Statistics For Business & Economics solution Chapter 14 – 14 – 1 © 2017 Cengage Learning. All Rights – Studocu
Mục Lục
14 – 1
© 2017 Cengage Learning. All Rights Reserved.
Chapter 14
Simple Linear Regression
Learning Objectives
- Understand how regression analysis can be used to develop an equation that estimates
mathematically how two variables are related.
- Understand the differences between the regression model, the regression equation, and the estimated
regression equation.
- Know how to fit an estimated regression equation to a set of sample data based upon the least-
squares method.
- Be able to determine how good a fit is provided by the estimated regression equation and compute
the sample correlation coefficient from the regression analysis output.
- Understand the assumptions necessary for statistical inference and be able to test for a significant
relationship.
- Know how to develop confidence interval estimates of y given a specific value of x in both the case
of a mean value of y and an individual value of y.
- Learn how to use a residual plot to make a judgement as to the validity of the regression
assumptions.
- Know the definition of the following terms:
independent and dependent variable
simple linear regression
regression model
regression equation and estimated regression equation
scatter diagram
coefficient of determination
standard error of the estimate
confidence interval
prediction interval
residual plot
Chapter 14
14 – 2
© 2017 Cengage Learning. All Rights Reserved.
Solutions:
1 a.
b. There appears to be a positive linear relationship between x and y.
c. Many different straight lines can be drawn to provide a linear approximation of the
relationship between x and y; in part (d) we will determine the equation of a straight line
that “best” represents the relationship according to the least squares criterion.
d.
15 40
3 8
55
ii
xy
xy
nn
2
()( )26 ()1 0
ii i
xxyy xx
1
2
()( ) 26
2.
() 10
ii
i
xxyy
b
xx
bybx
01
8263 02(.)().
yx ˆ0 2.
e. ˆ y 0 2(4) 10.
0
2
4
6
8
10
12
14
16
0123456
y
x
Chapter 14
14 – 4
© 2017 Cengage Learning. All Rights Reserved.
- a.
b.
50 83
10 16.
55
ii
xy
xy
nn
2
()( )171 ()1 90
ii i
xxyy xx
1 2
()( ) 171
0.
() 190
ii
i
xxyy
b
xx
01
bybx 16 (0)(10)7.
yx ˆ7 0.
c. ˆ y 7 0(6) 13
0
5
10
15
20
25
30
0 5 10 15 20 25
y
x
Simple Linear Regression
14 – 5
© 2017 Cengage Learning. All Rights Reserved.
- a.
b. There appears to be a positive linear relationship between the percentage of women working in the
five companies ( x ) and the percentage of management jobs held by women in that company ( y )
c. Many different straight lines can be drawn to provide a linear approximation of the
relationship between x and y ; in part (d) we will determine the equation of a straight line
that “best” represents the relationship according to the least squares criterion.
d.
300 215
60 43
55
xyii
xy
nn
2
( xxyyii )( ) 624 ( xxi ) 480
1 2
()( ) 624
1.
( ) 480
ii
i
xxyy
b
xx
bybx 01 43 1(60) 35
yx ˆ 35 1.
e. yx ˆ 35 1 35 1(60)43%
0
10
20
30
40
50
60
70
40 45 50 55 60 65 70 75
% Management
% Working
Simple Linear Regression
14 – 7
© 2017 Cengage Learning. All Rights Reserved.
- a.
b. The scatter diagram indicates a positive linear relationship between x = average number of passing
yards per attempt and y = the percentage of games won by the team.
c. xxn ii / 680 / 10 6 yyn / 464 / 1046.
2
( )( ) 121 ( ) 7.
ii i
xxyy xx
1 2
( )( ) 121.
17.
( ) 7.
ii
i
xxyy
b
xx
bybx 01 46 (17)(6)70.
yx ˆ70 17
d. The slope of the estimated regression line is approximately 17. So, for every increase of one yard
in the average number of passes per attempt, the percentage of games won by the team increases by
17%.
e. With an average number of passing yards per attempt of 6, the predicted percentage of games won
is y ˆ= -70 + 17(6) = 36%. With a record of 7 wins and 9 loses, the percentage of wins that
the Kansas City Chiefs won is 43 or approximately 44%. Considering the small data size, the
prediction made using the estimated regression equation is not too bad.
0
10
20
30
40
50
60
70
80
90
456789
Win%
Yds/Att
Chapter 14
14 – 8
© 2017 Cengage Learning. All Rights Reserved.
- a.
b. Let x = years of experience and y = annual sales ($1000s)
70 1080
7 108
10 10
xyii
xy
nn
2
() xxyyii ( )568 ()1 xxi 42
12
()( ) 568
4
() 142
ii
i
xxyy
b
xx
bybx 01 108 ()()4 7 80
yx 80 4
c. yx 80 4 80 4 9() 116 or $116,
50
60
70
80
90
100
110
120
130
140
150
02468101214
Annual Sales ($1000s)
Years of Experience
Chapter 14
14 – 10
© 2017 Cengage Learning. All Rights Reserved.
