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Chapter 13: Problem 1

Explain the meaning of the words simple and linear as used in simple linear regression.

Short Answer

Expert verified
In simple linear regression, 'simple' refers to the use of a single independent variable to predict the dependent variable, while 'linear' indicates that the relationship between these variables can be modelled using a straight line.

Step by step solution

01

Definition of Simple

In the context of simple linear regression, 'simple' refers to the number of independent variables in the model. A simple linear regression uses only one independent variable to predict the value of a dependent variable. This setting makes the model simpler compared to multiple regression, where multiple independent variables are used.
02

Definition of Linear

The term 'linear' in simple linear regression indicates that the relationship between the independent and dependent variables can be described by a straight line. Therefore, this relationship can be mathematically expressed as \(y = mx + c\), where \(y\) is the dependent variable, \(x\) is the independent variable, \(m\) is the gradient (also called the coefficient or parameter), and \(c\) is the y-intercept.
03

Combining 'Simple' and 'Linear' in Simple Linear Regression

Hence, a simple linear regression is a statistical method used for predicting a dependent variable given a single independent variable. It assumes a linear relationship between these two variables which is described by a straight line.

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Most popular questions from this chapter

The following data give the ages (in years) of husbands and wives for six couples. $$ \begin{array}{l|cccccc} \hline \text { Husband's age } & 43 & 57 & 28 & 19 & 35 & 39 \\ \hline \text { Wife's age } & 37 & 51 & 32 & 20 & 33 & 38 \\ \hline \end{array} $$ a. Do you expect the ages of husbands and wives to be positively or negatively related? b. Plot a scatter diagram. By looking at the scatter diagram, do you expect the correlation coefficient between these two variables to be close to zero, 1, or \(-1 ?\) c. Find the correlation coefficient. Is the value of \(r\) consistent with what you expected in parts a and b? d. Using a \(5 \%\) significance level, test whether the correlation coefficient is different from zero.

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?

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.

A population data set produced the following information. $$ N=250, \quad \Sigma x=9880, \quad \Sigma y=1456, \quad \Sigma x y=85,080, \quad \Sigma x^{2}=485,870 $$ 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.
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