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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
In deterministic models, the output is determined entirely by the input with no uncertainty, such as in the calculation of the area of a rectangle with given lengths and width. In contrast, probabilistic models involve uncertainty, where known input variables only influence the likelihood of certain outcomes, like in weather forecasting with multiple influencing factors.

Step by step solution

01

Understanding of Deterministic and Probabilistic Models

In a deterministic model, the output is completely determined by the input, meaning there is no randomness involved. The result is always the same for the same input. In contrast, in a probabilistic model, which involves probability, the output may vary despite the same input due to randomness.
02

Deterministic Model Example

An example that might model deterministically is the calculation of the area of a rectangle. Here, the dependent variable \(y\) (the area) is deterministically related to the length \(x_1\) and the width \(x_2\). For given values of length and width, the area is given by \(y = x_1 \times x_2\). There is no uncertainty involved, as the calculation will always yield the same result for the same length and width.
03

Probabilistic Model Example

An example that illustrates a probabilistic relationship is weather forecasting. Here, the dependent variable \(y\) (whether it rains or not tomorrow) is related to several independent variables such as today's temperature \(x_1\), current humidity level \(x_2\), wind speed \(x_3\), etc. Even though we might know these values, they do not determine with certainty whether it will rain tomorrow or not, instead they influence the likelihood of rain. Hence, this is a probabilistic relationship.

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Most popular questions from this chapter

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.

If we knew the width and height of cylindrical tin cans of food, could we predict the volume of these cans with precision and accuracy? a. Give the equation that would allow us to make such predictions. b. Is the relationship between volume and its predictors, height and width, a linear one? c. Should we use an additive multiple regression model to predict a volume of a can from its height and width? Explain. d. If you were to take logarithms of each side of the equation in Part (a), would the relationship be linear?

Obtain as much information as you can about the \(P\) -value for the \(F\) test for model utility in each of the following situations: a. \(k=2, n=21\), calculated \(F=2.47\) b. \(k=8, n=25\), calculated \(F=5.98\) c. \(k=5, n=26\), calculated \(F=3.00\) d. The full quadratic model based on \(x_{1}\) and \(x_{2}\) is fit, \(n=20\), and calculated \(F=8.25\). \mathrm{\\{} e . ~ \(k=5, n=100\), calculated \(F=2.33\)

A number of investigations have focused on the problem of assessing loads that can be manually handled in a safe manner. The article "Anthropometric, Muscle Strength, and Spinal Mobility Characteristics as Predictors in the Rating of Acceptable Loads in Parcel Sorting" (Ergonomics [1992]: \(1033-1044\) ) proposed using a regression model to relate the dependent variable \(y=\) individual's rating of acceptable load \((\mathrm{kg})\) to \(k=3\) independent (predictor) variables: \(x_{1}=\) extent of left lateral bending \((\mathrm{cm})\) $$ \begin{aligned} &x_{2}=\text { dynamic hand grip endurance (sec) } \\ &x_{3}=\text { trunk extension ratio }(\mathrm{N} / \mathrm{kg}) \end{aligned} $$ Suppose that the model equation is $$ y=30+.90 x_{1}+.08 x_{2}-4.50 x_{3}+e $$ and that \(\sigma=5\). a. What is the population regression function? b. What are the values of the population regression \(\underline{\mathrm{co}}\) efficients? c. Interpret the value of \(\beta_{1}\). d. Interpret the value of \(\beta_{3}\). e. What is the mean value of rating of acceptable load when extent of left lateral bending is \(25 \mathrm{~cm}\), dynamic hand grip endurance is \(200 \mathrm{sec}\), and trunk extension ratio is \(10 \mathrm{~N} / \mathrm{kg}\) ? f. If repeated observations on rating are made on different individuals, all of whom have the values of \(x_{1}, x_{2}\), and \(x_{3}\) specified in Part (e), in the long run approximately what percentage of ratings will be between \(13.5 \mathrm{~kg}\) and \(33.5 \mathrm{~kg} ?\)

Consider the dependent variable \(y=\) fuel efficiency of a car (mpg). a. Suppose that you want to incorporate size class of car, with four categories (subcompact, compact, midsize, and large), into a regression model that also includes \(x_{1}=\) age of car and \(x_{2}=\) engine size. Define the necessary dummy variables, and write out the complete model equation. b. Suppose that you want to incorporate interaction between age and size class. What additional predictors would be needed to accomplish this?
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