1S
lid
e©
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ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Slides by
JOHN
LOUCKS
St. Edward’s
University
2S
lid
e©
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ho
mso
n S
ou
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rn.
All
Rig
hts
Res
erv
ed
Ch
apte
r 13
, Par
t A
Ex
per
imen
tal
Des
ign
an
d A
nal
ysi
s o
f V
aria
nce
�In
tro
du
ctio
n t
o E
xp
erim
enta
l D
esig
n
and
An
aly
sis
of
Var
ian
ce
�A
nal
ysi
s o
f V
aria
nce
and
th
e C
om
ple
tely
Ran
do
miz
ed D
esig
n
�M
ult
iple
Co
mp
aris
on
Pro
ced
ure
s
3S
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e©
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8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�S
tati
stic
al s
tud
ies
can
be
clas
sifi
ed a
s b
ein
g e
ith
er
exp
erim
enta
l o
r o
bse
rvat
ion
al.
�In
an
ex
per
imen
tal
stu
dy
, on
e o
r m
ore
fac
tors
are
co
ntr
oll
ed s
o t
ha
t d
ata
can
be
ob
tain
ed a
bo
ut
ho
w t
he
fact
ors
in
flu
ence
th
e v
aria
ble
s o
f in
tere
st.
�In
an
ob
serv
atio
na
l st
ud
y, n
o a
ttem
pt
is m
ade
to
con
tro
l th
e fa
cto
rs.
�C
ause
-an
d-e
ffec
t re
lati
on
ship
sar
e ea
sier
to
est
abli
sh
in e
xp
erim
enta
l st
ud
ies
than
in
ob
serv
atio
nal
stu
die
s.
An
In
tro
du
ctio
n t
o E
xp
erim
enta
l D
esig
nan
d A
nal
ysi
s o
f V
ari
ance
�A
nal
ysi
s o
f v
aria
nce
(A
NO
VA
) ca
n b
e u
sed
to
an
aly
ze
the
da
ta o
bta
ined
fro
m e
xp
erim
enta
l o
r o
bse
rvat
ion
al
stu
die
s.
4S
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e©
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ho
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n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
An
In
tro
du
ctio
n t
o E
xp
erim
enta
l D
esig
nan
d A
nal
ysi
s o
f V
ari
ance
�In
th
is c
ha
pte
r th
ree
typ
es o
f ex
per
imen
tal
des
ign
s ar
e in
tro
du
ced
.
•a
com
ple
tely
ra
nd
om
ized
des
ign
•a
ran
do
miz
ed b
lock
des
ign
•a
fact
ori
al e
xp
erim
ent
5S
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e©
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n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
An
In
tro
du
ctio
n t
o E
xp
erim
enta
l D
esig
nan
d A
nal
ysi
s o
f V
ari
ance
�A
fac
tor
is a
var
iab
le t
hat
th
e ex
per
imen
ter
has
se
lect
ed f
or
inv
esti
ga
tio
n.
�A
tre
atm
ent
is a
lev
el o
f a
fact
or.
�E
xp
erim
enta
l u
nit
sar
e th
e o
bje
cts
of
inte
rest
in
th
e ex
per
imen
t.
�A
co
mp
lete
ly r
an
do
miz
ed d
esig
nis
an
ex
per
imen
tal
des
ign
in
wh
ich
th
e tr
eatm
ents
are
ran
do
mly
as
sig
ned
to
th
e ex
per
imen
tal
un
its.
6S
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e©
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n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
An
aly
sis
of
Var
ian
ce: A
Co
nce
ptu
al O
ver
vie
w
An
aly
sis
of
Var
ian
ce(A
NO
VA
) ca
n b
e u
sed
to
tes
tfo
r th
e eq
ual
ity
of
thre
e o
r m
ore
po
pu
lati
on
mea
ns.
Dat
a o
bta
ined
fro
m o
bse
rvat
ion
al o
r ex
per
imen
tal
stu
die
s ca
n b
e u
sed
fo
r th
e an
aly
sis.
We
wan
t to
use
th
e sa
mp
le r
esu
lts
to t
est
the
foll
ow
ing
hy
po
thes
es:
H0:
µ1
= µ
2 = µ
3 = .
. .
= µ
k
Ha:
No
t al
l p
op
ula
tio
n m
ean
s ar
e eq
ual
7S
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e©
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n S
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rn.
