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Chapter 13: Problem 6

Briefly explain the difference between a deterministic and a probabilistic regression model.

Short Answer

Expert verified
Deterministic regression models predict outcomes based solely on input values, yielding consistent outcomes. Probabilistic regression models, on the other hand, introduce an element of chance, giving outcomes based on probability distributions.

Step by step solution

01

Understand deterministic regression model

A deterministic regression model, in simple terms, is a model where the output or the predicted value is purely based on the input values. There's no involved randomness or unpredicted value in this model. Basically, by having the same input, you'll always get the same output.
02

Understand probabilistic regression model

On the contrary, a probabilistic regression model introduces a degree of uncertainty. It's implied that the output isn't a single value, but a probability distribution based on the input values. In this model, for the same input, the output varies according to a certain statistical distribution.
03

Comparison

In conclusion, the main difference lies in the fact that the deterministic regression model always gives the same output for the same input, whereas the probabilistic regression model yields different outputs that follow a statistical distribution. Thus, probabilistic regression models are more used when the data contains inherent randomness.

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Most popular questions from this chapter

The following table containing data on the aerobic exercise levels (running distance in miles) and blood sugar levels for 12 different days for a diabetic is reproduced from Exercise \(13.27 .\) $$ \begin{array}{l|rrrrrrrrrrr} \hline \text { Distance (miles) } & 2 & 2 & 2.5 & 2.5 & 3 & 3 & 3.5 & 3.5 & 4 & 4 & 4.5 & 4.5 \\ \hline \text { Blood sugar }(\mathrm{mg} / \mathrm{dL}) & 136 & 146 & 131 & 125 & 120 & 116 & 104 & 95 & 85 & 94 & 83 & 75 \\ \hline \end{array} $$ a. Find the standard deviation of errors. b. Compute the coefficient of determination. What percentage of the variation in blood sugar level is explained by the least squares regression of blood sugar level on the distance run? What percentage of this variation is not explained?

Consider the formulas for calculating a prediction interval for a new (specific) value of \(y\). For each of the changes mentioned in parts a through \(\mathrm{c}\) below, state the effect on the width of the confidence interval (increase, decrease, or no change) and why it happens. Note that besides the change mentioned in each part, everything else such as the values of \(a, b, \bar{x}, s_{e}\), and \(S S_{x x}\) remains unchanged. a. The confidence level is increased. b. The sample size is increased. c. The value of \(x_{0}\) is moved farther away from the value of \(\bar{x}\). d. What will the value of the margin of error be if \(x_{0}\) equals \(\bar{x}\) ?

Consider the data given in the following table. $$ \begin{array}{l|llllll} \hline x & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline y & 12 & 15 & 19 & 21 & 25 & 30 \\ \hline \end{array} $$ a. Find the least squares regression line and the linear correlation coefficient \(r\). b. Suppose that each value of \(y\) given in the table is increased by 5 and the \(x\) values remain unchanged. Would you expect \(r\) to increase, decrease, or remain the same? How do you expect the least squares regression line to change? c. Increase each value of \(y\) given in the table by 5 and find the new least squares regression line and the correlation coefficient \(r\). Do these results agree with your expectation in part b?

Explain the difference between a simple and a multiple regression model.

The following table gives the total daily U.S. crude oil imports (in millions of barrels, rounded to. the nearest million) for the years 1995 to 2008. $$ \begin{array}{l|rrrrrrr} \hline \text { Year } & 1995 & 1996 & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline \begin{array}{l} \text { Daily U.S. crude oil imports } \\ \text { (millions of barrels) } \end{array} & 7.23 & 7.51 & 8.23 & 8.71 & 8.73 & 9.07 & 9.33 \\ \hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \begin{array}{l} \text { Daily U.S. crude oil imports } \\ \text { (millions of barrels) } \end{array} & 9.14 & 9.66 & 10.08 & 10.13 & 10.12 & 10.03 & 9.78 \\ \hline \end{array} $$ a. Assign a value of 0 to 1995,1 to 1996,2 to 1997 , and so on. Call this new variable Time. Make a new table with the variables Time and Daily U.S. Crude Oil Imports. b. With time as an independent variable and the daily U.S. crude oil imports as the dependent variable, compute \(S S_{x w}, S S_{y v}\), and \(S S_{x v}\) c. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear positive relationship between time and daily U.S. crude oil imports? d. Find the least squares regression line \(\hat{y}=a+b x\). e. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{d}\). f. Compute the correlation coefficient \(r\) \(\mathrm{g}\). Predict the daily U.S. crude oil imports for \(x=20\). Comment on this prediction. h. Recalculate the correlation coefficient, ignoring the data for 2006,2007, and \(2008 .\) What happens to the value of the correlation coefficient? Create a scatter diagram of the data with time on the horizontal axis and imports on the vertical axis. Use the diagram to explain what happened to the value of \(r\).
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