/*! 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 12 The article "The Value and the L... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 12

The article "The Value and the Limitations of High-Speed Turbo-Exhausters for the Removal of Tar-Fog from Carburetted Water-Gas" (Society of Chemical Industry Joumal \([1946]: 166-168\) ) presented data on \(y=\) tar content (grains/100 \(\mathrm{ft}^{3}\) ) of a gas stream as a function of \(x_{1}=\) rotor speed \((\mathrm{rev} / \mathrm{min})\) and \(x_{2}=\) gas inlet temperature ( F). A regression model using \(x_{1}, x_{2}, x_{3}=x_{2}^{2}\) and \(x_{4}=x_{1} x_{2}\) was suggested: $$ \begin{aligned} \text { mean } y \text { value }=& 86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} \\ &+.001 x_{4} \end{aligned} $$ a. According to this model, what is the mean \(y\) value if \(x_{1}=3200\) and \(x_{2}=57 ?\) b. For this particular model, does it make sense to interpret the value of any individual \(\beta_{i}\left(\beta_{1}, \beta_{2}, \beta_{3}\right.\), or \(\left.\beta_{4}\right)\) in the way we have previously suggested? Explain.

Short Answer

Expert verified
a. Substitute given values into the model equation to calculate the mean 'y' value. b. The interpretation of each individual \(\beta_{i}\) is not straightforward or necessarily meaningful in this model due to the interaction term and squared term included in the model.

Step by step solution

01

Substitution for mean y-value

Substitute \(x_{1}=3200\) and \(x_{2}=57\) into the model along with \(x_{3}=x_{2}^{2}=57^{2}\) and \(x_{4}=x_{1} x_{2}=3200 \times 57\).
02

Calculation

Calculate the mean y-value using these substitutions. The operation of calculation involves simple algebraic multiplication and addition.
03

Interpretation of co-efficients

Discuss the meaning of each coefficient in the regression model. It should be noted that interpretation will depend on the context and nature of the data. The \(\beta_{i}\) coefficients represent the effects of the corresponding 'x' variables on the mean 'y' value, taking interaction and non-linearity into account, and their interpretation may not be straightforward.

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.

Statistical Regression
Statistical regression is a powerful tool used to analyze the relationship between a dependent variable and one or more independent variables. It forms the backbone of predictive modeling. By establishing a mathematical equation, regression allows us to summarize and study relationships among variables.

For example, consider a regression model designed to predict the tar content in a gas stream based on rotor speed and gas inlet temperature. It incorporates not just these variables but also their square and interaction terms to account for more complex relationships. Such a model can help researchers and engineers to understand and optimize processes like tar-fog removal.

In essence, regression analysis can reveal significant trends and patterns, forming a basis for informed decision-making and future predictions.
Modeling Relationships
Modeling relationships is at the core of regression analysis. By analyzing how variables interact with one another, you can create a model that captures the essence of these relationships.

In our example, the relationship between tar content, rotor speed, and gas inlet temperature, along with their interaction and squared terms, is encapsulated in a regression equation. This equation not only highlights direct effects but also the transformative effect of temperature (squared term) and the combined effect of speed and temperature (interaction term).
  • This deepens our understanding of the problem,
  • Allows us to predict outcomes under different conditions,
  • Helps in identifying the most influential factors.
Coefficient Interpretation
Interpreting coefficients in regression analysis involves understanding the impact of each predictor variable on the dependent variable. As each coefficient represents the expected change in the dependent variable per one-unit change in the predictor, it's crucial to comprehend them within the context of the data.

In the suggested model, coefficients indicate how rotor speed and temperature separately and jointly influence tar content. However, it's important to note that the interaction coefficient (尾4) and the squared term coefficient (尾3) indicate more complex nonlinear effects, which may not lend themselves to simple interpretations. When looking at such regression outputs, assess the influence of each variable carefully, taking into account the possible interdependencies reflected in the model.
Predictive Analytics
Predictive analytics uses statistical techniques, such as regression analysis, to make predictions about future outcomes based on historical data. It can be highly sophisticated, incorporating complicated mathematical models to predict trends and behaviors.

In practical scenarios, like predicting tar content from rotor speed and gas temperature, predictive analytics helps in estimation and foresight, leading to better control over industrial processes. By fitting the regression model to existing data, we can forecast the mean tar content under varying conditions, aiding in strategic planning and risk management.

