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Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c.
Chap 8
-1
Ch
ap
ter
8
Confide
nce I
nte
rval E
stim
ation
Busin
ess S
tatistics:
A F
irst C
ou
rse
5th
Ed
itio
n
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-2
Lea
rnin
g O
bje
ctives
In t
his
ch
ap
ter,
yo
u l
ea
rn:
�T
o c
onstr
uct and inte
rpre
t confidence inte
rval estim
ate
s
for
the m
ean a
nd t
he p
roport
ion
�H
ow
to d
ete
rmin
e t
he s
am
ple
siz
e n
ecessary
to
deve
lop a
confid
ence
inte
rval fo
r th
e m
ean o
r
pro
port
ion
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-3
Cha
pte
r O
utlin
e
Co
nte
nt
of
this
ch
ap
ter
�C
onfidence Inte
rvals
for
the P
opula
tion
Mean, µ
�w
hen P
opu
lation S
tan
dard
Devia
tion σ
is K
now
n
�w
hen P
opu
lation S
tan
dard
Devia
tion σ
is U
nkno
wn
�C
onfidence Inte
rvals
for
the P
opula
tion
Pro
port
ion, π
�D
ete
rmin
ing the R
equired S
am
ple
Siz
e
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-4
Poin
t and I
nte
rval E
stim
ate
s
�A
po
int
estim
ate
is a
sin
gle
nu
mb
er
�a
co
nfid
en
ce
in
terv
alp
rovid
es a
dd
itio
na
l in
form
atio
n a
bo
ut
the
va
ria
bili
ty o
f th
e e
stim
ate
Po
int
Esti
mate
Lo
wer
Co
nfi
de
nce
Lim
it
Up
pe
r
Co
nfi
de
nce
Lim
it
Wid
th o
f co
nfi
den
ce i
nte
rval
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-5
We c
an e
stim
ate
a
Pop
ula
tio
n P
ara
mete
r …
Poin
t E
stim
ate
s
with a
Sam
ple
Sta
tistic
(a P
oin
t E
stim
ate
)
Me
an
Pro
po
rtio
np
π
Xµ
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-6
Confid
ence I
nte
rvals
�H
ow
mu
ch
un
ce
rta
inty
is a
sso
cia
ted
with
a
po
int
estim
ate
of
a p
op
ula
tio
n p
ara
me
ter?
�A
n in
terv
al e
stim
ate
pro
vid
es m
ore
in
form
atio
n a
bo
ut
a p
op
ula
tio
n c
ha
racte
ristic
tha
n d
oe
s a
po
int e
stim
ate
�S
uch
in
terv
al e
stim
ate
s a
re c
alle
d c
on
fid
en
ce
in
terv
als
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-7
Confid
ence I
nte
rval E
stim
ate
�A
n inte
rval giv
es a
range
of valu
es:
�T
ake
s in
to c
on
sid
era
tio
n v
ari
atio
n in
sa
mp
le
sta
tistics fro
m s
am
ple
to
sam
ple
�B
ase
d o
n o
bse
rva
tio
ns f
rom
1 s
am
ple
�G
ive
s in
form
atio
n a
bo
ut
clo
se
ne
ss t
o
un
kn
ow
n p
op
ula
tio
n p
ara
me
ters
�S
tate
d in
te
rms o
f le
ve
l o
f con
fid
en
ce
�e.g
. 95%
confid
ent,
99%
confid
ent
�C
an n
ever
be 1
00
% c
onfid
ent
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-8
Confid
ence I
nte
rval E
xam
ple
Cere
al fill
exam
ple
�P
opula
tio
n h
as µ
= 3
68 a
nd σ
= 1
5.
�If y
ou t
ake a
sam
ple
of siz
e n
= 2
5 y
ou k
now
�3
68
±1
.96
* 1
5 /
=
(3
62
.12
, 3
73
.88
) co
nta
ins 9
5%
of
the
sa
mp
le m
ea
ns
�W
he
n y
ou
do
n’t k
no
w µ
, yo
u u
se
X t
o e
stim
ate
µ�
If X
= 3
62
.3 t
he
in
terv
al is
36
2.3
±1
.96
* 1
5 /
= (
35
6.4
2,
36
8.1
8)
�S
ince
35
6.4
2 ≤
µ≤
36
8.1
8,
the
in
terv
al b
ase
d o
n th
is s
am
ple
ma
ke
s a
co
rre
ct sta
tem
en
t a
bo
ut µ
.
