A’

LEVELGEOGRAPHY

GEO-
STATI
STI
CALANALYSI
SANDPRESENTATI
ON

Pr
epar
edf
orMEGTA2017AnnualConf
erence

ByAmosTendaiMunzar
a(0773245970)

1.
0Level
sofMeasur
ement

Measur sdoneonvar
ementi iabl
es.Avar
iabl
eisachar
act
eri
sti
cunderst
udywhi
ch
assumesdi
ff
erentval
uesf
ordi
ff
erentel
ement
sofapopul
ati
onorsampl
e.

Thel
evel
sofmeasur
ementar
e:

 nomi
naldat
a,consi
stssol
elyofnamesorl
abel
se.
gfar
mingr
egi
ons,
cl
assi
fi
cat
ionofset
tl
ement
srur
alorur
ban.

 or
dinaldat
a,consi
stsofobser
vat
ionst
hatar
erankedi
nter
msofsi
zeor
i
mpor
tancee.
gcl
assi
fi
cat
ionofset
tl
ement
sasA1orA2;orashi
gh,
medi
um
orl
owdensi
tysubur
bs.Magni
tudesofdi
ff
erencesbet
weenr
anksar
enot
cl
earcut
.

 i
nter
valdat
a,measur
edonascal
ethatdoesnothaveameani
ngf
ulzer
o;t
he
zer
oisassi
gnedar
bit
rar
il
yhencer
ati
osofi
nter
valscal
eval
ueshaveno
meani
nge.
gtemper
atur
e.At
emper
at eof0oCdoesnoti
ur mpl
ytheabsence
ofheat
.At
emper
at eof60oCi
ur snott
wiceashotasat
emper
at eof30oC.
ur

 r
ati
odat
a,hasameani
ngf
ulzer
oandt
her
ati
osofmeasur
ement
sar
e
consi
stenti
rr
espect
iveofuni
tsofmeasur
ement
.Theabsol
utezer
ois
uni
quel
ydef
ined.Forexampl
e,i
fthedi
stancebet
weent
wopoi
ntsi
s0cm,
it
cl
ear
lyi
mpl
iest
het
wopoi
ntsar
enotsepar
atedf
rom eachot
her
.Adi
stance
of20km i
stwi
ceasl
ongasadi
stanceof10km.Ot
hergoodexampl
esar
e
amountofr
ainf
all
,popul
ati
ondat
a,mi
grat
iondat
a.

 cycl
icdat
a,dat
aconsi
sti
ngofdi
rect
ionsort
imesi
nwhi
cht
hemeasur
ement
scal
eiscycl
ice.
gaf
ter3590comes00,
aft
er31Decembercomes1Januar
y.

Rangeofst
ati
sti
calanal
ysespossi
bleoneachdat
atype:

Page1of1
Nomi
nalandor
dinal
dat
a;ar
it
hmet
icoper
ati
onsof+,
-,xand/notpossi
ble

I
nter
valandr
ati
o;ar
it
hmet
icoper
ati
onsof+,
-,xand/andal
someasur
esofcent
ral
t
endencyanddi
sper
sionar
epossi
bleandmeani
ngf
ul.

2.
0Uni
var
iat
eSt
ati
sti
cs

Uni
var
iat
eSt
ati
sti
csr
efer
stost
ati
sti
csper
tai
ningt
oonevar
iabl
e.

2.
1Measur
esofcent
ralt
endency

Thesear
eaver
ages(
mean,
mode,
medi
an)whi
chl
ocat
ethecent
reofadat
asetand
ar
eknowncol
lect
ivel
yasmeasur
esofcent
ralt
endency.

Themodeofadat asetist
hatval
uewitht
hehi
ghestf
requency,t
hati
s,t
heval
ue
thatoccurt
hegr
eat
estnumberoft
imes.

Themediani
st hemi
ddleobser
vationinanorder
eddataset.Thedatai
susual
ly
order
edbyar
rangi
ngval
uesinascendingor
der
.Lettheor
der
edvaluesofadat
aset
be , , …, wher
e i
sthenumberofobser
vat
ions.Themedi
ani
sgi
venby

a. i
f i
sodd

b. i
f i
seven.

