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Chapter 13: Problem 14

Briefly explain the assumptions of the population regression model.

Short Answer

Expert verified
The five assumptions of the Population Regression Model include: Linearity (linear relationship between independent and dependent variables), Independence (residuals are independent), Homoscedasticity (error term has a constant variance), Normality (error term is normally distributed), and Absence of Multicollinearity (independent variables are not highly correlated).

Step by step solution

01

Assumption 1: Linearity

The first assumption of the PRM is linearity. It assumes a linear relationship between the independent and dependent variables. This means the line of best fit through the data points is a straight line rather than a curve or some sort of grouping.
02

Assumption 2: Independence

The second assumption is independence. The residuals (i.e., error terms) are independent. In particular, there is no correlation between consecutive errors in the case of time series data.
03

Assumption 3: Homoscedasticity

The third assumption is homoscedasticity. The error term has a constant variance. This means that the 'spread' of residuals remains constant (does not systematically increase or decrease) for different levels of the independent variable.
04

Assumption 4: Normality

The fourth assumption is normality. It implies that the error term is normally distributed and symmetrical around zero. This means that the distribution of the error term (its 'shape') is symmetrical and has a peak in the middle.
05

Assumption 5: No Multicollinearity

The fifth assumption is the absence of multicollinearity. This essentially means that the independent variables are not highly correlated with each other. A high correlation between two or more independent variables can result in the inability to separate the distinct effect of each variable on the dependent variable.

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Most popular questions from this chapter

The following table gives information on the limited tread warranties (in thousands of miles) and the prices of 12 randomly selected tires at a national tire retailer as of July 2009. $$ \begin{array}{l|llllllllllll} \hline \text { Warranty (thousands of miles) } & 60 & 70 & 75 & 50 & 80 & 55 & 65 & 65 & 70 & 65 & 60 & 65 \\ \hline \text { Price per tire }(\$) & 95 & 70 & 94 & 90 & 121 & 70 & 84 & 80 & 92 & 79 & 66 & 95 \\ \hline \end{array} $$ a. Taking warranty length as an independent variable and price per tire as a dependent variable, compute \(\mathrm{SS}_{x x}, \mathrm{SS}_{y y}\), and \(\mathrm{SS}_{x y}\) b. Find the regression of price per tire on warranty length. c. Briefly explain the meaning of the values of \(a\) and \(b\) calculated in part \(\mathrm{b}\). d. Calculate \(r\) and \(r^{2}\) and explain what they mean. e. Plot the scatter diagram and the regression line. f. Predict the price of a tire with a warranty length of 73,000 miles. g. Compute the standard deviation of errors. h. Construct a \(95 \%\) confidence interval for \(B\). i. Test at the \(5 \%\) significance level if \(B\) is positive. j. Using \(\alpha=.025\), can you conclude that the linear correlation coefficient is positive?

Bob's Pest Removal Service specializes in removing wild creatures (skunks, bats, reptiles, etc.) from private homes. He charges \(\$ 70\) to go to a house plus \(\$ 20\) per hour for his services. Let \(y\) be the total amount (in dollars) paid by a household using Bob's services and \(x\) the number of hours Bob spends capturing and removing the animal(s). The equation for the relationship between \(x\) and \(y\) is $$ y=70+20 x $$ a. Bob spent 3 hours removing a coyote from under Alice's house. How much will he be paid? b. Suppose nine persons called Bob for assistance during a week. Strangely enough, each of these jobs required exactly 3 hours. Will each of these clients pay Bob the same amount, or do you expect each one to pay a different amount? Explain. c. Is the relationship between \(x\) and \(y\) exact or nonexact?

For a sample data set on two variables, the value of the linear correlation coefficient is zero. Does this mean that these variables are not related? Explain

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

Two variables \(x\) and \(y\) have a negative linear relationship. Explain what happens to the value of \(y\) when \(x\) increases.
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