1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 13, Part A Experimental Design and Analysis of Variance Introduction to Experimental Design and Analysis of Variance Analysis of Variance and the Completely Randomized Design Multiple Comparison Procedures 3 Slide © 2008 Thomson South-Western. All Rights Reserved Statistical studies can be classified as being either experimental or observational. In an experimental study , one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest. In an observational study , no attempt is made to control the factors. Cause-and-effect relationships are easier to establish in experimental studies than in observational studies. An Introduction to Experimental Design and Analysis of Variance Analysis of variance (ANOVA) can be used to analyze the data obtained from experimental or observational studies. 4 Slide © 2008 Thomson South-Western. All Rights Reserved An Introduction to Experimental Design and Analysis of Variance In this chapter three types of experimental designs are introduced. • a completely randomized design • a randomized block design • a factorial experiment
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1S
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All
Rig
hts
Res
erv
ed
Slides by
JOHN
LOUCKS
St. Edward’s
University
2S
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All
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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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rn.
All
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hts
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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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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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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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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
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All
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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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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
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e©
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ho
mso
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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
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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
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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ho
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ou
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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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ho
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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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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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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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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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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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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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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
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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
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e©
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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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n S
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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
lid
e©
200
8 T
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
no
w
if t
her
e is
an
y s
ign
ific
an
t d
iffe
ren
ce i
n
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
.
An
aly
sis
of
var
ian
ce h
as p
rov
ided
stat
isti
cal
evid
ence
to
rej
ect
the
nu
ll
hy
po
thes
is o
f eq
ual
po
pu
lati
on
mea
ns.
Fis
her
’s l
east
sig
nif
ica
nt
dif
fere
nce
(L
SD
) p
roce
du
re
can
be
use
d t
o d
eter
min
e w
her
e th
e d
iffe
ren
ces
occ
ur.
43S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
Fo
r α
= .0
5 an
d n
T-k
= 1
5 –
3 =
12
deg
rees
of
free
do
m, t.0
25 =
2.1
79
LS
D=
+=
217
925
667
15
15
698
..
()
.L
SD=
+=
217
925
667
15
15
698
..
()
.
/2
11
LS
DM
SE
()
ij
tn
nα
=+
MS
E v
alu
e w
asco
mp
ute
d e
arli
er
Fis
her
’s L
SD
Pro
ced
ure
Bas
ed o
n t
he
Tes
t S
tati
stic
xi-xj
44S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�L
SD
fo
r P
lan
ts 1
an
d 2
Fis
her
’s L
SD
Pro
ced
ure
Bas
ed o
n t
he
Tes
t S
tati
stic
xi-xj
•C
on
clu
sio
n
•T
est
Sta
tist
ic
−1
2x
x=
|55
−68
| =
13
Rej
ect H
0if
> 6
.98
−1
2x
x
•R
ejec
tio
n R
ule
µµ
−0
12
: H
µµ
≠1
2:
aH
µµ
≠1
2:
aH
•H
yp
oth
eses
(A
)
Th
e m
ean
nu
mb
er o
f h
ou
rs w
ork
ed a
t P
lan
t 1
is
no
t eq
ual
to t
he
mea
n n
um
ber
wo
rked
at
Pla
nt
2.
45S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�L
SD
fo
r P
lan
ts 1
an
d 3
Fis
her
’s L
SD
Pro
ced
ure
Bas
ed o
n t
he
Tes
t S
tati
stic
xi-xj
•C
on
clu
sio
n
•T
est
Sta
tist
ic
−1
3x
x=
|55
−57
| =
2
Rej
ect H
0if
> 6
.98
−1
3x
x
•R
ejec
tio
n R
ule
µµ
−0
13
: H
µµ
≠1
3:
aH
•H
yp
oth
eses
(B
)
Th
ere
is n
o s
ign
ific
ant
dif
fere
nce
bet
wee
n t
he
mea
n
nu
mb
er o
f h
ou
rs w
ork
ed a
t P
lan
t 1
and
th
e m
ean
nu
mb
er o
f h
ou
rs w
ork
ed a
t P
lan
t 3.
46S
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
hts
Res
erv
ed
�L
SD
fo
r P
lan
ts 2
an
d 3
Fis
her
’s L
SD
Pro
ced
ure
Bas
ed o
n t
he
Tes
t S
tati
stic
xi-xj
•C
on
clu
sio
n
•T
est
Sta
tist
ic
−2
3x
x=
|68
−57
| =
11
Rej
ect H
0if
> 6
.98
−2
3x
x
•R
ejec
tio
n R
ule
µµ
−0
23
: H
µµ
≠2
3:
aH
•H
yp
oth
eses
(C
)
Th
e m
ean
nu
mb
er o
f h
ou
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
lid
e©
200
8 T
ho
mso
n S
ou
th-W
este
rn.
All
Rig
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
lid
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
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