Togett
hemeanofyouhavet
ofi
ndt
hesum ofal
ltheobser
vat
ionsandt
hendi
vide
t
hesum byt
henumberofobser
vat
ions.

Thepopul
ati
onmeanofasetofNmeasur
ement
sisgi
venbyt
hef
ormul
a:

wher
e ar
etheobser
vat
ions, i
sthepopul
ati
onsi
ze.

Thesymbol i
scal
ledsi
gmaandi
tmeansadd.

Page2of2
Thesampl
emeani
sobt
ainedusi
ngt
hef
ormul
a:

wher
e ar
etheobser
vat
ionsandni
sthesampl
esi
ze.

Exampl
e1

Theamountofr
ainf
all(
inmm)r
ecor
dedat15di
ff
erentl
ocat
ionsi
nthecount
ryona
par
ti
cul
ardayar
e:

44 5935414625476054463546413419

Fi
ndt
he(
a)mode,(
b)medi
anand(
c)meanoft
heamountofr
ainf
allr
ecei
vedon
t
hatpar
ti
cul
arday.

Sol
uti
on1

(
a)Mode=46mm

(
b)Medi
an

Rear
rangi
ng:192534353541414446464647545960

Si
ncen=15i
sodd,
medi
ancor
respondst
o

(
c)mean=

=42.
13333333

=42.
13mm

Achoi
ceof
tenhast
obemadeonwhi
chmeasur
eofl
ocat
iont
ousebet
weent
he
mode,medi
anandmean.Whenchoosi
ngasui
tabl
eaver
aget
ouse,t
hef
oll
owi
ng

Page3of3
f
act
orsar
econsi
der
ed:

 Thet
ypeofdat
abei
ngdeal
twi
th.Themodei
ssui
tabl
ewhenyouar
edeal
ing
wi
thdi
scr
etedat
amadeupofar
elat
ivel
yfew di
ff
erentval
ues.I
tisnot
sui
tabl
eforcont
inuousdat
a

 Theshapeoft
hedi
str
ibut
ion.Themeani
saf
fect
edbyout
li
ers(
ext
remel
y
hi
ghorl
owval
ues)whi
let
hemedi
ani
snotaf
fect
ed.Themedi
ani
sther
efor
e
appr
opr
iat
eifyouhaveaskeweddi
str
ibut
ionwhi
lstt
hemeani
ssui
tabl
efor
anear
-symmet
ri
caldi
str
ibut
ion.

 Usef
urt
herst
ati
sti
calanal
ysi
s.Themeanhasamor
eext
ensi
ver
olewi
thi
n
St
ati
sti
cscompar
edt
othemedi
anandt
hemode,
andt
her
efor
eiti
spr
efer
red
wher
efur
ther st
ati
sti
cal anal
ysi
sis needed e.
g conf
idence i
nter
val
est
imat
ionandhypot
hesi
stest
ing.

2.
2Measur
esofdi
sper
sion

Ther
angegi
vesasi
mpl
eindi
cat
oroft
hevar
iabi
li
tyofasetofobser
vat
ions.The
r
angeofasetofobser
vat
ionsi
sthedi
ff
erencebet
weent
hel
argestobser
vat
ionand
t
hesmal
lestobser
vat
ion.

Range=hi
ghestobser
vedval
ue–l
owestobser
vedval
ue

Al
thoughi
tisver
yeasyt
ouseandunder
stand,
ther
angei
snotar
eli
abl
ewayof

measur
ingt
hespr
eadofdat
abecausei
tisonl
ybasedononl
ytwoobser
vat
ions

whi
char
ethehi
ghestandl
owestval
ues.I
foneoft
heset
woval
uesi
sanout
li
er,

t
hent
hespr
eadofdat
aisr
atherexagger
ated.

Thevar
ianceandst
andar
ddevi
ati
onal
low ust
oavoi
dtheshor
tcomi
ngsoft
he
r
ange as a measur
e ofdi
sper
sion because t
hey t
ake i
nto accountal
lthe
obser
vat
ionsi
nthedat
asetasopposedt
ojustsel
ect
ingaf
ew.

