Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
Chi-square: Exampl-e
Observed Outcome
REGION
Centra] Suburbs Rural_
Cltv
PARTY
Repub l i cans 126 61 38 225 143 .4%)Democra ts 11 , 93 69 233 (45 .0%)
Independen ts 19 14 2 j 60 (11 .6%)
2 1 6 1 6 8 r 3 4 s 1 8 ( 1 0 0 2 )
Expected Percentage when there is no refat ionship between REGIONand PARTY
REGION
Centraf Suburbs Rura.l Rowa i ] . \ r:=_:2PARTY
Repu-bf icans
Democrat s
fndependents
4 3 . 4
4 5 . 0
1 1 . 6
1 0 0
Expected Outcome
REGION
Centraf Suburbs Rura.I
Ci tv
R e p u b l i c a n s 9 3 . 8 7 3 . 0 5 8 - 2 2 2 5 ( 4 3 . 4 2 )
D e m o c r a t s 9 1 . 2 7 5 . 5 6 0 . 3 2 3 3 ( 4 5 . 0 % )
I n d e p e n d e n t s 2 5 . 0 f 9 . 5 1 5 . 5 6 0 ( 1 1 . 6 % )
PARTY
2 L 6 1 6 8 L 3 4 s 1 B ( 1 0 0 % )
aLl O61
te6
7l
6xo
f i -g
?7. z
o-E32.. >
-?16,2rl. 0t4
7 , a lZ
I
{
tI
&,tz 1
Wy*trWA, W
'lI
{
t
t l ,rf .s(d-tpe." rihf z7
9r-^
1V*
4s ,4 { r
Observed Outcome
PARTYRepubl icansDemoc rat sfndependent s
REGIONSuburbs Rural_Cent. ral-
L Z O' 7L
L 9
2 2 5 ( 4 3 . 4 * ' )2 3 3 ( 4 s . 0 E )
6 0 ( 1 1 . 6 t )
o_L
Y J
I 4
3 8o v2'7
z ), tt 1 6 8 L34
^,ir >
s 1 8 ( 1 0 0 8 )
a
o, t
0 , ,O o
Oz, Ot,
0.2 On
Os )rt
( l , l tLw}1'
rao 1
tT,-r;TtA/5, 53
ghi*eua1g&S!;_Ex3mple:Question: The following shows the voting pattern by region in one particular year.
REGIONCentral Suburbs .@!eily
PARl]Yl(epublicans 725 61. 38Democrats 71. 93 69Independents 19 74 27
134 518
Concluct a test to find out if voting preference is associatecl with region. Set the level of
2252336 0 '
21e 168
Ho: There is no associationship between the voting Prefernce and region'
,Ha: There is an associationsNp betwrlen the voting prefernce and region'
The statisticai test that can be used to test the above two hypotheses is chi-square test.
We set up "(.
= 0.05 and from Table 5, we reject Ho whenT/'> 9.49, given
significance at 0.05.
Answen We first set up:
d.f.= (c-1) (r-1) = z x2 = 4.t L
cornputarions tor/': ,lf (, +:) _ r I/ L . s , r j .
-to'o uo'1s ' ( hu' U+/'z'*' or'* )
o u,- Q,6lld + ar:tL + -....t?;"t
= --lTT- ?7'L
0 ^ r ' u ,o [ t r l " , , , - r l "
* , "7 " - l
Since 43.665 is >.9.49 we reject Ho.conclusion: There is an ass-ociationship between the voting Patteln region'
Procedures of Statistical Hypothesis Testing Based on Some Observed Data (usually from a
sample)
a) Set up Ho (null hypothesis) to be protected based on the existing wisdom; this also
implies a competing Ha, alternative hypothesis.
b) If Ho is true, what do we expect from the observed data?
c) Choose a risk level, (significance level) -- the risk of making a type I error (that we are
willing to accept)
d) Select an appropriate statistical technique (“test”) for verifying the data
e) Set up a decision rule based on the expected outcome from the sample if Ho is true.
(This involves knowing the characteristics of the sample, etc).
f) Compare the expected outcome under Ho and the observed result.
g) Draw a conclusion: if the expected outcome happens, then retain Ho;
otherwise, reject Ho (indirectly accepting Ha) (this is based on "negative inference")
$ggEE
geSsegsE
$E$E
sEs933c$g$gE
BB
$cgEc5s;s5g5$55$$553g38gE
$$$B$E
5s$EE
sgsgEE
s$ggggjgSB
$$EB
$EsE
$lgeE
Sg
ssr s SS::s33E
EggE
5l5s! st::g
$I
6q
6E
6d
iEt8E.