But
what
about
the inte
rvals
fro
m o
ther
possib
le s
am
ple
s
of
siz
e 2
5?
25
25
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-9
Confid
ence I
nte
rval E
xam
ple
(continu
ed)
Sam
ple
#X
Lo
wer
Lim
it
Up
per
Lim
it
Co
nta
in
µ?
1362.3
0356.4
2368.1
8Y
es
2369.5
0363.6
2375.3
8Y
es
3360.0
0354.1
2365.8
8N
o
4362.1
2356.2
4368.0
0Y
es
5373.8
8368.0
0379.7
6Y
es
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-10
Confid
ence I
nte
rval E
xam
ple
�In
pra
ctice
yo
u o
nly
ta
ke o
ne
sa
mp
le o
f siz
e n
�In
pra
ctice
yo
u d
o n
ot
kn
ow
µso
yo
u d
o n
ot
kn
ow
if
the
in
terv
al a
ctu
ally
co
nta
ins µ
�H
ow
eve
r yo
u d
o k
no
w t
ha
t 9
5%
of
the in
terv
als
form
ed
in
th
is m
an
ne
r w
ill c
on
tain
µ
�T
hu
s,
ba
se
d o
n t
he
on
e s
am
ple
, yo
u a
ctu
ally
se
lecte
d y
ou
ca
n b
e 9
5%
co
nfid
en
t yo
ur
inte
rva
l
will
co
nta
in µ
(th
is is a
95
% c
on
fid
en
ce
in
terv
al)
(continu
ed)
No
te: 9
5%
co
nfid
ence
is b
ase
d o
n the
fa
ct th
at w
e u
se
d Z
= 1
.96
.
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-11
Estim
ation P
rocess
(mean
, µ
, is
u
nkn
ow
n)
Po
pu
latio
n
Ra
nd
om
Sa
mp
le
Mean
X =
50
Sam
ple
I am
95
%
co
nfi
den
t th
at
µis
betw
een
40 &
60.
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-12
Gen
era
l F
orm
ula
�T
he g
enera
l fo
rmula
for
all
confidence
inte
rvals
is:
Po
int
Es
tim
ate
±(C
riti
ca
l V
alu
e)(
Sta
nd
ard
Err
or)
Wh
ere
:•P
oin
t E
stim
ate
is the
sa
mple
sta
tistic e
stim
atin
g th
e p
op
ula
tio
n
pa
ram
ete
r o
f in
tere
st
•Cri
tica
l V
alu
eis
a tab
le v
alu
e b
ased
on th
e s
am
plin
g
dis
trib
utio
n o
f th
e p
oin
t estim
ate
an
d th
e d
esire
d c
on
fid
ence
le
ve
l
•Sta
nd
ard
Err
or
is th
e s
tand
ard
de
via
tion
of th
e p
oin
t e
stim
ate
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-13
Confid
ence L
evel
�C
onfidence L
evel
�T
he c
onfidence that th
e inte
rval
will
conta
in the u
nknow
n
popula
tion p
ara
mete
r
�A
perc
enta
ge (
less than 1
00%
)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-14
Confid
ence L
evel, (
1-α
)
�S
up
po
se
co
nfid
en
ce
le
ve
l =
95
%
�A
lso
wri
tte
n (
1 -
α)
= 0
.95
, (s
o α
= 0
.05
)
�A
re
lative
fre
qu
en
cy in
terp
reta
tio
n:
�95%
of
all
the c
onfid
ence inte
rva
ls t
hat
can b
e
constr
ucte
d w
ill c
onta
in t
he u
nkn
ow
n t
rue
para
mete
r
�A
sp
ecific
in
terv
al e
ith
er
will
co
nta
in o
r w
ill
no
t co
nta
in t
he
tru
e p
ara
me
ter
�N
o p
rob
ab
ility
involv
ed in a
specific
inte
rval(c
on
tin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-15
Confid
ence I
nte
rvals
Po
pu
lati
on
M
ean σ
Un
kn
ow
n
Co
nfi
den
ce
Inte
rvals
Po
pu
lati
on
Pro
po
rtio
n
σK
no
wn
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-16
Confid
ence I
nte
rval fo
r µ
(σK