All
Rig
hts
Res
erv
ed
H0:
µ1
= µ
2 = µ
3 = .
. .
= µ
k
Ha:
No
t al
l p
op
ula
tio
n m
ean
s ar
e eq
ual
If H
0is
rej
ecte
d, w
e ca
nn
ot
con
clu
de
that
all
po
pu
lati
on
mea
ns
are
dif
fere
nt.
Rej
ecti
ng
H0
mea
ns
that
at
leas
t tw
o p
op
ula
tio
n
mea
ns
hav
e d
iffe
ren
t v
alu
es.
An
aly
sis
of
Var
ian
ce: A
Co
nce
ptu
al O
ver
vie
w
8S
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e©
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n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
Fo
r ea
ch p
op
ula
tio
n,
the
resp
on
se (
dep
end
ent)
var
iab
le i
s n
orm
all
y d
istr
ibu
ted
.
Th
e v
aria
nce
of
the
resp
on
se v
aria
ble
, den
ote
d σ 2
,is
th
e sa
me
for
all
of
the
po
pu
lati
on
s.
Th
e o
bse
rvat
ion
s m
ust
be
ind
epen
den
t.
�A
ssu
mp
tio
ns
for
An
aly
sis
of
Var
ian
ce
An
aly
sis
of
Var
ian
ce: A
Co
nce
ptu
al O
ver
vie
w
9S
lid
e©
200
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ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�S
amp
lin
g D
istr
ibu
tio
n o
f
Giv
en H
0is
Tru
ex µ
1x 1x
3x 3x
2x 2x
Sam
ple
mea
ns
are
clo
se t
og
eth
erb
ecau
se t
her
e is
on
lyo
ne
sam
pli
ng
dis
trib
uti
on
wh
en H
0is
tru
e.
22 x
nσσ
=2
2 xnσ
σ=
An
aly
sis
of
Var
ian
ce: A
Co
nce
ptu
al O
ver
vie
w
10S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�S
amp
lin
g D
istr
ibu
tio
n o
f
Giv
en H
0is
Fal
sex
µ3
1x 1x
2x 2x
3x 3x
µ1
µ2
Sam
ple
mea
ns
com
e fr
om
dif
fere
nt
sam
pli
ng
dis
trib
uti
on
san
d a
re n
ot
as c
lose
to
get
her
wh
en H
0is
fal
se.
An
aly
sis
of
Var
ian
ce: A
Co
nce
ptu
al O
ver
vie
w
11S
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e©
200
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ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
An
aly
sis
of
Var
ian
ce
�B
etw
een
-Tre
atm
ents
Est
imat
e o
f P
op
ula
tio
n V
ari
an
ce
�W
ith
in-T
reat
men
ts E
stim
ate
of
Po
pu
lati
on
Var
ian
ce
�C
om
par
ing
th
e V
aria
nce
Est
imat
es: T
he F T
est
�A
NO
VA
Tab
le
12S
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e©
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ho
mso
n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
2
1
()
MS
TR
1
k
jj
j
nx
x
k
=
−
=−
∑
Bet
wee
n-T
reat
men
ts E
stim
ate
of
Po
pu
lati
on
Var
ian
ce σ 2
Den
om
ina
tor
is t
he
deg
rees
of
free
do
mas
soci
ated
wit
h S
ST
R
Nu
mer
ato
r is
cal
led
the
sum
of
squ
ares
du
eto
tre
atm
ents
(SS
TR
)
�T
he
esti
ma
te o
f σ 2
bas
ed o
n t
he
var
iati
on
of
the
sam
ple
mea
ns
is c
alle
d t
he
mea
n s
qu
are
du
e to
trea
tmen
tsan
d i
s d
eno
ted
by
MS
TR
.
13S
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ho
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n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
�T
he
esti
ma
te o
f σ 2
bas
ed o
n t
he
var
iati
on
of
the
sam
ple
ob
serv
atio
ns
wit
hin
eac
h s
amp
le i
s ca
lled
th
e m
ean
sq
uar
e er
ror
and
is
den
ote
d b
y M
SE
.
