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Chapter 10: Q3BSC (page 468)

a. What is a residual?

b. In what sense is the regression line the straight line that “best” fits the points in a scatterplot?

Short Answer

Expert verified

a. A residual represents the difference between the observed and predicted value of the dependent variable (y).

b. The regression line is the best fit as it describes the linear association between variables such that it has the minimum possible error.

Step by step solution

01

Define the term residual

a.

A residual is the difference between two values obtained for a value of the independent variable;observed and predicted value of the dependent variable y at x. Mathematically it is computed as,

\(\begin{array}{c}Residual = observed\;y - predicted\;y\\ = y - \hat y\end{array}\)

02

Discuss that the regression line is best fit.

b.

A regression line is obtained as the straight line which describes the relationship between a set of variables such that it has the lowest possible error.

The property of the regression line, that thesum of squares of the residuals is the lowest possible sum, makes it the best fit straight line.

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Most popular questions from this chapter

Interpreting r For the same two variables described in Exercise 1, if we find that r = 0, does that indicate that there is no association between those two variables?

The following exercises are based on the following sample data consisting of numbers of enrolled students (in thousands) and numbers of burglaries for randomly selected large colleges in a recent year (based on data from the New York Times).

The sample data result in a linear correlation coefficient of r= 0.499 and the regression equation\(\hat y = 3.83 + 2.39x\). What is the best predicted number of burglaries, given an enrollment of 50 (thousand), and how was it found?

Testing for a Linear Correlation. In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Weighing Seals with a Camera Listed below are the overhead widths (cm) of seals

measured from photographs and the weights (kg) of the seals (based on “Mass Estimation of Weddell Seals Using Techniques of Photogrammetry,” by R. Garrott of Montana State University). The purpose of the study was to determine if weights of seals could be determined from overhead photographs. Is there sufficient evidence to conclude that there is a linear correlation between overhead widths of seals from photographs and the weights of the seals?

Overhead Width

7.2

7.4

9.8

9.4

8.8

8.4

Weight

116

154

245

202

200

191

Finding Critical r Values Table A-6 lists critical values of r for selected values of n and a. More generally, critical r values can be found by using the formula

\(r = \frac{t}{{\sqrt {{t^2} + n - 2} }}\)

where the t value is found from the table of critical t values (Table A-3) assuming a two-tailed case with n - 2 degrees of freedom. Use the formula for r given here and in Table A-3 (with n - 2 degrees of freedom) to find the critical r values corresponding to \({H_1}:\rho \ne 0\), \(\alpha \)= 0.02, and n = 27.

Cigarette Nicotine and Carbon Monoxide Refer to the table of data given in Exercise 1 and use the amounts of nicotine and carbon monoxide (CO).

a. Construct a scatterplot using nicotine for the xscale, or horizontal axis. What does the scatterplot suggest about a linear correlation between amounts of nicotine and carbon monoxide?

b. Find the value of the linear correlation coefficient and determine whether there is sufficient evidence to support a claim of a linear correlation between amounts of nicotine and carbon monoxide.

c. Letting yrepresent the amount of carbon monoxide and letting xrepresent the amount of nicotine, find the regression equation.

d. The Raleigh brand king size cigarette is not included in the table, and it has 1.3 mg of nicotine. What is the best predicted amount of carbon monoxide?

Tar

25

27

20

24

20

20

21

24

CO

18

16

16

16

16

16

14

17

Nicotine

1.5

1.7

1.1

1.6

1.1

1.0

1.2

1.4
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