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Chapter 11: Q11-14E (page 653)

The following table is similar to Table 11.2. It is used for making the preliminary computations for finding the least-squares line for the given pairs of x- and y-values.

a. Complete the table.

b. FindSSxy.

c. Find SSxx.

d. Findβ1^.

e. FindX¯andY¯.

f. Findβ0^.

g. Find the least-squares line.

Short Answer

Expert verified

Answer

  1. Fig.1 Table
  2. -26.29
  3. 33.71
  4. -0.77
  5. 3.43, 4.43
  6. 1.78
  7. -16.7

Step by step solution

01

Introduction

The Least Squares Regression Line is the line that has the shortest vertical distance between the data points and the regression line. It's dubbed a "least squares" method because the optimum fit line reduces variance.

02

Complete the table

03

Find  SSxy

SSxy=∑xiyi−∑xi∑yin=80−24x317=80−7447=560−7447=−1847=−26.29

04

Find  SSxx

SSxy=∑xi2−(∑xi)2n=116−(24)27=116−5767=812−5767=2367= 33.71

05

Find β1^

β1^=SSxySSxx=−26.2933.71=−0.77

06

Find and X ¯and Y¯

x¯=∑xin=247= 3.43

y¯=∑yin=317= 4.43

07

Find β0^

β0^=y¯−β1^x¯= 4.43−(−0.77×3.43)= 4.43 + 2.6411= 1.78

08

Find the least-squares line

y^=β0^+β1^xi=1.78 +(−0.77×24)= 1.78 +(−18.48)=1.78−18.48=−16.7

Hence, the least-squares line is -16.7.

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Most popular questions from this chapter

Software millionaires and birthdays. Refer to Exercise 11.23 (p. 655) and the study of software millionaires and their birthdays. The data are reproduced on p. 663.

a. Find SSE s2and s for the simple linear regression model relating the number (y) of software millionaire birthdays in a decade to the total number (x) of U.S. births.

b. Find SSE s2and s for the simple linear regression model relating the number (y) of software millionaire birthdays in a decade to the number (x) of CEO birthdays.

c. Which of the two models' fit will have smaller errors of prediction? Why?

Decade

Total U.S. Births (millions)

Number of Software Millionaire Birthdays

Number of CEO Birthdays (in a random sample of 70 companies from the Fortune 500 list)

1920

28.582

3

2

1930

24.374

1

2

1940

31.666

10

23

1950

40.530

14

38

1960

38.808

7

9

1970

33.309

4

0

Repair and replacement costs of water pipes. Refer to the IHS Journal of Hydraulic Engineering (September 2012) study of water pipes, Exercise 11.21 (p. 655). Refer, again, to the Minitab simple linear regression printout (p. 655) relating y = the ratio of repair to replacement cost of commercial pipe to x = the diameter (in millimeters) of the pipe.

a. Locate the value of s on the printout.

b. Give a practical interpretation of s.

In each case, graph the line that passes through the given points.

a. (1, 1) and (5, 5) b. (0, 3) and (3, 0)

c. (-1, 1), and (4, 2) d. (-6, -3) and (2, 6)

Refer to Exercise 11.14 (p. 653). Calculate SSE and s for the least-squares line. Use the value of s to determine where most of the errors of prediction lie.

Do nice guys really finish last in business? Refer to the Nature (March 20, 2008) study of whether “nice guys finish last” in business, Exercise 11.18 (p. 653). Recall that college students repeatedly played a version of the game “prisoner’s dilemma,” where competitors choose cooperation, defection, or costly punishment. At the conclusion of the games, the researchers recorded the average payoff and the number of times punishment was used for each player. Based on a scatter plot of the data, the simple linear regression relating average payoff (y) to punishment use (x) resulted in SSE = 1.04.

a. Assuming a sample size of n = 28, compute the estimated standard deviation of the error distribution, s.

b. Give a practical interpretation of s.
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