8.0
aE
o
EE
t.8sE
=@
sE
EE
.
sg€
rd
\o a
!6
F\o
v|r|\o
F
6
N
rq
co ro
r 1
6 N
o
o
o
-.r'r
\6 6
F
\O I
rrt N
iO
O\ cO
F
t r
CO
\0 r|.,1
N
.: C
! O\ 6
F
\O
Ul
h rl
o h
h h
h !
! 9
rt i
r?r 6 rq l,r.n..r
(.) i{ N
c{
c{ (\l
i6\o
v
rvrh
\o6
c)
.|o\ r'r E
NF
.iio
-l\ !
Fo
*a.r o
c co F
\o r)
s i
rr o\ F
er i'r
<
c, 0\
ca F
\o vr rt
:t ..r
c
o
oo
oc
e
c
i d<
idd
d<
i d
c
i d
dd
cjd
d
dc
i
€ €
6 O
\ O.{
vr F
C
=
\O !? \O
O
F. v.| vt\O
€
F
* 0O
al F
..r .{ i
o o
c ro F
F
\o 6
: o\ co I
h !r
(.) at .l
i o
o a
rn6
c{e
{Ne
.rNN
Fro
|arc
{Ni
€
or i.'t\o
o
\..r F
F
.) a
r (. co
6t
?
\o co
i 6
o
+
o
rn
\or|h
r'.a
NN
F
<
o6
\os
6
c
r io
o\o
\ €c
oF
F
\oN
Nc
.|..lalc
{c{F
FF
ir:i
o
o\ H
ri !
co,o
q|\o
o
\ \o vr q
).o
-Q
e.| cO
r'r o\ F
.o !
i co \o !
c.ro o\
io
oo
oo
oo
oo
oo
oo
o
."r F!.i
6
vl\o
Fr r|\o
r.r -
N
v o
r rr
;-i-i;-ia;;,-,t,.j
\o\o
co
6
co ri
F
€
q| h
F
c) \o
No
e-i- i
9 9
9 n
!-1 !_1 !_1 u-.r s rq n
r
o\r
60
!r
.aF
hF
Fca t\
\o \o v) vr t
! !
n !.
t ri
6 rrl
-i -i
; ;
-i .i
;,i ;
a -i
^r -r
-i ^l
H.J
m.a
v) \o F a
o\ 9:
s : g
:pt
= g
R R
3 3
g S
g g
R g
g g
:
tqtN
sg
u.Er -
9C
'6o
T-?4
gv
.l
8s
6s
-ttE{-
ii'6o
1?
.9x
:ts
{ll>
:
.9:
EF
to
vr\o
F@
o\
o
vro
V
)ok
ro,'|o
h
o
ho
r)o
N
ar d r\
cr .i r.r r
{.fjr
h\o
F€
o\o
-c
|6r
t*
rr
rr
FN
cq
NN
N
h3
o-
o
.o..t F
\o
N
\o N
c{
co o\ \o
- (.l
c.r O
o\ h
i c.l N
O
F
.'l
@ al
c'r € o\
O
O
O
\9 -
(.r I
- r)
U U
h ;,'i
+ i'i ;j a
6.i
o\ h
:. F d
F. .., cd
in 6
c{ '.-.-: Q Q
.q e
A i:
1 q
9 n
aq
cl ad
di d
d.d
..i.+ $
F d
r..i +.d
F 6
g $
g g
S 3
$ ; S
X n
H fr g
F g
e :
X S
$F
Fi
-N
Nc
la
rN
60
s
\o
6q
)@
\OO
.O\
\O O
N
N
OF
a
!\OC
OO
O
O
cO
\O.a
Or
ql<
n
lt
!*-
o'r
c
lc{
o
rt
; 6
c. 6
F
ri
oi d
' h
- F
..r co
..r €
N i-
: r'
o
a
Q -
c] <l q
I
q
al I
+
n
q
ol n
r?