Thevar
ianceofasetofdat
aist
heaver
agesquar
eddevi
ati
onoft
hedat
apoi
nts

Page4of4
f
rom t
hei
rmean.Comput
ati
onal
ly,
thevar
iance( )ofasampl
eofnobser
vat
ions
i
sobt
ainedbyt
hef
ormul
a:

Thef
ormul
aforpopul
ati
onvar
iance i
sgi
venby:

Thest
andar
ddevi
ati
onofasetofobser
vat
ionsi
stheposi
ti
vesquar
erootoft
he

var
ianceoft
heset
.

Exampl
e2

Fi
ndt
he(
a)r
angeand(
b)st
andar
ddevi
ati
onf
ort
hedat
ainExampl
e1.

Sol
uti
on2

(
a)Range=60–19=41mm

(
b)St
andar
ddevi
ati
on

Page5of5
 =121.
0488889

St
andar
ddevi
ati
on=

=11.
00mm

2.
3Hypot
hesi
stest
ing

Ahypot
hesi
sisacl
aim orasser
ti
onmadeconcer
ningapopul
ati
onpar
amet
er.A

hypot
hesi
stesti
sat
estusedt
over
if
ywhet
hert
hecl
aimi
sli
kel
ytobet
rueorf
alse.

I
nthi
ssect
ion,
youwi
ll
bei
ntr
oducedt
otest
sconcer
ningt
hemeanofasi
ngl
e

popul
ati
on.

Hypot
hesi
stest
ingi
nvol
vesgat
her
ingevi
dencef
rom ar
andom sampl
edr
awnf
rom
t
hepopul
ati
onofi
nter
esti
nor
dert
odeci
dewhet
hert
henul
lhypot
hesi
sisl
ikel
yto
bet
rueorf
alse.Thehypot
hesi
sisr
eject
edi
fevi
dencef
rom t
hesampl
eisnot
consi
stentwi
tht
he st
ated hypot
hesi
s,ot
her
wise i
tis accept
ed.However
,the
accept
anceoft
hest
atedhypot
hesi
sdoesnotnecessar
il
yimpl
ythati
tist
rue,
rat
her
i
tisar
esul
tofi
nsuf
fi
cientevi
dencet
orej
ecti
t.

2.
3.1Typesofhypot
heses
Ther
ear
etwot
ypesofhypot
heseswhi
char
ecal
ledt
he:
 nul
lhypot
hesi
s,and
 al
ter
nat
ivehypot
hesi
s

Anul
lhypot
hesi
sisanasser
ti
onaboutt
heval
ueofapopul
ati
onpar
amet
er.I
tisa
f
ormalst
atementoft
hecl
aim bei
ngmadeconcer
ningapopul
ati
onmeasur
e.The
nul
lhypot
hesi
sisdenot
edbyH0.

Theal
ter
nat
ivehypot
hesi
s,denot
edbyH1,i
sthenegat
ionoft
henul
lhypot
hesi
s.For
exampl
e,anul
lhypot
hesi
smi
ghtasser
tthatt
hepopul
ati
onmeani
sequalt
oa
speci
fi
edval
ue .Wewr
it
ethi
sas .Theal
ter
nat
ivehypot
hesi
soppose

t
hisasser
ti
onandi
tiswr
it
tenas .I
nthi
scase,
theal
ter
nat
ivehypot
hesi
s

Page6of6
suggest
sthatt
hemeant
akesval
uest
hatar
eei
therbel
ow orabovei
t.Ther
efor
e,

t
oinvest
igat
eH0 weconductanon-
dir
ect
ionalt
estwhi
chi
sknownasat
wo-
tai
led
t
est
.

A nul
lhypot
hesi
smi
ghtasser
tthatt
hepopul
ati
onmeani
satl
eastequalt
oa
cer
tai
n speci
fi
ed val
ue .We wr
it
e .I
nthi
s case,t
he al
ter
nat
ive

hypot
hesi
swoul
dconsi
stofval
uesbel
ow ,t
hati
s, .Si
mil
arl
y,i
fanul
l

hypot
hesi
sasser
tthatt
hepopul
ati
onmeani
slesst
hanorequalt
oaspeci
fi
ed
val
ue ,t
hati
s, ,t
heal
ter
nat
ivewi
llbe .I
nbot
hthesecases,

si
nce t
he al
ter
nat
ive hypot
heses consi
stofval
ues ei
therbel
ow orabove t
he
speci
fi
edval
ue ,
weconductaone-
sidedt
estoraone-
tai
ledt
est
.

