Statistics for Business

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Chapter 20 - problem 38, 40

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Chapter 20
Curved Patterns
Question No. 38: Cellular Phones in Africa
(a) The scatterplot of two types of subscribers suggests a possible linear trend in the number of
landlines. The plot of Landline subscribers seems more Linear than that of Mobile
subscribers.

Mobile subsribes (000)

9000

300000

8000

250000

7000

200000

6000

150000

5000

100000

4000

50000

3000

0
1990
-50000
-100000

1995

2000

2005

2010

Year
Mobile Subscribers (Sub-Sahara, 000)

2000
2015
1000

Landline subscribers (000)

Mobile & Landline Subscribers Vs Year

350000

0

Land Line Subscribers (Sub-Sahara, 000)
Linear (Mobile Subscribers (Sub-Sahara, 000))
Linear (Land Line Subscribers (Sub-Sahara, 000))

(b)

Land Line Subscribers Vs Year
Land Line Subscribers(000)

9000

y = 374.51x - 744620
R² = 0.987

8000
7000
6000
5000

Land Line Subscribers (SubSahara, 000)

4000
3000

Linear (Land Line Subscribers
(Sub-Sahara, 000))

2000
1000
0
1990

1995

2000

2005
Year

2010

2015

The linear trend of number of land line subscribers has high regression fitted value, r2=0.987,
but it doesn’t seem to have a trend in the extremes or in the middle of the data.
(c) The regression equation is number of landline subscribers (in 1000s) = 374.51(year)-744620
The slope implies that there is an annual growth in the average number of landline
subscribers by 374510 and the negative intercept represents a large unrealistic extrapolation
for the 0th year.
(d)

Residual Plot
400
300
200
Residuals

100
0
-1001994

1996

1998

2000

2002

2004

2006

2008

2010

-200
-300
-400
-500

Year

There is no pattern that can be interpreted from the residual plot. The residuals represent a
poor fit, deviating from the linearity. The linear equation under-predicts in the edges of the
plot and over-predicts in the the middle of the plot.
(e)

log e Landlne subsribes (000)

Log Subscribers Vs Year
9.2
y = 0.0751x - 141.8
R² = 0.9723

9
8.8
8.6
8.4
8.2
8
7.8
1990

1995

2000

2005
Year

2010

2015

2012

Log trend line shows the bending pattern in the original plot . The residuals from this curve seems to
be random. So, the curve of ‘Estimated loge (Number of Subscribers) = b0 + b1 Year’ is not a better
summary of the growth of the use of landlines compared to that of ‘Number of Subscribers = b0 + b1
Year’ model.

(f)

Log Subscribers Vs Year
16

y = 0.5819x - 1156
R² = 0.9788

log e mobile subsribes (000)

14
12
10
8
6
4
2
0
1994

1996

1998

2000

2002

2004

2006

2008

2010

Year

The regression equation for the log is
log e (number of mobile subscribers) = 0.5819 (years) – 1156
16
14

y = 0.5819x - 1156
R² = 0.9788

12
10
8

y = 0.0751x - 141.8
R² = 0.9723

6
4
2
0
1994

1996

1998

2000

2002

2004

2006

2008

Log land Line

Log mobile

Linear (Log land Line)

Linear (Log mobile)

log inv (5819) = 1.789, log inv (0.0751) = 1.078

2010

2012

2012

This implies a high annual rate of growth as the growth rate in the number of mobile
subscribers is 1.789x1000, whereas the growth rate of the number of landline subscribers is
1.078x1000

Question Number 40: CO2

CO2 (million tons) Vs GDP (billion dollars)
8000
7000
y = 0.5094x + 55.537
R² = 0.5553

CO2 (million tons)

6000
5000
4000
3000
2000
1000
0
$0.00

$2,000.00

$4,000.00
$6,000.00
$8,000.00
GDP (billion dollars)

$10,000.00

$12,000.00

(a) The three prominent Outliers are People’s Republic of China, US and Japan
(b) The plot after removing the outliers

CO2 (million of tons)

CO2 Vs GDP
1800
1600
1400
1200
1000
800
600
400
200
0
$0.00

y = 0.4587x + 39.846
R² = 0.4041

$500.00

$1,000.00

$1,500.00

$2,000.00

GDP (billion dollars)
CO2 (million tons)

Linear (CO2 (million tons))

$2,500.00

The pattern in the plot says that the countries with low GDP have lower levels of CO2 emission. The
pattern in the plot is an exponential pattern
The equation to summarize the variation in the form of regression line :
CO2 (in millions of tons)=0.4587*GDP(in billion dollars)+39.846
(c)

Log CO2 Vs Log GDP
10
y = 0.879x + 0.2104
R² = 0.8043

8

Log CO2

6
4
2
0
-2

0

2

4

-2

6

8

10

Log GDP

The linear pattern is apparent in the scatterplot.
(d) The fitted equation for the plot is : Log CO2 = (Log GDP)*0.879+0.2104

Residual Plot
2.5
2
1.5

Residuals

1

-2

0.5
0
-0.5

0

2

4

-1
-1.5
-2

Log GDP

6

8

10

.
(e) The fitted equation implies that the fit seems to be appropriate as no pattern is found. The
variation over log GDP is also seems to be the equal
Fitted equation: Log CO2 = (Log GDP)*0.879+0.2104
(f)

Log 10 CO2 Vs Log 10 GDP
4.5
4

y = 0.879x + 0.0914
R² = 0.8043

3.5

Log10 CO2

3
2.5
2
1.5
1
0.5
0
-0.5 -0.5 0
-1

0.5

1

1.5

2

2.5

3

Log 10 GDP

Yes, there is change in the y-intercept and in the fitted regression line.

3.5

4

4.5

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