Wit
hin
-Tre
atm
ents
Est
imat
eo
f P
op
ula
tio
n V
aria
nce
σ 2
Den
om
ina
tor
is t
he
deg
rees
of
free
do
mas
soci
ated
wit
h S
SE
Nu
mer
ato
r is
cal
led
the
sum
of
squ
ares
du
e to
err
or
(SS
E)
MS
E=
−∑
−
=
()
ns
nk
jj
jk
T
12
1M
SE=
−∑
−
=
()
ns
nk
jj
jk
T
12
1
14S
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e©
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ho
mso
n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
Co
mp
arin
g t
he
Var
ian
ce E
stim
ates
: Th
e F
Tes
t
�If
th
e n
ull
hy
po
thes
is i
s tr
ue
and
th
e A
NO
VA
assu
mp
tio
ns
are
val
id, t
he
sam
pli
ng
dis
trib
uti
on
of
MS
TR
/M
SE
is
an F
dis
trib
uti
on
wit
h M
ST
R d
.f.
equ
al t
o k
-1
and
MS
E d
.f. e
qu
al t
o n
T-k.
�If
th
e m
ean
s o
f th
e k
po
pu
lati
on
s ar
e n
ot
equ
al, t
he
val
ue
of
MS
TR
/M
SE
wil
l b
e in
flat
ed b
ecau
se M
ST
Ro
ver
esti
mat
es σ 2
.
�H
ence
, we
wil
l re
ject
H0
if t
he
resu
ltin
g v
alu
e o
fM
ST
R/
MS
E a
pp
ears
to
be
too
lar
ge
to h
av
e b
een
sele
cted
at
ran
do
m f
rom
th
e ap
pro
pri
ate F
dis
trib
uti
on
.
15S
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e©
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ho
mso
n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
�S
amp
lin
g D
istr
ibu
tio
n o
f M
ST
R/
MS
E
Do
No
t R
ejec
t H
0
Rej
ect H
0
MS
TR
/M
SE
Cri
tica
l V
alu
e
Fα
Sam
pli
ng
Dis
trib
uti
on
of
MS
TR
/M
SE
α
Co
mp
arin
g t
he
Var
ian
ce E
stim
ates
: Th
e F
Tes
t
16S
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e©
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ho
mso
n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
MS
TR
SS
TR
-=k
1M
ST
RS
ST
R
-=k
1
MS
ES
SE -
=n
kT
MS
ES
SE -
=n
kT
MS
TR
MS
E
MS
TR
MS
E
So
urc
e o
fV
aria
tio
nS
um
of
Sq
uar
esD
egre
es o
fF
reed
om
Mea
nS
qu
are
F
Tre
atm
ents
Err
or
To
tal
k-
1
nT
-1
SS
TR
SS
E
SS
T
nT
-k
SS
T i
s p
arti
tio
ned
into
SS
TR
an
d S
SE
.
SS
T’s
deg
rees
of
free
do
m
(d.f
.) a
re p
arti
tio
ned
in
to
SS
TR
’sd
.f. a
nd
SS
E’s
d.f
.
AN
OV
A T
able
p-V
alu
e
17S
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e©
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n S
ou
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este
rn.
All
Rig
hts
Res
erv
ed
AN
OV
A T
able
SS
T d
ivid
ed b
y i
ts d
egre
es o
f fr
eed
om
nT
–1
is t
he
ov
eral
l sa
mp
le v
aria
nce
th
at
wo
uld
be
ob
tain
ed i
f w
etr
eate
d t
he
enti
re s
et o
f o
bse
rvat
ion
s as
on
e d
ata
set.
Wit
h t
he
enti
re d
ata
set
as o
ne
sam
ple
, th
e fo
rmu
lafo
r co
mp
uti
ng
th
e to
tal
sum
of
squ
ares
, SS
T, i
s:
2
11
SS
T(
)S
ST
RS
SE
jn
k
ijj
i
xx
==
=−
=+
∑∑
18S
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e©
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8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
AN
OV
A T
able
AN
OV
A c
an b
e v
iew
ed a
s th
e p
roce
ss o
f p
arti
tio
nin
gth
e to
tal
sum
of
squ
ares
an
d t
he
deg
rees
of
free
do
min
to t
hei
r co
rres
po
nd
ing
so
urc
es: t
reat
men
ts a
nd
err
or.
Div
idin
g t
he
sum
of
squ
ares
by
th
e ap
pro
pri
ate
deg
rees
of
free
do
m p
rov
ides
th
e v
aria
nce
est
ima
tes
and
th
e F
val
ue
use
d t
o t
est
the
hy
po
thes
is o
f eq
ual
po
pu
lati
on
mea
ns.
