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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

Best Multiple Regression Equation For the regression equation given in Exercise 1, the P-value is 0.000 and the adjusted \({R^2}\)value is 0.925. If we were to include an additional predictor variable of neck size (in.), the P-value becomes 0.000 and the adjusted\({R^2}\)becomes 0.933. Given that the adjusted \({R^2}\)value of 0.933 is larger than 0.925, is it better to use the regression equation with the three predictor variables of length, chest size, and neck size? Explain.

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).

Enrollment (thousands)

53

28

27

36

42

Burglaries

86

57

32

131

157

True or false: If the sample data lead us to the conclusion that there is sufficient evidence to support the claim of a linear correlation between enrollment and number of burglaries, then we could also conclude that higher enrollments cause increases in numbers of burglaries.

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Using the listed old/new mpg ratings, find the best predicted new

mpg rating for a car with an old rating of 30 mpg. Is there anything to suggest that the prediction is likely to be quite good?

In Exercises 5–8, we want to consider the correlation between heights of fathers and mothers and the heights of their sons. Refer to the

StatCrunch display and answer the given questions or identify the indicated items.

The display is based on Data Set 5 “Family Heights” in Appendix B.

A son will be born to a father who is 70 in. tall and a mother who is 60 in. tall. Use the multiple regression equation to predict the height of the son. Is the result likely to be a good predicted value? Why or why not?

Exercises 13–28 use the same data sets as Exercises 13–28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Use the pizza costs and subway fares to find the best predicted

subway fare, given that the cost of a slice of pizza is $3.00. Is the best predicted subway fare likely to be implemented?
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