2.
3.2TypeIandTypeI
IEr
ror
s
I
n deci
ding t
orej
ectoraccepta nul
lhypot
hesi
s,t
her
e wi
llbe chances f
or
er
roneousl
yrej
ect
ingoraccept
ingi
t.Sucher
ror
smaybeduet
ofaul
tysampl
ing
pr
ocedur
es.
At
ypeIer
rori
scommi
tt
edwhenat
ruenul
lhypot
hesi
sisr
eject
ed.Thepr
obabi
li
ty
ofcommi
tt
ingat
ypeIer
rori
scal
ledt
hel
evelofsi
gni
fi
canceandi
tisdenot
edby .
I
tiscommont
ouse1%,
5%and10%l
evelofsi
gni
fi
cancei
ncal
cul
ati
ons.
At
ypeI
Ier
rori
scommi
tt
edi
fweacceptt
henul
lhypot
hesi
swheni
tisf
alse.The
pr
obabi
li
tyt
hatt
het
estwi
llbeabl
etodet
ectaf
alsenul
lhypot
hesi
siscal
ledt
he
powerofat
est
.Inot
herwor
ds,
thepowerofat
esti
sthepr
obabi
li
tyofr
eject
ingH0
wheni
ndeedH0i
sfal
se.

2.
3.3St
epsFol
lowedi
nHypot
hesi
sTest
ing

Thef
oll
owi
ngst
epsshoul
dbef
oll
owedwhenconduct
ingahypot
hesi
stest
:

St
ep1:St
atet
heNul
landAl
ter
nat
iveHypot
hesi
s
The nul
land al
ter
nat
ive hypot
heses ar
e speci
fi
ed att
hisi
nit
ialst
age bef
ore
gat
her
inganyevi
dence.I
twoul
dbeunet
hicalandr
athermani
pul
ati
vet
ofor
mul
ate
t
heH0 andH1 atone’
sconveni
enceaf
tergat
her
ingevi
dence;apr
act
iset
hatwe

Page7of7
r
efert
oasdat
asnoopi
ng.

St
ep2:I
dent
if
ytheDi
str
ibut
ion
Choosebet
weent
wodi
str
ibut
ionsnamel
ythez-
dist
ri
but
ionandt
het
-di
str
ibut
ion.

Whent
est
ingf
ort
hepopul
ati
onmean weuse:
a)Thez-
dist
ri
but
ionwhent
he:
 popul
ati
onst
andar
ddevi
ati
on i
sknown
 popul
ati
onst
andar
ddevi
ati
oni
sunknownand i
slar
ge(
b)Thet
-di
str
ibut
ionwhent
hepopul
ati
onst
andar
ddevi
ati
on i
sunknownand
t
hesampl
esi
ze i
ssmal
l( .
St
ep3:Det
ermi
net
heRej
ect
ionandAccept
anceRegi
on
Dependi
ngont
hedi
str
ibut
ioni
dent
if
iedandt
hel
evelofsi
gni
fi
cancedesi
red,you
f
indaval
uef
rom st
ati
sti
calt
abl
eswhi
chwecal
lacr
it
icalval
ue.Thecr
it
icalval
ue
separ
atest
heaccept
ancer
egi
onf
rom t
her
eject
ionr
egi
on.Ther
eject
ionr
egi
oni
s
madeupofar
angeofval
uessucht
hati
fat
estst
ati
sti
ccal
cul
atedf
rom sampl
e
dat
afal
lsi
nitt
henul
lhypot
hesi
swoul
dber
eject
ed.Ther
eject
ionr
egi
onal
so
dependsont
henat
ureoft
heal
ter
nat
ivehypot
hesi
sasshowni
nthef
oll
owi
ng
f
igur
es.

ar
eaofr
eject
ion( ) ar
eaofr
eject
ion( )

cr
it
icalval
ue 0 cr
it
icalval
ue
Rej
ect
ionRegi
onf
or

Ar
eaofr
eject
ion

0 cr
it
icalval
ue
Page8of8
Rej
ect
ionRegi
onf
or

ar
eaofr
eject
ion

Cr
it
icalval
ue 0
Rej
ect
ionRegi
onf
or

St
ep4:Cal
cul
atet
heTestSt
ati
sti
c
At
estst
ati
sti
cisaval
uecal
cul
atedf
rom sampl
edat
athati
susedt
odeci
de
whet
herornott
orej
ectH0.Oncet
het
estst
ati
sti
cfal
lswi
thi
nther
eject
ionr
egi
on,
H0i
srej
ect
ed.

