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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

A population data set produced the following information. $$ N=460, \quad \Sigma x=3920, \quad \Sigma y=2650, \quad \Sigma x y=26,570, \quad \sum x^{2}=48,530 $$ Find the population regression line.

The following information is obtained from a sample data set. $$ n=10, \quad \Sigma x=100, \quad \Sigma y=220, \quad \Sigma x y=3680, \quad \Sigma x^{2}=1140 $$ Find the estimated regression line.

Will you expect a positive, zero, or negative linear correlation between the variables for each of the following examples? a. SAT scores and GPAs of students b. Stress level and blood pressure of individuals c. Amount of fertilizer used and yield of corn per acre d. Ages and prices of houses e. Heights of husbands and incomes of their wives

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. \(y=100+5 x\) b. \(y=400-4 x\)

Two variables \(x\) and \(y\) have a positive linear relationship. Explain what happens to the value of \(y\) when \(x\) increases. Give one example of a positive relationship between two variables.
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