19S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Tes
t fo
r th
e E
qu
alit
y o
f k
Po
pu
lati
on
Mea
ns
F=
MS
TR
/M
SE
H0:
µ1
= µ
2 = µ
3 = .
. .
= µ
k
Ha:
No
t al
l p
op
ula
tio
n m
ean
s ar
e eq
ual
�H
yp
oth
eses
�T
est
Sta
tist
ic
20S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Tes
t fo
r th
e E
qu
alit
y o
f k
Po
pu
lati
on
Mea
ns
�R
ejec
tio
n R
ule
wh
ere
the
val
ue
of Fα is
bas
ed o
n a
nF
dis
trib
uti
on
wit
h k
-1
nu
mer
ato
r d
.f.
and
nT
-k
den
om
ina
tor
d.f
.
Rej
ect H
0if
p-v
alu
e <α
p-v
alu
e A
pp
roac
h:
Cri
tica
l V
alu
e A
pp
roac
h:
Rej
ect H
0if
F>Fα
21S
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e©
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ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Au
toS
hin
e, I
nc.
is
con
sid
erin
g m
ark
etin
g a
lo
ng
-
last
ing
car
wax
. T
hre
e d
iffe
ren
t w
axes
(T
yp
e 1,
Ty
pe
2,
and
Ty
pe
3)
hav
e b
een
dev
elo
ped
.
�E
xam
ple
: A
uto
Sh
ine,
In
c.
In o
rder
to
tes
t th
e d
ura
bil
ity
of
thes
e w
axes
, 5 n
ew c
ars
wer
e
wax
ed w
ith
Ty
pe
1, 5
wit
h T
yp
e
2, a
nd
5 w
ith
Ty
pe
3. E
ach
car
was
th
en
rep
eate
dly
ru
n t
hro
ug
h a
n a
uto
mat
ic c
arw
ash
un
til
the
wax
co
atin
g s
ho
wed
sig
ns
of
det
erio
rati
on
.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
22S
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e©
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ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Th
e n
um
ber
of
tim
es e
ach
car
wen
t th
rou
gh
th
e
carw
ash
bef
ore
its
wax
det
erio
rate
d i
s sh
ow
n o
n t
he
nex
t sl
ide.
A
uto
Sh
ine,
In
c. m
ust
dec
ide
wh
ich
wax
to m
ark
et.
Are
th
e th
ree
wax
es
equ
ally
eff
ecti
ve?
�E
xam
ple
: A
uto
Sh
ine,
In
c.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
Fac
tor
. . .
C
ar w
ax
Tre
atm
ents
. .
. T
yp
e I,
Ty
pe
2, T
yp
e 3
Ex
per
imen
tal
un
its
. .
. C
ars
Res
po
nse
var
iab
le
. . .
Nu
mb
er o
f w
ash
es
23S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
1 2 3 4 5
27 30 29 28 31
33 28 31 30 30
29 28 30 32 31
Sam
ple
Mea
n
Sam
ple
Va
rian
ce
Ob
serv
atio
nW
axT
yp
e 1
Wax
Ty
pe
2W
axT
yp
e 3
2.5
3.3
2.5
29.0
30.4
30.0
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
24S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�H
yp
oth
eses
wh
ere:
µ1 =
mea
n n
um
ber
of
was
hes
usi
ng
Ty
pe
1 w
ax
µ2 =
mea
n n
um
ber
of
was
hes
usi
ng
Ty
pe
2 w
ax
µ3 =
mea
n n
um
ber
of
was
hes
usi
ng
Ty
pe
3 w
ax
H0:
µ1
= µ
2 = µ
3
Ha:
No
t al
l th
e m
ean
s ar
e eq
ual
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
25S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Bec
ause
th
e sa
mp
le s
izes
are
all
eq
ual
:
MS
E =
33.
2/(1
5 -
3) =
2.7
7
MS
TR
= 5
.2/
(3 -
1) =
2.6
SS
E =
4(2
.5)
+ 4
(3.3
) +
4(2
.5)
= 3
3.2
SS
TR
= 5
(29
–29.
8)2
+ 5
(30.
4–2
9.8)
2+
5(3
0–29
.8)2
= 5
.2
�M
ean
Sq
ua
re E
rro
r
�M
ean
Sq
ua
re B
etw
een
Tre
atm
ents
=+
+1
23
()/
3x
xxx
=+
+1
23
()/
3x
xxx
= (
29 +
30.
4 +
30)
/3
= 2
9.8
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
26S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�R
ejec
tio
n R
ule
wh
ere F
.05
= 3
.89
is b
ased
on
an
Fd
istr
ibu
tio
nw
ith
2 n
um
erat
or
deg
rees
of
free
do
m a
nd
12
den
om
inat
or
deg
rees
of
free
do
m
p-V
alu
e A
pp
roac
h:
Rej
ect H
0if
p-v
alu
e <
.05
Cri
tica
l V
alu
e A
pp
roac
h:
Rej
ect H
0if
F>
3.89
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
27S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�T
est
Sta
tist
ic
Th
ere
is i
nsu
ffic
ien
t ev
iden
ce t
o c
on
clu
de
that
the
mea
n n
um
ber
of
was
hes
fo
r th
e th
ree
wax
typ
es a
re n
ot
all
the
sam
e.
�C
on
clu
sio
nF =
MS
TR
/M
SE
= 2
.60/
2.77
= .9
39
Th
e p-
val
ue
is g
reat
er t
ha
n .1
0, w
her
e F
= 2
.81.
(Ex
cel
pro
vid
es a
p-v
alu
e o
f .4
2.)
Th
eref
ore
, we
can
no
t re
ject
H0.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
28S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
So
urc
e o
fV
aria
tio
nS
um
of
Sq
uar
esD
egre
es o
fF
reed
om
Mea
nS
qu
ares
F
Tre
atm
ents
Err
or
To
tal
2 14
5.2
33.2
38.4
12
2.60
2.77
.939
�A
NO
VA
Tab
le
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
A C
om
ple
tely
Ran
do
miz
ed E
xp
erim
enta
l D
esig
n
p-V
alu
e
.42
29S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�E
xam
ple
: R
eed
Man
ufa
ctu
rin
g
Jan
et R
eed
wo
uld
lik
e to
kn
ow
if
ther
e is
an
y s
ign
ific
ant
dif
fere
nce
in
the
mea
n n
um
ber
of
ho
urs
wo
rked
per
wee
k f
or
the
dep
artm
ent
ma
nag
ers
at h
er t
hre
e m
anu
fact
uri
ng
pla
nts
(in
Bu
ffal
o, P
itts
bu
rgh
, an
d D
etro
it).
An
Fte
st w
ill
be
con
du
cted
usi
ng
α=
.05.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
30S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�E
xam
ple
: R
eed
Man
ufa
ctu
rin
g
A s
imp
le r
and
om
sam
ple
of
fiv
e
man
ager
s fr
om
eac
h o
f th
e th
ree
pla
nts
was
tak
en a
nd
th
e n
um
ber
of
ho
urs
wo
rked
by
eac
h m
ana
ger
in
th
e
pre
vio
us
wee
k i
s sh
ow
n o
n t
he
nex
t
slid
e.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
Fac
tor
. . .
M
anu
fact
uri
ng
pla
nt
Tre
atm
ents
. .
. B
uff
alo
, Pit
tsb
urg
h, D
etro
it
Ex
per
imen
tal
un
its
. .
. M
ana
ger
s
Res
po
nse
var
iab
le
. . .
Nu
mb
er o
f h
ou
rs w
ork
ed
31S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
1 2 3 4 5
48 54 57 54 62
73 63 66 64 74
51 63 61 54 56
Pla
nt
1B
uff
alo
Pla
nt
2P
itts
bu
rgh
Pla
nt
3D
etro
itO
bse
rvat
ion
Sam
ple
Mea
n
Sam
ple
Va
rian
ce
5568
57
26.0
26.5
24.5
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
32S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
H0:
µ 1 = µ 2 = µ 3
Ha:
No
t al
l th
e m
ean
s ar
e eq
ual
wh
ere:
µ 1
= m
ean
nu
mb
er o
f h
ou
rs w
ork
ed p
erw
eek
by
th
e m
ana
ger
s at
Pla
nt
1µ 2
= m
ean
nu
mb
er o
f h
ou
rs w
ork
ed p
erw
eek
by
th
e m
anag
ers
at
Pla
nt
2 µ 3
= m
ean
nu
mb
er o
f h
ou
rs w
ork
ed p
erw
eek
by
th
e m
anag
ers
at
Pla
nt
3
1. D
evel
op
th
e h
yp
oth
eses
.