- a.
b. The scatter diagram indicates a positive linear relationship between x = cars in service (1000s) and y
= annual revenue ($millions).
c. / 43 / 6 7 / 462 / 6 77
xxn ii yyn
2
( )( ) 734 ( ) 56.
ii i
xxyy xx
1
2
( )( ) 734.
12.
( ) 56.
ii
i
xxyy
b
xx
bybx 01 77 (12)(7)17.
yx ˆ17 12
d. For every additional 1000 cars placed in service annual revenue will increase by 12 ($millions)
or $12,966,000. Therefor every additional car placed in service will increase annual revenue by
$12,966.
e. yx ˆ17 12 17 12(11) 125
A prediction of annual revenue for Fox Rent A Car is approximately $126 million.
0
20
40
60
80
100
120
140
160
0 2 4 6 8 10 12 14
Annual Revenue ($millions)
Cars in Service (1000s)
Simple Linear Regression
14 – 11
© 2017 Cengage Learning. All Rights Reserved.
10. a.
b. The scatter diagram indicates a positive linear relationship between x = percentage increase in the
stock price and y = percentage gain in options value. In other words, options values increase as stock
prices increase.
c. / 2939 / 10 293 / 6301 / 10 630.
xxn ii yyn
2
( xxyyii )( ) 314,501 ( xxi ) 115,842.
12
( )( ) 314,501.
2.
( ) 115,842.
ii
i
xxyy
b
xx
01
bybx 630 (2)(293)167.
yx ˆ167 2
d. The slope of the estimated regression line is approximately 2. So, for every percentage increase in
the price of the stock the options value increases by 2%.
e. The rewards for the CEO do appear to be based upon performance increases in the stock value.
While the rewards may seem excessive, the executive is being rewarded for his/her role in increasing
the value of the company. This is why such compensation schemes are devised for CEOs by boards
of directors. A compensation scheme where an executive got a big salary increase when the
company stock went down would be bad. And, if the stock price for a company had gone down
during the periods in question, the value of the CEOs options would also go down.
0
200
400
600
800
1000
1200
1400
0 100 200 300 400 500 600
% Gain in Options Value
% Increase in Stock Price
Simple Linear Regression
14 – 13
© 2017 Cengage Learning. All Rights Reserved.
12. a.
b. The scatter diagram indicates a positive linear relationship between x = hotel room rate and the
amount spent on entertainment.
c. / 945 / 9 105 / 1134 / 9 126
xxn ii yyn
2
( xxyyii )( ) 4237 ( xxi ) 4100
12
( )( ) 4237
1.
( ) 4100
ii
i
xxyy
b
xx
01
bybx 126 (1)(105) 17
yx ˆ17 1.
d. With a value of x = $128, the predicted value of y for Chicago is
yx ˆ 17 1 17 1(128) 150
Note: In The Wall Street Journal article the entertainment expense for Chicago was $146. Thus, the
estimated regression equation provided a good estimate of entertainment expenses for Chicago.
70
90
110
130
150
170
190
70 90 110 130 150 170
Entertainment ($)
Hotel Room Rate ($)
Chapter 14
14 – 14
© 2017 Cengage Learning. All Rights Reserved.
- a.
b. Let x = adjusted gross income and y = reasonable amount of itemized deductions
399 97.
57 13.
77
xyii
xy
nn
2
( xxyyii )( ) 1233 ( xxi ) 7648
12
()( )1233.
0.
() 7648
ii
i
xxyy
b
xx
bybx 01 13 (0)(57) 4.
yx 468 016..
c. yx 4 68 0 16.. ..(4 68 0 1652.). 5 13 08or approximately $13,080.
The agent’s request for an audit appears to be justified.
0.
5.
10.
15.
20.
25.
30.
0 20 40 60 80 100 120 140.
Reasonable Amount of Itemized
Deductions ($1000s)
Adjusted Gross Income ($1000s)
Chapter 14
14 – 16
© 2017 Cengage Learning. All Rights Reserved.
- a. The estimated regression equation and the mean for the dependent variable are:
ˆ yxi 68 3 y 35
The sum of squares due to error and the total sum of squares are
22
SSE ( ˆ) 230 SST ( ) 1850
yyii yyi
Thus, SSR = SST – SSE = 1850 – 230 = 1620
b. r
2
= SSR/SST = 1620/1850 =.
The least squares line provided an excellent fit; 87% of the variability in y has been explained by
the estimated regression equation.
c. rxy .876 .
Note: the sign for r is negative because the slope of the estimated regression equation is negative.
( b 1 = -3)
- The estimated regression equation and the mean for the dependent variable are:
ˆ yxi 7 .9 y 16.
The sum of squares due to error and the total sum of squares are
22
SSE ( ˆ) 127 SST ( ) 281.
ii i
yy yy
Thus, SSR = SST – SSE = 281 – 127 = 153.
r
2
= SSR/SST = 153.9/281 =.