now
n)
�A
ssum
ptio
ns
�P
op
ula
tio
n s
tan
dard
de
via
tio
n σ
is k
no
wn
�P
op
ula
tio
n is n
orm
ally
dis
trib
ute
d�
If p
opu
latio
n is n
ot n
orm
al, u
se
larg
e s
am
ple
�C
on
fid
ence
inte
rva
l estim
ate
:
wh
ere
is t
he
po
int
estim
ate
Zα
/2is
th
e n
orm
al d
istr
ibu
tio
n c
ritica
l valu
e f
or
a p
rob
ab
ility
of
α/2
in
ea
ch
ta
il
is t
he
sta
nd
ard
err
or
nσ/2
ZX
α±
X nσ
/
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-17
Fin
din
g t
he C
ritical V
alu
e,
Zα
/2
�C
onsid
er
a 9
5%
confidence inte
rval:
Zα
/2=
-1.9
6Zα
/2=
1.9
6
0.0
5
so 0
.95
1=
=−
αα
0.0
25
2=
α0.0
25
2=
α
Po
int
Es
tim
ate
Lo
we
r C
on
fid
en
ce
L
imit
Up
pe
rC
on
fid
en
ce
L
imit
Z u
nit
s:
X u
nit
s:
Po
int
Es
tim
ate
0
1.9
6/2
Z±
=α
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-18
Co
mm
on L
evels
of
Confid
ence
�C
om
mo
nly
use
d c
on
fid
en
ce
le
ve
ls a
re 9
0%
,
95
%,
an
d 9
9%
Co
nfi
de
nce
Le
ve
l
Co
nfi
de
nce
Co
eff
icie
nt,
Zα
/2v
alu
e
1.2
8
1.6
45
1.9
6
2.3
3
2.5
8
3.0
8
3.2
7
0.8
0
0.9
0
0.9
5
0.9
8
0.9
9
0.9
98
0.9
99
80
%
90
%
95
%
98
%
99
%
99
.8%
99
.9%
α−
1
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-19
µµ
x=
Inte
rva
ls a
nd L
evel of
Con
fidence
Co
nfid
ence
In
terv
als
Inte
rva
ls
exte
nd
fro
m
to
(1-α
)x10
0%
of
inte
rvals
constr
ucte
d
conta
in µ
;
(α)x
10
0%
do
not.
Sam
plin
g D
istr
ibution
of th
e M
ean
nσ2
/α
ZX
−
nσ2
/α
ZX
+
x
x1
x2
/2α
/2α
α−
1
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-20
Exam
ple
�A
sa
mp
le o
f 1
1 c
ircu
its f
rom
a la
rge
no
rma
l
po
pu
latio
n h
as a
me
an
re
sis
tan
ce
of 2
.20
o
hm
s.
We
kn
ow
fro
m p
ast
testin
g t
ha
t th
e
po
pu
latio
n s
tan
da
rd d
evia
tio
n is 0
.35
oh
ms.
�D
ete
rmin
e a
95
% c
on
fid
en
ce
in
terv
al fo
r th
e
tru
e m
ea
n r
esis
tan
ce
of th
e p
op
ula
tio
n.
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-21
2.4
06
8
1.9
93
2
0.2
06
8
2
.20
)11
(0.3
5/
1.9
6
2
.20
nσ/2
ZX
≤≤
±=
±=
±
µ
α
Exam
ple
�A
sa
mp
le o
f 1
1 c
ircu
its f
rom
a la
rge
no
rma
l
po
pu
latio
n h
as a
me
an
re
sis
tan
ce
of 2
.20
o
hm
s.
We
kn
ow
fro
m p
ast
testin
g t
ha
t th
e
po
pu
latio
n s
tan
da
rd d
evia
tio
n is 0
.35
oh
ms.
�S
olu
tion:
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-22
Inte
rpre
tation
�W
e a
re 9
5%
confident th
at th
e tru
e m
ean
resis
tance is b
etw
een 1
.9932 and 2.4
068
ohm
s
�A
lthough the tru
e m
ean m
ay o
r m
ay n
ot be
in this
inte
rval, 9
5%
of in
terv
als
form
ed in
this
manner
will
conta
in the tru
e m
ean
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-23
Confid
ence I
nte
rvals
Po
pu
lati
on
M
ean σ
Un
kn
ow
n
Co
nfi
den
ce
Inte
rvals
Po
pu
lati
on
Pro
po
rtio
n
σK
no
wn
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-24
Do Y
ou E
ver
Tru
ly K
now
σ?
�P
roba
bly
not!
�In
virtu
ally
all
rea
l w
orl
d b
usin
ess s
ituatio
ns, σ
is n
ot
kno
wn.
�If there
is a
situation w
here
σis
kno
wn t
hen µ
is a
lso
kno
wn (
sin
ce t
o c
alc
ula
te σ
you n
eed t
o k
now
µ.)
�If y
ou tru
ly k
no
w µ
there
wou
ld b
e n
o n
ee
d to g
ath
er
a
sam
ple
to e
stim
ate
it.
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-25
�If the p
opula
tion s
tandard
devia
tion σ
is
unknow
n, w
e c
an s
ubstitu
te the s
am
ple
sta
ndard
devia
tion, S
�T
his
intr
oduces e
xtr
a u
ncert
ain
ty, sin
ce
S is
variable
fro
m s
am
ple
to s
am
ple
�S
o w
e u
se the t d
istr
ibution
inste
ad o
f th
e
norm
al dis
trib
ution
Confid
ence I
nte
rval fo
r µ
(σU
nknow
n)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-26
�A
ssu
mp
tio
ns
�P
op
ula
tio
n s
tandard
devia
tio
n is u
nkn
ow
n
�P
op
ula
tio
n is n
orm
ally
dis
trib
ute
d
�If p
opula
tio
n is n
ot
norm
al, u
se la
rge s
am
ple
�U
se
Stu
de
nt’s t
Dis
trib
utio
n
�C
on
fid
en
ce
In
terv
al E
stim
ate
:
(wh
ere
tα
/2is
the c
ritical valu
e o
f th
e t
dis
trib
ution w
ith n
-1 d
egre
es
of
freedom
and
an a
rea o
f α
/2 in e
ach t
ail)
Confid
ence I
nte
rval fo
r µ
(σU
nknow
n)
nSt
X2
/α
±
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-27
Stu
de
nt’s t
Dis
trib
ution
�T
he t
is a
fa
mily
of
dis
trib
ution
s
�T
he tα
/2valu
e d
epe
nds o
n d
egre
es o
f fr
ee
do
m (
d.f
.)
�N
um
ber
of ob
serv
atio
ns th
at a
re fre
e to
va
ry a
fte
r sam
ple
mean
ha
s b
een
ca
lcu
late
d
d.f
. =
n -
1
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-28
If the m
ean o
f th
ese t
hre
e
valu
es is 8
.0,
then X
3m
ust be 9
(i.e
., X
3is
not fr
ee to v
ary
)
Deg
rees o
f F
ree
do
m (
df)
He
re, n
= 3
, so
de
gre
es o
f fr
eed
om
=
n –
1 =
3 –
1 =
2
(2 v
alu
es c
an b
e a
ny n
um
be
rs, bu
t th
e thir
d is n
ot fr
ee
to
va
ry
for
a g
ive
n m
ea
n)
Idea:
Num
ber
of
observ
ations t
hat
are
fre
e to v
ary
aft
er
sam
ple
mea
n h
as b
een c
alc
ula
ted
Exam
ple
:S
up
pose t
he m
ean o
f 3 n
um
bers
is 8
.0
Le
t X
1=
7
Le
t X
2=
8
Wh
at is
X3?
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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Chap 8
-29
Stu
de
nt’s t
Dis
trib
ution
t0
t (
df =
5)
t (
df =
13)
t-d
istr
ibutio
ns a
re b
ell-
sh
ap
ed
an
d s
ym
me
tric
, b
ut
ha
ve
‘fa
tter’
tails
th
an
th
e
no
rma
l
Sta
nd
ard
N
orm
al
(t w
ith
df =
∞)
No
te:
t
Z
as
n
incre
ase
s
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-30
Stu
de
nt’s t
Table
Up
pe
r T
ail
Are
a
df
.25
.10
.05
11
.00
03
.07
86
.31
4
20
.81
71
.88
62
.92
0
30
.76
51
.63
82
.35
3
t0
2.9
20
Th
e b
od
y o
f th
e tab
le
co
nta
ins t v
alu
es, n
ot
pro
ba
bili
tie
s
Let:
n =
3
df
= n
-1 =
2
α=
0.1
0α
/2 =
0.0
5
α/2
= 0
.05
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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Chap 8
-31
Sele
cte
d t
dis
trib
ution v
alu
es
With
co
mp
ari
so
n t
o t
he
Z v
alu
e
Co
nfi
den
ce t
t
t
Z
Level
(10 d
.f.)
(20 d
.f.)
(30 d
.f.)
(∞
d.f
.)
0.8
0
1.3
72 1
.325 1
.310 1
.28
0.9
0 1
.812 1
.725 1
.697 1
.645
0.9
5 2
.228 2
.086 2
.042 1
.96
0.9
9 3
.169 2
.845 2
.750 2
.58
Note
: t Z
as n in
cre
ases
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-32
Exa
mp
le o
f t d
istr
ibu
tio
n
co
nfid
en
ce
in
terv
al
A r
andom
sam
ple
of n =
25 h
as X
= 5
0 a
nd
S =
8. F
orm
a 9
5%
confidence inte
rval fo
r µ
�d
.f. =
n –
1 =
24
, s
o
The c
onfid
ence inte
rval is
2.0
639
0.0
25
t/2
==
αt
25
8(2
.063
9)
50
nS/2
±=
±αt
X
46.6
98 ≤µ≤
53.3
02
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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Chap 8
-33
Exa
mp
le o
f t d
istr
ibu
tio
n
co
nfid
en
ce
in
terv
al
�In
terp
retin
g t
his
in
terv
al re
qu
ire
s t
he
assu
mp
tio
n t
ha
t th
e p
op
ula
tio
n y
ou
are
sa
mp
ling
fro
m is a
pp
roxim
ate
ly a
no
rma
l d
istr
ibu
tio
n (
esp
ecia
lly s
ince n
is o
nly
25
).
�T
his
co
nd
itio
n c
an
be
ch
ecke
d b
y c
rea
tin
g a
:
�N
orm
al pro
bab
ility
plo
t or
�B
oxp
lot
(continu
ed)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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Chap 8
-34
Confid
ence I
nte
rvals
Po
pu
lati
on
M
ean σ
Un
kn
ow
n
Co
nfi
den
ce
Inte
rvals
Po
pu
lati
on
Pro
po
rtio
n
σK
no
wn
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-35
Confid
ence I
nte
rvals
for
the
Popula
tion P
rop
ort
ion, π
�A
n inte
rval estim
ate
for
the p
opula
tion
pro
port
ion (
π)
can b
e c
alc
ula
ted b
y
addin
g a
n a
llow
ance for
uncert
ain
ty to
the s
am
ple
pro
port
ion (
p )
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-36
Confid
ence I
nte
rvals
for
the
Popula
tion P
rop
ort
ion, π
�R
ecall
that th
e d
istr
ibution o
f th
e s
am
ple
pro
port
ion is a
ppro
xim
ate
ly n
orm
al if the
sam
ple
siz
e is larg
e,
with s
tandard
devia
tion
�W
e w
ill e
stim
ate
this
with s
am
ple
data
:(co
ntin
ue
d)
n
p)
p(1
−
n
)(1
σp
ππ
−=
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-37
Confid
ence I
nte
rval E
ndp
oin
ts
�U
pp
er
an
d lo
we
r co
nfid
en
ce
lim
its f
or
the
p
op
ula
tio
n p
rop
ort
ion
are
ca
lcu
late
d w
ith
th
e
form
ula
�w
here
�
Zα
/2is
th
e s
tan
da
rd n
orm
al va
lue
fo
r th
e le
ve
l o
f co
nfid
en
ce
de
sir
ed
�p
is t
he
sa
mp
le p
rop
ort
ion
�n
is t
he
sa
mp
le s
ize
�N
ote
: m
ust have n
p >
5 a
nd n
(1-p
) >
5n
p)
p(1
/2Z
p−
±α
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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Chap 8
-38
Exam
ple
�A
random
sam
ple
of 100 p
eople
show
s that 25 a
re left-h
anded.
�F
orm
a 9
5%
confidence inte
rval fo
r
the tru
e p
roport
ion o
f le
ft-h
anders
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-39
Exam
ple
�A
ra
nd
om
sa
mp
le o
f 1
00
pe
op
le s
ho
ws
tha
t 2
5 a
re le
ft-h
an
de
d.
Fo
rm a
95
%
co
nfid
en
ce
in
terv
al fo
r th
e t
rue
pro
po
rtio
n
of
left
-han
de
rs.
/1
00
0.2
5(0
.75)
1.9
62
5/1
00
p)/
np(1
/2Z
p
±=
−±α
0.3
349
0.1
651
(0.0
43
3)
1.9
6
0.2
5
≤≤
±=
π
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-40
Inte
rpre
tation
�W
e a
re 9
5%
co
nfid
en
t th
at th
e t
rue
p
erc
en
tag
e o
f le
ft-h
an
de
rs in
th
e p
op
ula
tio
n
is b
etw
ee
n 16
.51
% a
nd
33
.49
%.
�A
lth
ou
gh
th
e in
terv
al fr
om
0.1
65
1 t
o 0
.33
49
m
ay o
r m
ay n
ot
con
tain
th
e tru
e p
rop
ort
ion
, 9
5%
of
inte
rva
ls f
orm
ed
fro
m s
am
ple
s o
f siz
e 1
00
in
th
is m
an
ne
r w
ill c
on
tain
th
e t
rue
p
rop
ort
ion
.
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-41
Dete
rmin
ing
Sam
ple
Siz
e
Fo
r th
e
Mean
Dete
rmin
ing
Sam
ple
Siz
e
Fo
r th
eP
rop
ort
ion
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
c..
Chap 8
-42
Sam
plin
g E
rror
�T
he
re
qu
ire
d s
am
ple
siz
e c
an
be
fo
un
d t
o r
ea
ch
a d
esir
ed
ma
rgin
of
err
or
(e)
with
a s
pe
cifie
d
leve
l o
f co
nfid
en
ce
(1
-α ααα
)
�T
he
ma
rgin
of
err
or
is a
lso
ca
lled
sa
mp
ling
err
or
�th
e a
mount
of im
pre
cis
ion in t
he e
stim
ate
of th
e
pop
ula
tion p
ara
mete
r
�th
e a
mount
added a
nd
subtr
acte
d to the p
oin
t
estim
ate
to form
the c
onfid
ence inte
rval
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-43
Dete
rmin
ing
Sam
ple
Siz
e
Fo
r th
e
Mean
Dete
rmin
ing
Sam
ple
Siz
e
nσ2
/α
ZX
±
nσ2
/α
Ze
=Sam
plin
g e
rror
(marg
in o
f err
or)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-44
Dete
rmin
ing
Sam
ple
Siz
e
Fo
r th
e
Mean
Dete
rmin
ing
Sam
ple
Siz
e
nσ2
/α
Ze
=
(co
ntin
ue
d)
2
2σ
22
/ e
Zn
α=
No
w s
olv
e
for
n to
get
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-45
Dete
rmin
ing
Sam
ple
Siz
e
�T
o d
ete
rmin
e t
he
re
qu
ire
d s
am
ple
siz
e f
or
the
me
an
, yo
u m
ust
kno
w:
�T
he d
esire
d leve
l of
confid
ence (
1 -
α ααα),
whic
h
dete
rmin
es the c
ritical valu
e,
Zα
/2
�T
he a
ccepta
ble
sam
plin
g e
rror,
e
�T
he s
tanda
rd d
evia
tio
n, σ
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-46
Req
uir
ed S
am
ple
Siz
e E
xam
ple
If σ
= 4
5, w
ha
t sa
mp
le s
ize
is n
ee
de
d t
o
estim
ate
th
e m
ea
n w
ith
in ±
5 w
ith
90
%
co
nfid
en
ce
?