�p
-Val
ue
and
Cri
tica
l V
alu
e A
pp
roac
hes
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
33S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
2. S
pec
ify
th
e le
vel
of
sig
nif
ican
ce.
α =
.05
�p
-Val
ue
and
Cri
tica
l V
alu
e A
pp
roac
hes
3. C
om
pu
te t
he
val
ue
of
the
test
sta
tist
ic.
MS
TR
= 4
90/
(3 -
1) =
2
45
SS
TR
= 5
(55
-60
)2+
5(6
8 -
60)2
+ 5
(57
-60
)2=
490
= (
55 +
68
+ 5
7)/
3 =
60
xx(Sam
ple
siz
es a
re a
ll e
qu
al.)
Mea
n S
qu
are
Du
e to
Tre
atm
ents
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
34S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
3. C
om
pu
te t
he
val
ue
of
the
test
sta
tist
ic.
MS
E =
308
/(1
5 -
3) =
2
5.66
7
SS
E =
4(2
6.0)
+ 4
(26.
5) +
4(2
4.5)
= 3
08
Mea
n S
qu
are
Du
e to
Err
or
(co
n’t
.)
F=
MS
TR
/M
SE
= 2
45/
25.6
67 =
9.
55
�p
-Val
ue
and
Cri
tica
l V
alu
e A
pp
roac
hes
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
35S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Tre
atm
ent
Err
or
To
tal
490
308
798
2 12 14
245
25.6
67
So
urc
e o
fV
aria
tio
nS
um
of
Sq
uar
esD
egre
es o
fF
reed
om
Mea
nS
qu
are
9.55F
�A
NO
VA
Tab
le
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
p-V
alu
e
.003
3
36S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
5. D
eter
min
e w
het
her
to
rej
ect H
0.
We
hav
e su
ffic
ien
t ev
iden
ce t
o c
on
clu
de
that
th
e m
ean
nu
mb
er o
f h
ou
rs w
ork
ed p
er w
eek
by
d
epar
tmen
t m
ana
ger
s is
no
t th
e sa
me
at a
ll 3
pla
nt.
Th
e p-
val
ue
<.0
5,so
we
reje
ct H
0.
Wit
h 2
nu
mer
ato
r d
.f. a
nd
12
den
om
ina
tor
d.f
.,th
e p-
val
ue
is .0
1 fo
r F
= 6
.93.
T
her
efo
re, t
he
p-v
alu
e is
les
s th
an .0
1 f
or F
= 9
.55.
�p
–Val
ue
Ap
pro
ach
4. C
om
pu
te t
he p
–va
lue.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
37S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
5. D
eter
min
e w
het
her
to
rej
ect H
0.
Bec
ause
F=
9.5
5 >
3.89
, we
reje
ct H
0.
�C
riti
cal
Val
ue
Ap
pro
ach
4. D
eter
min
e th
e cr
itic
al v
alu
e an
d r
ejec
tio
n r
ule
.
Rej
ect H
0if
F>
3.89
We
hav
e su
ffic
ien
t ev
iden
ce t
o c
on
clu
de
that
th
e m
ean
nu
mb
er o
f h
ou
rs w
ork
ed p
er w
eek
by
d
epar
tmen
t m
ana
ger
s is
no
t th
e sa
me
at a
ll 3
pla
nt.
Bas
ed o
n a
n F
dis
trib
uti
on
wit
h 2
nu
mer
ato
rd
.f. a
nd
12
den
om
inat
or
d.f
., F
.05
= 3
.89.
Tes
tin
g f
or
the
Eq
ual
ity
of k
Po
pu
lati
on
Mea
ns:
An
Ob
serv
atio
nal
Stu
dy
38S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Mu
ltip
le C
om
par
iso
n P
roce
du
res
�S
up
po
se t
hat
an
aly
sis
of
var
ian
ce h
as p
rov
ided
st
atis
tica
l ev
iden
ce t
o r
ejec
t th
e n
ull
hy
po
thes
is o
f eq
ual
po
pu
lati
on
mea
ns.
�F
ish
er’s
lea
st s
ign
ific
an
t d
iffe
ren
ce (
LS
D)
pro
ced
ure
ca
n b
e u
sed
to
det
erm
ine
wh
ere
the
dif
fere
nce
s o
ccu
r.
