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Chapter 14: Problem 1

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.

Short Answer

Expert verified
Deterministic models contain variables that precisely determine an output. For example, in the equation \(F = ma\), the force (\(F\)) is entirely determined by the mass (\(m\)) and acceleration (\(a\)), with no randomness involved. Probabilistic models, on the other hand, incorporate randomness. An example is predicting the probability of rain based on temperature and humidity levels. Even if we know the temperature and the humidity, we can only predict a probability of rain, not a certain outcome.

Step by step solution

01

Understanding Deterministic Models

In a deterministic model, the output is completely determined by the input variables, meaning there is no randomness involved in the model. For instance, a mathematical equation such as \(y = 2x + 3z\) is deterministic. Where \(y\) is the dependent variable, and \(x\) and \(z\) are the independent variables.
02

Example of Deterministic Model

An example of a deterministic model might involve a physics equation like \(F = ma\), where force (\(F\)) depends entirely on mass (\(m\)) and acceleration (\(a\)). Here, \(F\) is the dependent variable, while \(m\) and \(a\) are the independent variables. Changing either \(m\) or \(a\) will change \(F\) in a predictable and consistent way.
03

Understanding Probabilistic Models

A probabilistic model, on the other hand, involves randomness. That is, even if you know the values of the independent variables, the output (dependent variable) can still vary. One common example of a probabilistic model is logistic regression. The outcome is not strictly determined, but has a probability associated with it.
04

Example of Probabilistic Model

An example of a probabilistic model can be predicting the chance of rain (\(y\)) based on the temperature (\(x\)) and humidity level (\(z\)). Even with the same temperature and humidity levels, it might not rain all the time. Thus, we can only predict the 'probability' of rain, not the actual occurrence. In this case, \(y\) is the dependent variable and \(x\) and \(z\) are the independent variables.

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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.

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.

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 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.
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