Thecal
cul
ati
onoft
het
estst
ati
sti
cdependsonwhet
hert
hepopul
ati
onst
andar
d
devi
ati
on i
sknownorunknownandal
soont
hesampl
esi
zeassummar
isedi
n
Tabl
e1bel
ow.

Tabl
e1:TestSt
ati
sti
cforTest
ing
When i
sknown When i
sunknown
CaseI
:ni
slar
georsmal
l CaseI
I:ni
slar
ge

~N(
0,1) ~N(
0,1)

CaseI
II
:ni
ssmal
l

~t
(n-
1)

St
ep5:Deci
deWhet
herornott
oRej
ectH0

Page9of9
Thedeci
sioni
smadeont
hebasi
sofacompar
isonbet
weent
heval
ueoft
het
est
st
ati
sti
candt
hecr
it
icalval
ue.I
fthet
estst
ati
sti
cisgr
eat
ert
hant
hecr
it
icalval
uei
n
absol
utet
erms,
itwi
ll
fal
lint
her
eject
ionr
egi
ont
husl
eadi
ngt
other
eject
ionofH0.

St
ep6:MakeaConcl
usi
on
I
fH0 i
srej
ect
ed,weconcl
udet
hatH1 i
spr
obabl
ytr
ue.I
fwef
ailt
orej
ectH0,we
concl
udet
hatt
heevi
dencegat
her
edi
sinsuf
fi
cientt
owar
rantt
her
eject
ionofH0.

Exampl
e3

Ani
mmi
grat
ionof
fi
cercl
aimst
hatatl
east40peopl
ecr
osst
hebor
deri
ll
egal
lyi
nto
Sout
hAf
ri
caever
yday.Tot
estt
hecl
aim,t
henumberofpeopl
ecr
ossi
ngi
ll
egal
ly
i
ntoSout
hAf
ri
cawasnot
edon12r
andoml
ysel
ect
eddaysasf
oll
ows:
284136501739216426304212

Testt
hecl
aim madebyt
hei
mmi
grat
ionof
fi
cerata5%l
evelofsi
gni
fi
cance.

Sol
uti
on3
1.

2.Thepopul
ati
onst
andar
ddevi
ati
on i
sunknown,
butt
hesampl
esi
ze =12i
s
smal
l,
soweuset
het
-di
str
ibut
ion.

3. and i
t i
s a one-
tai
led t
est
. The cr
it
ical val
ue i
s

-
1.80 0
Wewoul
drej
ectH0i
f 1.
80

Page10of10
 4.Youshoul
dver
if
ythat ands=14.
6339

=-
0.2762

5.Si
nce =0.
2762<1.
80,
wef
ailt
orej
ectH0

6.Thedat
adoesnotpr
ovi
desuf
fi
cientevi
denceat5% l
evelofsi
gni
fi
cancet
o
r
ejectH0,
ther
efor
eweconcl
udet
hatt
hecl
aimi
sli
kel
ytobet
rue.

3.
0Bi
var
iat
eSt
ati
sti
cs
Bi
var
iat
eSt
ati
sti
csr
efer
stost
ati
sti
csper
tai
ningt
otwovar
iabl
es.
3.
1Cor
rel
ati
onanal
ysi
s
Cor
rel
ati
onanal
ysi
sisast
ati
sti
calpr
ocedur
ethati
susedt
omeasur
etheext
entt
o
whi
cht
wovar
iabl
esar
erel
ated.Twovar
iabl
esar
ehi
ghl
ycor
rel
atedi
ftheymove
wel
ltoget
hersucht
hati
tispossi
blet
opr
edi
ctt
hemovementofonevar
iabl
eusi
ng
knowl
edgeaboutt
hemovementoft
heot
hervar
iabl
e.Forexampl
e,goodr
ainsar
e
associ
atedwi
thgoodhar
vest
s(ot
hercondi
ti
onshel
dconst
ant
).
Thecor
rel
ati
oncoef
fi
cientdenot
edby i
sanumer
icalmeasur
eoft
hest
rengt
hof
t
hel
inearr
elat
ionshi
pbet
weent
wovar
iabl
esX andY.A cor
rel
ati
oncoef
fi
cient
t
akesval
uesbet
ween-
1and+1i
ncl
usi
ve,
thati
s .

