefficient indicate that there is a strong linear relationship? Explain why or why not.

An auction house released a list of 25 recently sold paintings. The artist's name and the sale price of each painting appear on the list. Would the correlation coefficient be an appropriate way to summarize the relationship between artist and sale price? Why or why not?

The relationship between hospital patient-to-nurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following response variables, indicate whether you expect the slope of the least squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of quality of patient care (higher values indicate higher quality)

For a given data set, the sum of squared deviations from the line \(y=40+6 x\) is \(529.5 .\) For this same data set, which of the following could be the sum of squared deviations from the least squares regression line? Explain your choice. i. 308.6 ii. 529.6 iii. 617.4

For each of the following pairs of variables, indicate whether you would expect a positive correlation, a negative correlation, or a correlation close to \(0 .\) Explain your choice. a. Interest rate and number of loan applications b. Height and \(\mathrm{IQ}\) c. Height and shoe size d. Minimum daily temperature and cooling cost
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.