'':F
d..i
+G
od
ci.j
di .i.o
od
d..:.i+
r; ito
oo
-N
:t f''o
.oo
\o.r.i
I c
n -
< \o
€o
' ;::;
i -.{.{
N N
N c
! - F
).o ..l.'j i.1
r'i i 't
+
t *
sr !t \t '..)V
)ql9
c-
c
o
Fc
l rt
i F
r
co o\ i co o\ c- F
N
c{ o. n cc o
- o
o\ l'. r'r o\ \t
co <
9 \o €
6 o\ 5
r) @
.i: ('r N
-
; &
6
&
5 co Q
6 io c.r c- d
.b ; r)
c) * -
: i.
o N
.a o
..1 I q
.! v-1 09 9
.'1 a'l 1 c)'1
oq(j d
i - riri<
icd
d
-:l.i + d
ir oic
iri..i $
\oF
c
oo
i.r
rr"v)\o
€
o\o
c
'r\oc
oQ
Ni
ri.)- -
- .;;:;a
rNN
N
N..r c
.r (.|r4..irirn
in.r
+
{ {
<
l {
i +
tl\o
F c
oo
:N
rn
ss
$s
5s
5;s
;s3
s€
33
fi D
:ss
Siq
iiqE
Eq
qfrq
ifici ri
Fl o
i; c.i'*
ri<j
cd o
'.j ^i.d
vi 6
F
co
o
- c{ ro
rr ro c'v1
o
i
N 'i
vt t'- o\ o
F
"r ir
'' '''
- -;;';;;
- N
N c
.l ..l N i\ &
6 ..| i'i i'i ..i 6
iir.i;rra{r|\o
.''<t\o
-''l
F3
RF
xS
sH
ES
n3
53
5;F
eR
i$6
ES
Eq
iqq
SE
iqq
q{.i +'o
F o
i d..i -i + : : : : i S
f X n
F K
R 3
$ g
g n
H S
g 3
6 S
F E
3 5
=
i a
.t - V
h
\O
t-
co
Cr
O
6.\.i
rr h
\O
F
co
d
\ O
F
ci
6
V
r) Q
l_
co
O
\ O
O
O
O
O
O
O
O
- -
- -
- -
:r :
: F
N
i.i
ci cr
cl c.r N
N
N
t'rI
h
\ot''
co
o\
o
Eo
B
ooo@=ofE'6.9-=
ontar io
Prairie
1
a
J
t z 3 + ,
Ra,l x
s+3
sg
Y
Per capitaEnerqvconsiinption , X
l-
5
2
3
Per Cap i taIncome
I
1
3
4
2
5
-vsa(
Ra""k x
Exampre Z nank (]o.rest:t. hiqhest-S)
At lan t ic
Quebec
Ontarr-o
Pra i r ie
fa"k
l 5
+
+3
2
I
rank:" X
6r
+3n
I
t z 3 + 5 6
I z 3 + r b
(,
5
+32
L
L z 3 + r L
/ z 3 4 3 - 6
of the following area, test the hypotheFis
Example 3
a) The following are ten matchedcollected recently. Find out if there is a
of moisture and temp.
Moisture, X Te4P, Y
47
J 5
z a
z t
4 4
5 5
6 0
, )
J - O
9
z )
, ' 7
? n
J J
4 8
4 9
3 2
b) lf the above data represent a randorn
that there isternp In
iemperature ft soitstween the ranks
Accept Ho
Reject Ho
Ho is true
Conect decision
Ho is false
Type II enor
Conect decision
a)
b)
Set up Fl (null hypothesis) (also impli,rs a
If LIo is tnre, what do we expect from
Choose o (siglificance level), the ri a fype I enor (and we are willing tor
accept)
d) Select an appropriate statistical techniqlue
Set gp,a ileoision rule basqd on the e+p€oted
involves knowing the test statistic, its san
a one-tailecl or two-tailed test, and conring
Do ssmercaloulations to see if what
erqectations.
Draw a conclusion: if the expected outoome
(indirectly accepting H") (this is based on
from the sample if Ho is true. (This
and deciding whether it is
with a critic'al vqlue,)
thp sample confofins to ihe
then retain Ho; otherwise, reject ft
inference")
Example 3B (Rank Correlation)
Question: Given before
Solution:
Set up hypotheses:
We shall test Ho against Ha, where
Ho: There is no linear relationship between the ranks of moisture and temperature (r = 0)
Ha: There is a linear relationship between the ranks ofmoisture and temperature (r "0)
Decision rule: Set the sig. level
The test statistic (from the sample) is r.
Reject Ho when r
When d.f.= no ofpairs
Computations
Conclusion: