Confidence Intervals Confidence interval for sample mean
嚜澧onf
i
denc
eI
nt
er
v
a
l
s
Conf
i
denc
ei
nt
er
v
a
l
f
ors
a
mpl
emea
n
Bec
a
us
et
hea
r
eaundert
hes
t
a
nda
r
dnor
ma
l
c
ur
v
ebet
ween每
1
.
96a
nd1
.
96i
s.
95,
wek
now:
TheCL
Tt
el
l
sus
:
a
st
hes
a
mpl
es
i
z
eni
nc
r
ea
s
es
,
t
hes
a
mpl
emea
ni
sa
ppr
ox
i
ma
t
el
yNor
ma
l
wi
t
h
nds
t
a
nda
r
ddev
i
a
t
i
on
mea
n? a
Thi
si
sequi
v
a
l
entt
o:
Thus
,
weha
v
eas
t
a
nda
r
dnor
ma
l
v
a
r
i
a
bl
e
whi
c
hc
a
nbei
nt
er
pr
et
eda
st
hepr
oba
bi
l
i
t
yt
ha
tt
hei
nt
er
v
a
l
I
ft
heunder
l
yi
ngpopul
a
t
i
oni
sNor
ma
l
l
ydi
s
t
r
i
but
ed,
wedon＊
tneedCLTorl
a
r
g
e
s
a
mpl
es
i
z
ef
ort
hes
a
mpl
emea
nt
obeNor
ma
l
l
ydi
s
t
r
i
but
ed每nor
ma
l
i
t
yi
s
g
ua
r
a
nt
eed.
s95%.
i
nc
l
udest
het
r
uemea
n? i
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
1
Conf
i
denc
ei
nt
er
v
a
l
f
ors
a
mpl
emea
n
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
2
Conf
i
denc
ei
nt
er
v
a
l
f
ors
a
mpl
emea
n
TheCI
i
nt
er
v
a
l
i
sc
ent
er
eda
tt
hes
a
mpl
emea
na
ndex
t
ends 1
.
96
Thei
nt
er
v
a
l
t
oea
c
hs
i
deoft
hes
a
mpl
emea
n.
Thust
hei
nt
er
v
a
l
＊
swi
dt
hi
s2(
1
.
96)???? a
ndi
snotr
a
ndom;
onl
yt
hei
nt
er
v
a
l
bounda
r
i
esa
r
er
a
ndom
i
st
husc
a
l
l
edt
he95% c
onf
i
denc
ei
nt
er
v
a
l
f
ort
hemea
n.
Thi
si
nt
er
v
a
l
v
a
r
i
esf
r
om s
a
mpl
et
os
a
mpl
e,
a
st
hes
a
mpl
emea
nv
a
r
i
es
.
Sot
hei
nt
er
v
a
l
i
t
s
el
fi
sar
a
ndom i
nt
er
v
a
l
:
i
t
sboundsa
r
er
a
ndom v
a
r
i
a
bl
es
.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
3
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
4
Ba
s
i
cPr
oper
t
i
esofConf
i
denc
eI
nt
er
v
a
l
s
I
nt
er
pr
et
i
ngaConf
i
denc
eLev
el
Wes
t
a
r
t
edwi
t
ha
nev
ent(
t
ha
tt
her
a
ndom i
nt
er
v
a
l
c
a
pt
ur
est
het
r
uev
a
l
ue??)
whos
epr
oba
bi
l
i
t
ywa
s.
95
Forag
i
v
ens
a
mpl
e,
t
heCI
c
a
nbeex
pr
es
s
edei
t
hera
s
i
e
swi
t
hi
nt
hi
sf
i
x
edi
nt
er
v
a
l
wi
t
hpr
oba
bi
l
i
t
y0.
95.
I
ti
st
empt
i
ngt
os
a
yt
ha
t? l
ora
s
? i
sac
ons
t
a
nt(
unf
or
t
una
t
el
yunk
nownt
ous
)
.
I
ti
st
her
ef
or
ei
nc
or
r
e
c
tt
owr
i
t
et
he
s
t
a
t
ement
?l
i
esi
n(
a,
b)
)
=0.
95
P(
t
heri
si
n(
a,
b)
ori
s
n＊
t
.
-s
i
nc
e? ei
Ac
onc
i
s
eex
pr
es
s
i
onf
ort
hei
nt
er
v
a
l
i
sx? 1
.
96
wher
e每g
i
v
est
hel
ef
tendpoi
nt(
l
owerl
i
mi
t
)
a
nd+g
i
v
est
her
i
g
htendpoi
nt(
upper
l
i
mi
t
)
.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
Ba
s
i
c
a
l
l
y,
?i
snotr
a
ndom (
i
t
＊
sac
ons
t
a
nt
)
,
s
oi
tc
a
n＊
tha
v
eapr
oba
bi
l
i
t
ya
s
s
oc
i
a
t
ed
wi
t
hi
t
sbeha
v
i
or
.
5
I
nt
er
pr
et
i
ngaConf
i
denc
eLev
el
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
6
I
nt
er
pr
et
i
ngaConf
i
denc
eLev
el
I
ns
t
ea
d,
ac
or
r
ec
ti
nt
er
pr
et
a
t
i
onof※
95% c
onf
i
denc
e§r
el
i
esont
hel
ong
r
unr
el
a
t
i
v
e
f
r
equenc
yi
nt
er
pr
et
a
t
i
onofpr
oba
bi
l
i
t
y.
Ex
a
mpl
e:
t
hev
er
t
i
c
a
l
l
i
nec
ut
st
hemea
s
ur
ementa
x
i
sa
tt
het
r
ue(
butunk
nown)
v
a
l
ueof?.
Tos
a
yt
ha
ta
nev
entAha
spr
oba
bi
l
i
t
y.
95i
st
os
a
yt
ha
ti
ft
hes
a
meex
per
i
menti
s
per
f
or
medov
era
ndov
era
ga
i
n,
i
nt
hel
ongr
unAwi
l
l
oc
c
ur95% oft
het
i
me.
Sot
her
i
ghti
nt
er
pr
et
a
t
i
oni
st
os
a
yt
ha
ti
nr
epea
t
eds
a
mpl
i
ng
,
95% oft
he
Theot
her
c
onf
i
denc
ei
nt
er
v
a
l
sobt
a
i
nedf
r
om a
l
l
s
a
mpl
eswi
l
l
a
c
t
ua
l
l
yc
ont
a
i
n? .
5% oft
hei
nt
er
v
a
l
swi
l
l
not
.
One hundred 95% CIs (asterisks identify intervals that do not include ?).
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
7
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
8
I
nt
er
pr
et
i
ngaConf
i
denc
eLev
el
Ot
herLev
el
sofConf
i
denc
e
Not
i
c
et
ha
t7oft
he1
00i
nt
er
v
a
l
ss
hownf
a
i
l
t
oc
ont
a
i
n?.
Pr
oba
bi
l
i
t
yof1每? i
sa
c
hi
ev
edbyus
i
ngz
npl
a
c
eof1
.
96
?/2 i
I
nt
hel
ongr
un,
onl
y5% oft
hei
nt
er
v
a
l
ss
oc
ons
t
r
uc
t
edwoul
df
a
i
l
t
oc
ont
a
i
n?.
Ac
c
or
di
ngt
ot
hi
si
nt
er
pr
et
a
t
i
on,
t
hec
onf
i
denc
el
ev
el
i
snotas
t
a
t
ementa
bouta
ny
pa
r
t
i
c
ul
a
ri
nt
er
v
a
l
,
eg(
79.
3,
80.
7)
.
I
ns
t
ea
di
tper
t
a
i
nst
owha
twoul
dha
ppeni
fav
er
yl
a
r
g
enumberofl
i
k
ei
nt
er
v
a
l
s
wer
et
obec
ons
t
r
uc
t
edus
i
ngt
hes
a
meCI
f
or
mul
a
.
P(每z?/2 ? Z < z?/2) = 1 每 ?
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
9
Ot
herL
ev
el
sofConf
i
denc
e
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
10
Ex
a
mpl
e
A1
00(
1每?)
%c
onf
i
denc
ei
nt
er
v
a
l
f
ort
hemea
n? whent
hev
a
l
ueof? i
sk
nowni
s
g
i
v
enby
As
a
mpl
eof40uni
t
si
ss
el
ec
t
eda
nddi
a
met
ermea
s
ur
edf
orea
c
hone.
Thes
a
mpl
e
mea
ndi
a
met
eri
s5.
426mm,
a
ndt
hes
t
a
nda
r
ddev
i
a
t
i
onofmea
s
ur
ement
si
s0.
1
mm.
Let
＊
sc
a
l
c
ul
a
t
eac
onf
i
denc
ei
nt
er
v
a
l
f
ort
r
uea
v
er
a
g
edi
a
met
erus
i
ngac
onf
i
denc
el
ev
el
1
0.
of90%.
Thi
sr
equi
r
est
ha
t1
00(
1每?)
=90,
f
r
om whi
c
h? =.
Us
i
ngqnorm(0.05)
or
,
equi
v
a
l
ent
l
y,
by
.
645
z
?/2=z
.
05=1
(
c
or
r
es
pondi
ngt
oac
umul
a
t
i
v
ez
c
ur
v
ea
r
eaof.
95)
.
Thedes
i
r
edi
nt
er
v
a
l
i
st
hen
Thef
or
mul
af
ort
heCI
c
a
na
l
s
obeex
pr
es
s
edi
nwor
dsa
s
Poi
nte
s
t
i
mat
e? (z critical value) (standard error).
zc
r
i
t
i
c
al
v
al
u
e
)
(z critical value) (standard error).
s
t
andar
de
r
r
or
)
.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
11
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
12
I
nt
er
v
a
l
wi
dt
h
Si
nc
et
he95% i
nt
er
v
a
l
ex
t
ends1
.
96
x
,
t
hewi
dt
hoft
hei
nt
er
v
a
l
i
s2(
1
.
96)
Sa
mpl
es
i
z
ec
omput
a
t
i
on
t
oea
c
hs
i
deof
Forea
c
hdes
i
r
edc
onf
i
denc
el
ev
el
a
ndi
nt
er
v
a
l
wi
dt
h,
wec
a
ndet
er
mi
net
he
nec
es
s
a
r
ys
a
mpl
es
i
z
e.
=3.
92
Ex
a
mpl
e:
Ar
es
pons
et
i
mei
sNor
ma
l
l
ydi
s
t
r
i
but
edwi
t
hs
t
a
nda
r
ddev
i
a
t
i
on25
mi
l
l
i
s
ec
.Anews
ys
t
em ha
sbeeni
ns
t
a
l
l
ed,
a
ndwewi
s
ht
oes
t
i
ma
t
et
het
r
ue
a
v
er
a
ger
es
pons
et
i
me? f
ort
henewenv
i
r
onment
.
Si
mi
l
a
r
l
y,
t
hewi
dt
hoft
he99% i
nt
er
v
a
l
i
s(
us
i
ngqnorm(0.005))
2(
2.
58)
=5.
1
6
wha
t
As
s
umi
ngt
ha
tr
es
pons
et
i
mesa
r
es
t
i
l
l
nor
ma
l
l
ydi
s
t
r
i
but
edwi
t
h? =25,
s
a
mpl
es
i
z
ei
snec
es
s
a
r
yt
oens
ur
et
ha
tt
her
es
ul
t
i
ng95% CI
ha
sawi
dt
hof(
a
tmos
t
)
1
0?
Weha
v
emor
ec
onf
i
denc
et
ha
tt
he99% i
nt
er
v
a
l
i
nc
l
udest
het
r
uev
a
l
uepr
ec
i
s
el
ybec
a
us
ei
ti
swi
der
.
Thehi
g
hert
hedes
i
r
eddeg
r
eeofc
onf
i
denc
e,
t
hewi
dert
her
es
ul
t
i
ngi
nt
er
v
a
l
wi
l
l
be.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
Ex
a
mpl
e
13
cont＊d
Thes
a
mpl
es
i
z
enmus
ts
a
t
i
s
f
y
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
14
Unk
nownmea
na
ndv
a
r
i
a
nc
e
Wek
nowt
ha
t
-aCI
f
ort
hemea
n? ofanor
ma
l
di
s
t
r
i
but
i
on
ora
nydi
s
t
r
i
but
i
on
-al
a
r
g
es
a
mpl
eCI
f
or? f
Rea
r
r
a
ng
i
ngt
hi
sequa
t
i
ong
i
v
es
%i
s
:
wi
t
hac
onf
i
denc
el
ev
el
of1
00(
1每?)
=2(
1
.
96)
(
25)
/1
0=9.
80
So
c
hwi
l
l
r
a
r
el
ybek
nown.
I
ns
t
ea
dwewor
k
Apr
a
c
t
i
c
a
l
di
f
f
i
c
ul
t
yi
st
hev
a
l
ueof?,whi
wi
t
ht
hes
t
a
nda
r
di
z
edv
a
r
i
a
bl
e
2=9
n=(
9.
80)
6.
04
Si
nc
enmus
tbea
ni
nt
eg
er
,
as
a
mpl
es
i
z
eof97i
sr
equi
r
ed.
Wher
et
hes
a
mpl
es
t
a
nda
r
ddev
i
a
t
i
onSha
sr
epl
a
c
ed?.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
15
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
16
Unk
nownmea
na
ndv
a
r
i
a
nc
e
ALa
r
g
eSa
mpl
eI
nt
er
v
a
l
f
or?
I
fni
ss
uf
f
i
c
i
ent
l
yl
a
r
g
e,
t
hes
t
a
nda
r
di
z
edv
a
r
i
a
bl
e
Pr
ev
i
ous
l
y,
t
her
ewa
sr
a
ndomnes
sonl
yi
nt
henumer
a
t
orofZbyv
i
r
t
ueof
es
t
i
ma
t
or
.
I
nt
henews
t
a
nda
r
di
z
edv
a
r
i
a
bl
e,
bot
h
a
not
her
.
,
t
he
andSv
a
r
yi
nv
a
l
uef
r
om ones
a
mpl
et
o
ha
sa
ppr
ox
i
ma
t
el
yas
t
a
nda
r
dnor
ma
l
di
s
t
r
i
but
i
on.
Thi
si
mpl
i
est
ha
t
Thust
hedi
s
t
r
i
but
i
onoft
hi
snewv
a
r
i
a
bl
es
houl
dbewi
dert
ha
nt
heNor
ma
l
t
or
ef
l
ec
t
t
heex
t
r
aunc
er
t
a
i
nt
y.
Thi
si
si
ndeedt
r
uewhenni
ss
ma
l
l
.
ddsl
i
t
t
l
eex
t
r
av
a
r
i
a
bi
l
i
t
y,
s
ot
hi
s
Howev
er
,
f
orl
a
r
g
ent
hes
ubs
t
i
t
ut
i
onofSf
or? a
v
a
r
i
a
bl
ea
l
s
oha
sa
ppr
ox
i
ma
t
el
yas
t
a
nda
r
dnor
ma
l
di
s
t
r
i
but
i
on.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
t
hc
onf
i
denc
el
ev
el
a
ppr
ox
i
ma
t
el
y
i
sal
a
r
ges
a
mpl
ec
onf
i
denc
ei
nt
er
v
a
l
f
or? wi
%.
1
00(
1每?)
Thi
sf
or
mul
ai
sv
a
l
i
dr
eg
a
r
dl
es
soft
hepopul
a
t
i
ondi
s
t
r
i
but
i
on.
17
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
18
Sma
l
l
s
a
mpl
ei
nt
er
v
a
l
sf
ort
hemea
n
ALa
r
g
eSa
mpl
eI
nt
er
v
a
l
f
or?
I
nwor
ds
,
t
heCIi
s
zc
r
i
t
i
c
a
l
v
a
l
ue)
(
es
t
i
ma
t
eds
t
a
nda
r
der
r
oroft
hemea
n)
.
poi
ntes
t
i
ma
t
eof? ? (
Gener
a
l
l
ys
pea
k
i
ng
,
n>40wi
l
l
bes
uf
f
i
c
i
entt
oj
us
t
i
f
yt
heus
eoft
hi
si
nt
er
v
a
l
.
Thi
si
ss
omewha
tmor
ec
ons
er
v
a
t
i
v
et
ha
nt
her
ul
eoft
humbf
ort
heCLTbec
a
us
eof
t
hea
ddi
t
i
ona
l
v
a
r
i
a
bi
l
i
t
yi
nt
r
oduc
edbyus
i
ngSi
npl
a
c
eof?.
?TheCI
es
ent
edi
nea
r
l
i
ers
ec
t
i
oni
sv
a
l
i
dpr
ov
i
dedt
ha
tni
sl
a
r
g
e
f
or? pr
? Rul
eoft
humb:
n>
40
? Ther
es
ul
t
i
ngi
nt
er
v
a
l
c
a
nbeus
edwha
t
ev
ert
hena
t
ur
eoft
hepopul
a
t
i
on
di
s
t
r
i
but
i
on.
?TheCLTc
a
nnotbei
nv
ok
edwhenni
ss
ma
l
l
? Needt
odos
omet
hi
ngel
s
ewhenn<
40
?Whenn<
40,
weha
v
et
o
? ma
k
eas
pec
i
f
i
ca
s
s
umpt
i
ona
boutt
hef
or
m oft
hepopul
a
t
i
ondi
s
t
r
i
but
i
on
a
nd
? t
hender
i
v
eaCI
t
a
i
l
or
edt
ot
ha
ta
s
s
umpt
i
on.
?Forex
hepopul
a
t
i
oni
sdes
c
r
i
bedbya
a
mpl
e,
wec
oul
ddev
el
opaCI
f
or? whent
Nor
ma
l
,
org
a
mmadi
s
t
r
i
but
i
on,
oraWei
bul
l
di
s
t
r
i
but
i
on,
a
nds
oon.
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
19
___________________________________________________________________________________
Copyr
i
ghtPr
of
.
Va
nj
aDuk
i
c
,
Appl
i
edMa
t
hema
t
i
c
s
,
CUBoul
der
STAT4000/5000
20
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- confidence interval of the mean calculator
- confidence interval for t
- confidence interval for proportion in excel
- confidence interval for population mean calculator
- confidence interval for population proportion calculator
- confidence interval for proportions formula
- confidence interval calculator sample mean
- confidence interval for 2 proportions
- confidence interval for mean calculator
- confidence interval for proportion calculator
- confidence interval two sample means
- confidence interval for difference of means