39S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Fis
her
’s L
SD
Pro
ced
ure
11
MS
E(
)
ij
ij
xx
t
nn
−=
+
�T
est
Sta
tist
ic
�H
yp
oth
eses
µµ
−0
: i
jH
µµ
≠:
ai
jH
40S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Fis
her
’s L
SD
Pro
ced
ure
wh
ere
the
val
ue
of t a
/2 is
bas
ed o
n a
td
istr
ibu
tio
n w
ith
nT
-k
deg
rees
of
free
do
m.
�R
ejec
tio
n R
ule
Rej
ect H
0if
p-v
alu
e <α
p-v
alu
e A
pp
roac
h:
Cri
tica
l V
alu
e A
pp
roac
h:
Rej
ect H
0if
t<
-t a
/2 o
r t
> ta/
2
41S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�T
est
Sta
tist
ic
Fis
her
’s L
SD
Pro
ced
ure
Bas
ed o
n t
he
Tes
t S
tati
stic
xi-xj
__
/2
11
LS
DM
SE
()
ij
tn
nα
=+
wh
ere
−i
jx
x
Rej
ect H
0if
> L
SD
−i
jx
x
�H
yp
oth
eses
�R
ejec
tio
n R
ule
µµ
−0
: i
jH
µµ
≠:
ai
jH
42S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Fis
her
’s L
SD
Pro
ced
ure
Bas
ed o
n t
he
Tes
t S
tati
stic
xi-xj
�E
xam
ple
: R
eed
Man
ufa
ctu
rin
g
Rec
all
that
Jan
et R
eed
wan
ts t
o k
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w
if t
her
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an
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ific
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t d
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the
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ma
nag
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at h
er t
hre
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fact
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pla
nts
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An
aly
sis
of
var
ian
ce h
as p
rov
ided
stat
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cal
evid
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to
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ect
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nu
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hy
po
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ual
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Fis
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east
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her
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ces
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ur.
43S
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All
Rig
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Res
erv
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Fo
r α
= .0
5 an
d n
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deg
rees
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do
m, t.0
25 =
2.1
79
LS
D=
+=
217
925
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15
15
698
..
()
.L
SD=
+=
217
925
667
15
15
698
..
()
.
/2
11
LS
DM
SE
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ij
tn
nα
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MS
E v
alu
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Fis
her
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Pro
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Bas
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All
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�L
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on
clu
sio
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est
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tist
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−1
2x
x=
|55
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Rej
ect H
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x
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12
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)
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t eq
ual
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he
mea
n n
um
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wo
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nt
2.
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All
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�L
SD
fo
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lan
ts 1
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d 3
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her
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SD
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Bas
ed o
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xi-xj
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Rej
ect H
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n R
ule
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oth
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)
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ific
ant
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fere
nce
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wee
n t
he
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n
nu
mb
er o
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t P
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e m
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nu
mb
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rs w
ork
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t P
lan
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n S
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All
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�L
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xi-xj
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on
clu
sio
n
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est
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3x
x=
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| =
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Rej
ect H
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.98
−2
3x
x
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ejec
tio
n R
ule
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−0
23
: H
µµ
≠2
3:
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yp
oth
eses
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)
Th
e m
ean
nu
mb
er o
f h
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rs w
ork
ed a
t P
lan
t 2
is
no
t eq
ual
to t
he
mea
n n
um
ber
wo
rked
at
Pla
nt
3.
47S
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200
8 T
ho
mso
n S
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rn.
All
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hts
Res
erv
ed
�T
he
exp
erim
ent-
wis
e T
yp
e I
erro
r ra
te g
ets
larg
er f
or
pro
ble
ms
wit
h m
ore
po
pu
lati
on
s (l
arg
er k
).
Ty
pe
I E
rro
r R
ates
αE
W=
1 –
(1 –α
)(k –
1)!
�T
he
com
par
iso
n-w
ise
Ty
pe
I er
ror
rateα
ind
icat
es
the
lev
el o
f si
gn
ific
ance
ass
oci
ated
wit
h a
sin
gle
p
airw
ise
com
par
iso
n.
�T
he
exp
erim
ent-
wis
e T
yp
e I
erro
r ra
teα
EW
is t
he
pro
bab
ilit
y o
f m
akin
g a
Ty
pe
I er
ror
on
at
leas
t o
ne
of
the
(k–
1)!
pai
rwis
eco
mp
aris
on
s.
48S
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e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
En
d o
f C
hap
ter
13, P
art
A