Page11of11
Possi
bleval
uesof ar
eint
erpr
etedasf
oll
ows:
 i
sequal
tozer
o,i
ndi
cat
est
her
eisnocor
rel
ati
onbet
weent
hevar
iabl
es.
 i
ndi
cat
esaper
fectposi
ti
vecor
rel
ati
onbet
weent
hevar
iabl
es.
 i
ndi
cat
esaper
fectnegat
ivecor
rel
ati
onbet
weent
hevar
iabl
es
 cl
oset
o+1i
ndi
cat
esast
rongposi
ti
vecor
rel
ati
on
 cl
oset
o-1i
ndi
cat
esast
rongnegat
ivecor
rel
ati
on
 cl
oset
ozer
oimpl
iesaweakcor
rel
ati
onbet
weent
hevar
iabl
es

Thef
oll
owi
ngadj
ect
ivesmayhel
pyout
odescr
ibet
hedegr
eeofl
inearassoci
ati
on
bet
weent
wovar
iabl
es:

Val
uesof Sui
tabl
eadj
ect
ives
+0.
7to+1.
0 St
rong,
posi
ti
ve
+0.
4 t
o Fai
r/moder
ate,
+0.
69 posi
ti
ve
+0.
3 t
o Weak,
posi
ti
ve
+0.
39
0.
0to+0.
29 Negl
igi
ble/
scant
posi
ti
ve
0.
0to-
0.29 Negl
igi
ble/
scant
negat
ive
-
0.3t
o-0.
39 Weak,
negat
ive
-
0.4t
o-0.
69 Fai
r/moder
ate,
negat
ive
-
0.7t
o-1.
0 St
rong,
negat
ive

Ther
ear
etwocommonl
yusedcor
rel
ati
oncoef
fi
cient
swhi
char
e:
a)Pear
son’
spr
oductmomentcor
rel
ati
oncoef
fi
cient
()
b)Spear
man’
srankcor
rel
ati
oncoef
fi
cient( )

3.
1.1Pear
son’
spr
oductmomentcor
rel
ati
oncoef
fi
cient
Pear
son’
scor
rel
ati
oncoef
fi
cienti
scal
cul
atedf
orr
ati
odat
aori
nter
valdat
a.I
tis

Page12of12
based on t
hemean and st
andar
d devi
ati
on and t
her
efor
ecan beaf
fect
ed by
ext
reme val
ues. Pear
son’
s coef
fi
cient i
s obt
ained usi
ng t
he f
oll
owi
ng
comput
ati
onalf
ormul
a:

Exampl
e4
Thef
oll
owi
ngdat
aar
ear
andom sampl
eofi
ndexedpr
icesofgol
dandpl
ati
num
overasi
xyearper
iod:

Gol
d(X) 1 1 1 1 1 9
2 0 4 1 2
Pl
ati
num 1 1 2 1 2 1
(
Y) 8 7 3 9 0 5

Cal
cul
ateandi
nter
prett
hecor
rel
ati
oncoef
fi
cientf
ort
hedat
a.

Sol
uti
on4

Acor
rel
ati
oncoef
fi
cientof0.
9474i
scl
oset
o+1andi
tindi
cat
esast
rongposi
ti
ve
cor
rel
ati
onbet
weent
hepr
icesofgol
dandpl
ati
num.Whenpl
ati
num pr
icesar
e
goi
ngup,
gol
dpr
icesar
eal
soexpect
edt
ogoup,
othert
hingsbei
ngequal
.

Page13of13
3.
1.2Spear
man’
srankcor
rel
ati
oncoef
fi
cient
TheSpear
man’
scor
rel
ati
oncoef
fi
cientdenot
edby i
scal
cul
atedf
orr
ankeddat
a.

