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Chapter 13: Problem 7

Why is the random error term included in a regression model?

Short Answer

Expert verified
The random error term is included in a regression model to account for the variation in the dependent variable that is not explained by the independent variables. It captures the unknown or unpredictable fluctuations, and the influences of factors not explicitly included in the model. Its randomness implies that it varies and is unpredictable, ensuring more reliable predictions on average.

Step by step solution

01

Conceptual Understanding of Regression Model

A regression model is a statistical tool used to understand the relationship between variables. In its simplest form, a linear regression model can be represented as Y = a + bX + e. Here, Y is the dependent variable, X is the independent variable, a is the y-intercept, b is the slope and e is the random error term.
02

Role of the Random Error Term

The random error term in a regression model serves a few important purposes. It accounts for the variation in the dependent variable Y that is not explained by the independent variable X. It also captures the impact of variables not included in the model and represents the inaccuracy of the predictions made by the model.
03

Significance of Random Error Term

The error term is considered random because it varies and is unpredictable. The ideal assumption is that these error terms are normally distributed and have a mean of zero. It reflects that on average, the regression predictions are correct. Further, it is assumed that variance of errors remains constant across all levels of predictor variables, giving reliable results.

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Most popular questions from this chapter

Explain the difference between linear and nonlinear relationships between two variables.

An auto manufacturing company wanted to investigate how the price of one of its car models depreciates with age. The research department at the company took a sample of eight cars of this model and collected the following information on the ages (in years) and prices (in hundreds of dollars) of these cars. $$ \begin{array}{l|rrrrrrrr} \hline \text { Age } & 8 & 3 & 6 & 9 & 2 & 5 & 6 & 3 \\ \hline \text { Price } & 45 & 210 & 100 & 33 & 267 & 134 & 109 & 235 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between ages and prices of cars? b. Find the regression line with price as a dependent variable and age as an independent variable. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\). d. Plot the regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the regression line. e. Predict the price of a 7 -year-old car of this model. \(\mathbf{f}\). Estimate the price of an 18 -year-old car of this model. Comment on this finding.

Explain the difference between exact and nonexact relationships between two variables. Give one example of each.

Plot the following straight lines. Give the values of the \(y\) -intercept and slope for each of these lines and interpret them. Indicate whether each of the lines gives a positive or a negative relationship between \(x\) and \(y\) a. \(y=-60+8 x\) b. \(y=300-6 x\)

A car rental company charges $$\$ 50$$ a day and 20 cents per mile for renting a car. Let \(y\) be the total rental charges (in dollars) for a car for one day and \(x\) be the miles driven. The equation for the relationship between \(x\) and \(y\) is $$ y=50+.20 x $$ a. How much will a person pay who rents a car for one day and drives it 100 miles? b. Suppose each of 20 persons rents a car from this agency for one day and drives it 100 miles. Will each of them pay the same amount for renting a car for a day or do you expect each person to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?
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