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.

(b)

0

1000

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4000

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7000

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9000

-100000

-50000

0

50000

100000

150000

200000

250000

300000

350000

1990 1995 2000 2005 2010 2015

Lan

dlin

e s

ub

scri

be

rs (

00

0)

Mo

bile

su

bsr

ibe

s (0

00

)

Year

Mobile & Landline Subscribers Vs Year

Mobile Subscribers (Sub-Sahara, 000)

Land Line Subscribers (Sub-Sahara, 000)

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

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

y = 374.51x - 744620 R² = 0.987

0

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2000

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7000

8000

9000

1990 1995 2000 2005 2010 2015

Lan

d L

ine

Su

bsc

rib

ers

(00

0)

Year

Land Line Subscribers Vs Year

Land Line Subscribers (Sub-Sahara, 000)

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

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)

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)

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0

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1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Re

sid

ual

s

Year

Residual Plot

y = 0.0751x - 141.8 R² = 0.9723

7.8

8

8.2

8.4

8.6

8.8

9

9.2

1990 1995 2000 2005 2010 2015

log

e L

and

lne

su

bsr

ibe

s (0

00

)

Year

Log Subscribers Vs Year

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)

The regression equation for the log is

log e (number of mobile subscribers) = 0.5819 (years) – 1156

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

y = 0.5819x - 1156 R² = 0.9788

0

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1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

log

e m

ob

ile s

ub

srib

es

(00

0)

Year

Log Subscribers Vs Year

y = 0.0751x - 141.8 R² = 0.9723

y = 0.5819x - 1156 R² = 0.9788

0

2

4

6

8

10

12

14

16

1994 1996 1998 2000 2002 2004 2006 2008 2010 2012

Log land Line Log mobile

Linear (Log land Line) Linear (Log mobile)

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

(a) The three prominent Outliers are People’s Republic of China, US and Japan

(b) The plot after removing the outliers

y = 0.5094x + 55.537 R² = 0.5553

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$0.00 $2,000.00 $4,000.00 $6,000.00 $8,000.00 $10,000.00 $12,000.00

CO

2 (

mill

ion

to

ns)

GDP (billion dollars)

CO2 (million tons) Vs GDP (billion dollars)

y = 0.4587x + 39.846 R² = 0.4041

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$0.00 $500.00 $1,000.00 $1,500.00 $2,000.00 $2,500.00

CO

2 (

mill

ion

of

ton

s)

GDP (billion dollars)

CO2 Vs GDP

CO2 (million tons) Linear (CO2 (million tons))

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)

The linear pattern is apparent in the scatterplot.

(d) The fitted equation for the plot is : Log CO2 = (Log GDP)*0.879+0.2104

y = 0.879x + 0.2104 R² = 0.8043

-2

0

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10

-2 0 2 4 6 8 10

Log

CO

2

Log GDP

Log CO2 Vs Log GDP

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

-2 0 2 4 6 8 10

Re

sid

ual

s

Log GDP

Residual Plot

.

(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)

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

y = 0.879x + 0.0914 R² = 0.8043

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4.5

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Log1

0 C

O2

Log 10 GDP

Log 10 CO2 Vs Log 10 GDP