We see that 54% of the variability in y has been explained by the least squares line.
.547.
xy
r
- a. / 600 / 6 100 / 330 / 6 55
ii
xxn yyn
22
SST = ( ) 1800 SSE = ( ˆ) 287.
yyii yyi
SSR = SST – SSR = 1800 – 287 = 1512.
b.
2 SSR 1512.
.
SST 1800
r
c.
2
rr .84.
Simple Linear Regression
14 – 17
© 2017 Cengage Learning. All Rights Reserved.
- a. The estimated regression equation and the mean for the dependent variable are:
y ˆ= 80 + 4 x y = 108
The sum of squares due to error and the total sum of squares are
22
SSE ( ˆ) 170 SST ( ) 2442
yyii yyi
Thus, SSR = SST – SSE = 2442 – 170 = 2272
b. r
2
= SSR/SST = 2272/2442 =.
We see that 93% of the variability in y has been explained by the least squares line.
c. .93.
xy
r
- a. / 160 / 10 16 / 55,500 / 10 5550
ii
xxn yyn
2
( xxyyii )( ) 31, 284 ( xxi ) 21.
12
()( )31,
1439
() 2 1.
ii
i
xxyy
b
xx
01
bybx 5550 ( 1439)(16)28,
yx ˆ28, 574 1439
b. SST = 52,120,800 SSE = 7,102,922.
SSR = SST – SSR = 52,120,800 – 7,102,922 = 45,017,
2
r = SSR/SST = 45,017,877/52,120,800 =.
The estimated regression equation provided a very good fit.
c. yx ˆ 28,574 1439 28,574 1439(15) 6989
Thus, an estimate of the price for a bike that weighs 15 pounds is $6989.
- a.
3450 33, 700
575 5616.
66
ii
xy
xy
nn
2
( xxyyii )( ) 712, 500 ( xxi ) 93, 750
12
()( )712, 500
7.
() 93, 750
ii
i
xxyy
b
xx
bybx 01 5616 (7)(575) 1246
yx 1246 67 7 6..
Simple Linear Regression
14 – 19
© 2017 Cengage Learning. All Rights Reserved.
e. MSR = SSR / 1 = 67.
F = MSR / MSE = 67 / 4 = 16.
Using F table (1 degree of freedom numerator and 3 denominator), p -value is between .025 and.
Using Excel or Minitab, the p -value corresponding to F = 16 is .0272.
Because p -value, we reject H 0 : 1 = 0
Source
of Variation
Sum
of Squares
Degrees
of Freedom
Mean
Square
F
p -value
Regression 67 1 67 16.
Error 12 3 4.
Total 80 4
- a. s
2
= MSE = SSE/( n – 2) = 230/3 = 76.
b. s MSE 76 8.
c.
2
()1 80
i
xx
12
8.
0.
() 180
b
i
s
s
xx
d.
1
1 3
4.
.
b
b
t
s
Using t table (3 degrees of freedom), area in tail is less than .01; p -value is less than.
Using Excel or Minitab, the p -value corresponding to t = -4 is .0193.
Because p -value, we reject H 0 : 1 = 0
e. MSR = SSR/1 = 1620
F = MSR/MSE = 1620/76 = 21.
Using F table (1 degree of freedom numerator and 3 denominator), p -value is less than.
Using Excel or Minitab, the p -value corresponding to F = 21 is .0193.
Because p -value, we reject H 0 : 1 = 0
Source
of Variation
Sum
of Squares
Degrees
of Freedom
Mean
Square
F
p -value
Regression 1620 1 1620 21.
Error 230 3 76.
Total 1850 4
Chapter 14
14 – 20
© 2017 Cengage Learning. All Rights Reserved.
- a. s
2
= MSE = SSE/( n – 2) = 127/3 = 42.
s MSE 42 6.
b.
2
()1 xxi 90
12
6.
0.
() 190
b
i
s
s
xx
1
1
.
1.
.
b
b
t
s
Using t table (3 degrees of freedom), area in tail is between .05 and.
p -value is between .10 and.
Using Excel or Minitab, the p -value corresponding to t = 1 is .1530.
Because p -value >, we cannot reject H 0 : 1 = 0; x and y do not appear to be related.
c. MSR = SSR/1 = 153 /1 = 153.
F = MSR/MSE = 153.9/42 = 3.
Using F table (1 degree of freedom numerator and 3 denominator), p -value is greater than.
Using Excel or Minitab, the p -value corresponding to F = 3 is .1530.
Because p -value >, we cannot reject H 0 : 1 = 0; x and y do not appear to be related.
- a. In the statement of exercise 18, y ˆ= 23 + .318 x
In solving exercise 18, we found SSE = 287.
2
sn MSE = SSE/( -2) =287 / 4 71.
s MSE 71 8.
2
()1 xx 4,
12
8.
.
()14, 950
b
s
s
xx
1
1
.
4.
.
b
b
t
s
Using t table (4 degrees of freedom), area in tail is between .005 and.
p -value is between .01 and.
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