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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

What does a linear correlation coefficient tell about the relationship between two variables? Within what range can a correlation coefficient assume a value?

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.

The following table gives information on the amount of sugar (in grams) and the calorie count in one serving of a sample of 13 varieties of Kellogg's cereal. $$ \begin{array}{l|rrrrrrrrrrrrr} \hline \text { Sugar (grams) } & 4 & 15 & 12 & 11 & 8 & 6 & 7 & 2 & 7 & 14 & 20 & 3 & 13 \\ \hline \text { Calories } & 120 & 200 & 140 & 110 & 120 & 80 & 190 & 100 & 120 & 190 & 190 & 110 & 120 \\ \hline \end{array} $$ a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between the amount of sugar and the number of calories per serving? b. Find the predictive regression equation of the number of calories on the amount of sugar. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the predictive regression line on the scatter diagram of part a and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Calculate the predicted calorie count for a cereal with 16 grams of sugar per serving. f. Estimate the calorie count for a cereal with 52 grams of sugar per serving. Comment on this finding.

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

The following table contains information on the amount of time that each of 12 students spends each day (on average) on social networks (Facebook, Twitter, etc.) and the Internet for social or entertainment purposes and his or her grade point average (GPA). $$ \begin{array}{l|rrrrrrrrrrrr} \hline \text { Time (hours per day) } & 4.4 & 6.2 & 4.2 & 1.6 & 4.7 & 5.4 & 1.3 & 2.1 & 6.1 & 3.3 & 4.4 & 3.5 \\ \hline \text { GPA } & 3.22 & 2.21 & 3.13 & 3.69 & 2.7 & 2.2 & 3.69 & 3.25 & 2.66 & 2.89 & 2.71 & 3.36 \\ \hline \end{array} $$a. Construct a scatter diagram for these data. Does the scatter diagram exhibit a linear relationship between grade point average and time spent on social networks and the Internet? b. Find the predictive regression line of GPA on time. c. Give a brief interpretation of the values of \(a\) and \(b\) calculated in part b. d. Plot the predictive regression line on the scatter diagram of part a, and show the errors by drawing vertical lines between scatter points and the predictive regression line. e. Calculate the predicted GPA for a college student who spends \(3.8\) hours per day on social networks and the Internet for social or entertainment purposes. f. Calculate the predicted GPA for a college student who spends 16 hours per day on social networks and the Internet for social or entertainment purposes. Comment on this finding.
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