Thi
sis dat
a measur
ed on t
he or
dinalscal
e.The cor
rel
ati
on coef
fi
cient i
s

i
nter
pret
ed i
nthe same way as t
he Pear
son’
s cor
rel
ati
on coef
fi
cient .The
comput
ati
onalf
ormul
afor i
sgi
venby

wher
e i
sthedi
ff
erencei
nranksobt
ainedbysubt
ract
ingt
her
anksofyval
ues
f
rom t
her
anksofxval
uesf
oreachpai
rofobser
vat
ions.

Exampl
e5
Apaneloft
woj
udgesr
ankedt
hequal
it
yofpot
atoeshar
vest
edf
rom 5A1f
arms(
A,
B,
C,DandE)asf
oll
ows:

A1 A B C D E
f
arm
Judge 4 5 1 3 2
1
Judge 5 4 2 3 1
2

I
sther
eagr
eementi
nthemanneri
nwhi
cht
hej
udgesper
cei
vet
hequal
it
yof
pot
atoesf
rom t
hef
ivef
arms?
Sol
uti
on5
Judge Judge di di2
1 2
4 5 - 1
1
5 4 1 1
1 2 - 1
1
3 3 0 0

Page14of14
 2 1 1 1

=0.
8

Thecor
rel
ati
oncoef
fi
cienti
sfai
rl
yhi
ghandposi
ti
veshowi
ngt
hatt
hej
udgesdonot
di
ff
ermuchi
nthewayt
heyper
cei
vet
hequal
it
yofpot
atoesf
rom t
he5f
arms.

3.
2Regr
essi
onAnal
ysi
s
I
nreall
if
esi
tuat
ionswear
eusual
lyi
nter
est
edi
nrel
ati
onshi
psbet
weenvar
iabl
es.
Si
mpl
eli
nearr
egr
essi
oni
susedt
oest
abl
isht
hef
unct
ionalf
orm ofar
elat
ionshi
p
bet
weent
wovar
iabl
es.

3.
2.1TypesofVar
iabl
es
Ther
ear
etwot
ypesofvar
iabl
est
hatwedealwi
thi
nsi
mpl
eli
nearr
egr
essi
on
anal
ysi
s.Thevar
iabl
ethathasadependencer
elat
ionshi
pont
heot
hervar
iabl
eis
cal
ledt
her
esponsevar
iabl
eordependentvar
iabl
e,denot
edbyY.Theot
hervar
iabl
e
i
scal
ledt
heexpl
anat
oryori
ndependentvar
iabl
edenot
edbyX.

3.
2.2Scat
terPl
ots

Random pai
rsofobser
vat
ionsf
ort
hei
ndependentanddependentvar
iabl
esar
e
pl
ott
edonagr
apht
oshowt
heki
ndofr
elat
ionshi
pthatexi
stsbet
weent
hevar
iabl
es.
Fr
om t
hescat
terpl
otyoucant
elli
fther
elat
ionshi
pcanbemodel
ledbyast
rai
ght
l
ineequat
ionbef
orepr
oceedi
ngt
odol
inearr
egr
essi
onanal
ysi
s.

Page15of15
Exampl
e6
Ar
esear
cherwoul
dli
ket
oest
abl
isht
hei
mpactofpopul
ati
ongr
owt
honeconomi
c
devel
opment
.He col
lect
ed t
he f
oll
owi
ng cr
oss-
sect
ionaldat
afr
om si
x SADC
count
ri
es.
Popul
ati
ongr
owt
hrat
e(%) AnnualGDP(
US$bi
ll
ion)
7 18
Dr
aw a 9 35 scat
ter pl
ot
t
o 5 12 r
epr
esentt
he
dat
a. 15 50 Commenton
t
he ki
nd 12 36 of

6 24 r
elat
ionshi
p
bet
ween popul
ati
on
gr
owt
handeconomi
cdevel
opment
.

Sol
uti
on6

Thepoi
ntsseem t
obef
oll
owi
ngal
inewi
thposi
ti
vegr
adi
ent
.Ifyoui
nser
tal
ineof
bestf
itt
hrought
hepoi
nts,
youwi
llseet
hatt
hepoi
ntsdonotdevi
atemuchf
rom t
he
l
ine.Wecant
her
efor
econcl
udet
hatt
her
eisast
rongposi
ti
vel
inearr
elat
ionshi
p

Page16of16
bet
weenpopul
ati
ongr
owt
handeconomi
cdevel
opment
.