As powerful as it is, predictive analytics also requires caution; the predictions are only as reliable as the assumptions of the model and the quality of the data fed into it.

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 article "The Undrained Strength of Some Thawed Permafrost Soils" (Canadian Geotechnical Journal \([1979]: 420-427\) ) contained the accompanying data (see page 658) on \(y=\) shear strength of sandy soil \((\mathrm{kPa})\), \(x_{1}=\) depth \((\mathrm{m})\), and \(x_{2}=\) water content \((\%)\). The predicted values and residuals were computed using the estimated regression equation $$ \begin{aligned} \hat{y}=&-151.36-16.22 x_{1}+13.48 x_{2}+.094 x_{3}-.253 x_{4} \\ &+.492 x_{5} \\ \text { where } x_{3} &=x_{1}^{2}, x_{4}=x_{2}^{2} \text { , and } x_{5}=x_{1} x_{2} \end{aligned} $$ a. Use the given information to compute SSResid, SSTo, and SSRegr. b. Calculate \(R^{2}\) for this regression model. How would you interpret this value? c. Use the value of \(R^{2}\) from Part (b) and a \(.05\) level of significance to conduct the appropriate model utility test.

The article "Readability of Liquid Crystal Displays: A Response Surface" (Human Factors \([1983]: 185-190)\) used the estimated regression equation to describe the relationship between \(y=\) error percentage for subjects reading a four-digit liquid crystal display and the independent variables \(x_{1}=\) level of backlight, \(x_{2}=\) character subtense, \(x_{3}=\) viewing angle, and \(x_{4}=\) level of ambient light. From a table given in the article, SSRegr \(=19.2\), SSResid \(=\) \(20.0\), and \(n=30\). a. Does the estimated regression equation specify a useful relationship between \(y\) and the independent variables? Use the model utility test with a \(.05\) significance level. b. Calculate \(R^{2}\) and \(s_{e}\) for this model. Interpret these values. c. Do you think that the estimated regression equation would provide reasonably accurate predictions of error rate? Explain.

Explain the difference between a deterministic and a probabilistic model. Give an example of a dependent variable \(y\) and two or more independent variables that might be related to \(y\) deterministically. Give an example of a dependent variable \(y\) and two or more independent variables that might be related to \(y\) in a probabilistic fashion.

The article "The Caseload Controversy and the Study of Criminal Courts" (Journal of Criminal Law and Criminology [1979]: \(89-101\) ) used a multiple regression analysis to help assess the impact of judicial caseload on the processing of criminal court cases. Data were collected in the Chicago criminal courts on the following variables: \(\begin{aligned} y &=\text { number of indictments } \\ x_{1} &=\text { number of cases on the docket } \end{aligned}\) \(x_{2}=\) number of cases pending in criminal court trial system The estimated regression equation (based on \(n=367\) observations) was $$ \hat{y}=28-.05 x_{1}-.003 x_{2}+.00002 x_{3} $$ where \(x_{3}=x_{1} x_{2}\). a. The reported value of \(R^{2}\) was \(.16 .\) Conduct the model utility test. Use a \(.05\) significance level. b. Given the results of the test in Part (a), does it surprise you that the \(R^{2}\) value is so low? Can you think of a possible explanation for this? c. How does adjusted \(R^{2}\) compare to \(R^{2}\) ?

The article "The Influence of Temperature and Sunshine on the Alpha-Acid Contents of Hops" (Agricul. \mathrm{\\{} t u r a l ~ M e t e o r o l o g y ~ [ 1 9 7 4 ] : ~ \(375-382\) ) used a multiple regression model to relate \(y=\) yield of hops to \(x_{1}=\) mean temperature \(\left({ }^{\circ} \mathrm{C}\right)\) between date of coming into hop and date of picking and \(x_{2}=\) mean percentage of sunshine during the same period. The model equation proposed is $$ y=415.11-6060 x_{1}-4.50 x_{2}+e $$ a. Suppose that this equation does indeed describe the true relationship. What mean yield corresponds to a temperature of 20 and a sunshine percentage of 40 ? b. What is the mean yield when the mean temperature and percentage of sunshine are \(18.9\) and 43 , respectively? c. Interpret the values of the population regression coefficients.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.