(Alw
ays r
ou
nd
up)
219.1
95
(45)
(1.6
45)
e
σZ
n2
22
2
22
==
=
So
th
e r
eq
uir
ed
sa
mp
le s
ize
is n
= 2
20
Basic
Busin
ess S
tatistics, 11e ©
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rentice-H
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Chap 8
-47
If σ
is u
nkno
wn
�If u
nknow
n, σ
can b
e e
stim
ate
d w
hen
usin
g the r
equired s
am
ple
siz
e form
ula
�U
se
a v
alu
e f
or
σth
at
is e
xp
ecte
d t
o b
e
at
lea
st
as la
rge
as t
he
tru
e σ
�S
ele
ct
a p
ilot
sa
mp
le a
nd
estim
ate
σ
with
the
sa
mp
le s
tan
da
rd d
evia
tio
n,
S
Basic
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ess S
tatistics, 11e ©
2009 P
rentice-H
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Chap 8
-48
Dete
rmin
ing
Sam
ple
Siz
e
Dete
rmin
ing
Sam
ple
Siz
e
Fo
r th
eP
rop
ort
ion
2
2
e
)(1
Zn
ππ
−=
No
w s
olv
e
for
n to
get
n
)(1
Ze
ππ
−=
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
all,
In
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Chap 8
-49
Dete
rmin
ing
Sam
ple
Siz
e
�T
o d
ete
rmin
e t
he r
equired s
am
ple
siz
e f
or
the
pro
port
ion, you m
ust
kno
w:
�T
he
desire
d le
ve
l o
f co
nfide
nce
(1
-α ααα
), w
hic
h d
ete
rmin
es the
cri
tica
l va
lue
, Zα
/2
�T
he
acce
pta
ble
sa
mp
ling
err
or,
e
�T
he
tru
e p
ropo
rtio
n o
f e
vents
of in
tere
st, π
�π
ca
n b
e e
stim
ate
d w
ith
a p
ilot sa
mp
le if n
ece
ssary
(or
co
nserv
ative
ly u
se
0.5
as a
n e
stim
ate
of π)
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
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rentice-H
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In
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Chap 8
-50
Req
uir
ed S
am
ple
Siz
e E
xam
ple
Ho
w larg
e a
sam
ple
wou
ld b
e n
ece
ssary
to e
stim
ate
the
tru
e p
rop
ort
ion d
efe
ctive
in
a larg
e p
opu
lation w
ith
in ±
3%
,w
ith
95
%
co
nfide
nce
?
(Assum
e a
pilo
t sam
ple
yie
lds p
= 0
.12)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-51
Req
uir
ed S
am
ple
Siz
e E
xam
ple
So
lutio
n:
Fo
r 9
5%
co
nfid
en
ce
, u
se
Zα
/2=
1.9
6
e =
0.0
3
p =
0.1
2, so
use
th
is t
o e
stim
ate
π
So u
se
n =
451
45
0.7
42
(0.0
3)
0.1
2)
(0.1
2)(
12
(1.9
6)
2e
)(1
2/2
Zn
=−
=−
=π
πα
(co
ntin
ue
d)
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-52
Eth
ical Is
sues
�A
co
nfid
en
ce
in
terv
al e
stim
ate
(re
fle
ctin
g
sa
mp
ling
err
or)
sh
ou
ld a
lwa
ys b
e in
clu
de
d
wh
en
re
po
rtin
g a
po
int
estim
ate
�T
he
le
ve
l o
f co
nfid
en
ce
sh
ou
ld a
lwa
ys b
e
rep
ort
ed
�T
he
sa
mp
le s
ize
sh
ou
ld b
e r
ep
ort
ed
�A
n in
terp
reta
tio
n o
f th
e c
on
fid
en
ce
in
terv
al
estim
ate
sh
ou
ld a
lso
be
pro
vid
ed
Basic
Busin
ess S
tatistics, 11e ©
2009 P
rentice-H
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In
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Chap 8
-53
Cha
pte
r S
um
mary
�In
troduced t
he c
oncep
t of confide
nce inte
rva
ls�
Dis
cusse
d p
oin
t estim
ate
s�
Develo
ped c
onfid
ence
inte
rval estim
ate
s�
Cre
ate
d c
onfide
nce in
terv
al estim
ate
s f
or
the m
ean
(σknow
n)
�D
ete
rmin
ed c
onfide
nce inte
rval estim
ate
s for
the
mean (σ
unkno
wn)
�C
reate
d c
onfide
nce in
terv
al estim
ate
s f
or
the
pro
port
ion
�D
ete
rmin
ed r
equir
ed s
am
ple
siz
e f
or
mean a
nd
pro
port
ion s
ett
ings
�A
ddre
ssed c
onfid
ence
inte
rval estim
atio
n a
nd e
thic
al
issu
es