3.
2.3TheSi
mpl
eLi
nearRegr
essi
onModel

Thesi
mpl
eli
nearr
egr
essi
onmodeli
sgi
venby

wher
e i
sthedependentvar
iabl
e, i
sthei
ndependentvar
iabl
e, i
stheer
ror
t
erm, and ar
ecal
led popul
ati
on r
egr
essi
on coef
fi
cient
srepr
esent
ing t
he

i
nter
ceptand sl
oper
espect
ivel
y.Themodelr
epr
esent
stheunknownpopul
ati
on
r
elat
ionshi
pbet
weent
het
wovar
iabl
es.Theer
rort
ermi
sincl
udedi
nthemodelt
o
capt
uret
heef
fectofot
heri
mpor
tantvar
iabl
est
hatar
eomi
tt
edi
nthemodel
.

Ther
egr
essi
onequat
ioni
sest
imat
edby:

wher
e

and

i
stheest
imat
edr
egr
essi
onequat
ionconnect
ingvar
iabl
esX andY,
wher
e and ar
eest
imat
esoft
hepopul
ati
oni
nter
cept andpopul
ati
onsl
ope

oft
hel
iner
espect
ivel
y.

Exampl
e7
Est
imat
ether
egr
essi
onequat
ionf
ort
hedat
aofExampl
e6

Sol
uti
on7
Usi
ngt
het
wovar
iabl
est
ati
sti
calmodeonyourcal
cul
ator
,youwi
llobt
aint
he
f
oll
owi
ngr
esul
ts:

Page17of17
 =3.
405405405

=-
1.481981978

Ther
egr
essi
onequat
ioni
s

3.
2.4I
nter
pret
ati
onof and
Thepar
amet
er i
sthei
nter
ceptont
hedependentvar
iabl
e.I
tist
heval
uet
hatt
he
var
iabl
e i
spr
edi
ctedt
oassumei
fthevar
iabl
e hasazer
oval
ue.Youshoul
d
guar
dagai
nsti
nter
pret
ingt
hei
nter
cepti
nter
msoft
hedependentvar
iabl
eift
he
r
angeof -
val
uesusedt
oconst
ructt
hemodeldonoti
ncl
udezer
o.

Thepar
amet
er r
epr
esent
sther
ateofchangeof wi
thr
espectt
o .Thust
he
val
ueof showst
hecor
respondi
ngchangei
ntheval
ueof f
orever
yuni
tchange
i
ntheval
ueof .

I
ntheequat
ionobt
ainedabove,
theval
ue cannotbei
nter
pret
edi
n
t
ermsofeconomi
cdevel
opmentsi
ncet
he val
uesusedt
oconst
ructt
heequat
ion
di
dnoti
ncl
udezer
o.
Theval
ue r
epr
esent
sthecor
respondi
ngi
ncr
easei
nGDPf
orever
y
uni
tincr
easei
nthepopul
ati
ongr
owt
hrat
e.

3.
2.5Est
imat
ingVal
uesoft
heDependentVar
iabl
e
Ther
egr
essi
onmodelcanbeusedt
oest
imat
eval
uesoft
hedependentvar
iabl
e
gi
venval
uesoft
hei
ndependentvar
iabl
e.Agi
venval
ueofXi
ssubst
it
utedi
ntot
he

Page18of18
r
egr
essi
onequat
iont
oobt
aint
hecor
respondi
ngval
ueofY.Themodelcangi
ve
r
eli
abl
eest
imat
esofval
uesofYf
orX-
val
ueswi
thi
nther
angeofval
uesusedt
o
const
ructt
hemodel
.Out
sidet
her
angeofX-
val
uesusedi
ntheconst
ruct
ionoft
he
model
,themodelbecomesunpr
edi
ctabl
eandmaygi
vemi
sleadi
ngr
esul
ts.Fort
his
r
eason,
youshoul
dguar
dagai
nstext
rapol
ati
oni
nregr
essi
onanal
ysi
s.

Exampl
e8
Use t
he modelobt
ained i
n Exampl
e7t
o est
imat
ethe l
evelofeconomi
c
devel
opmenti
fthepopul
ati
onofacount
rygr
owsby10%.

Sol
uti
on8

=US$32.
57207207bi
ll
ion

Page19of19