TESTIN - Brunswick School Department · made b y male s i s les s tha n wha t th e marketin g...
23
SIS TESTING USING SAMPLE will look at the basics of setting up and carrying out sing a univariate data set. Then we will use this w conclusions about some unknown population anytime we make a decision based on sample data, ror, so we will discuss what types of errors might g hypotheses. and carry out a hypothesis test about a population and carry out a hypothesis test about a population and Type II errors in context. factors that affect the power of a test. ST PROCEDURES is 3rd ed. pages 525-529/4th ed. pages 578-581) sed on sample data helps us evaluate claims about chers and analysts use hypothesis testing methods
Transcript of TESTIN - Brunswick School Department · made b y male s i s les s tha n wha t th e marketin g...
HYPO
THES
IS T
ESTIN
G US
ING
A SI
NGLE
SAM
PLE
In th
is s
ectio
n, w
e w
ill lo
ok a
t the
bas
ics
of s
ettin
g up
and
car
ryin
g ou
ta
hypo
thes
is t
est
usin
g a
univ
aria
te d
ata
set.
The
n w
e w
ill u
se t
his
info
rmat
ion
to d
raw
co
nclu
sion
s ab
out
som
e un
know
n po
pula
tion
para
met
er.
Fina
lly, a
nytim
e w
e m
ake
a de
cisi
on b
ased
on
sam
ple
data
,th
ere
is a
ris
k of
err
or,
so w
e w
ill d
iscu
ss w
hat
type
s of
err
ors
mig
htbe
mad
e w
hen
test
ing
hypo
thes
es.
OBJE
CTIV
ES•
Cor
rect
ly s
et u
p an
d ca
rry
out a
hyp
othe
sis
test
abo
ut a
pop
ulat
ion
mea
n.•
Cor
rect
ly s
et u
p an
d ca
rry
out a
hyp
othe
sis
test
abo
ut a
pop
ulat
ion
prop
ortio
n.•
Des
crib
e T
ype
I and
Typ
e II
err
ors
in c
onte
xt.
« U
nder
stan
d th
e fa
ctor
s tha
t af
fect
the
pow
er o
f a te
st.
HYPO
THES
ES A
ND T
EST
PROC
EDUR
ES(In
trod
uctio
n to
Sta
tistic
s &
Dat
a A
naly
sis
3rd
ed. p
ages
525
-529
/4th
ed
. pag
es 5
78-5
81)
Mak
ing
deci
sion
s ba
sed
on s
ampl
e da
ta h
elps
us
eval
uate
cla
ims
abou
ta
popu
latio
n. R
esea
rche
rs a
nd a
naly
sts
use
hypo
thes
is t
estin
g m
etho
dsin
a v
arie
ty o
f se
tting
s to
cho
ose
betw
een
two
com
petin
g cl
aim
s ab
out
a po
pula
tion
cha
ract
eris
tic.
'•..
."T
he fi
rst
step
in c
arry
ing
out
a hy
poth
esis
tes
t is
dev
elop
ing
the
null
and
alte
rnat
ive
hypo
thes
es.
The
se a
re s
tate
men
ts t
hat
will
be
used
inth
e de
cisi
on-m
akin
g pr
oces
s. F
irst,
the
rese
arch
er f
orm
s a
hypo
thes
isba
sed
on
som
e in
itial
cl
aim
. Fo
r ex
ampl
e,
supp
ose
an
auto
man
ufac
ture
r pu
rcha
ses
off-
road
tir
es t
hat
are
supp
osed
to
have
am
ean
trea
d th
ickn
ess
of 0
.312
5 in
. The
aut
o m
anuf
actu
rer w
ill a
ssum
e
213
PGroves
Text Box
Chapter 10 STUDY GUIDE -------------------- 1) Reading 2) Practice MC 3) Practice Short Answer 4) Answer Key
214 *
Cha
pter
9
the
new
tir
es h
ave
been
man
ufac
ture
d as
spe
cifi
ed.
Aft
er a
ll, th
e ti
reco
mpa
ny w
ould
n't
stay
in b
usin
ess
for
long
if th
ey d
idn'
t pr
ovid
e w
hat
the
initi
al c
laim
abo
ut t
he m
ean
trea
d th
ickn
ess
that
the
aut
o co
mpa
nybe
lieve
s to
be
fact
. T
his
initi
al a
ssum
ptio
n is
cal
led
the
null
hyp
othe
sis.
We
wri
te t
he n
ull h
ypot
hesi
s as
:H
0:ft
= 0.
3125
.w
here
,H
0 st
ands
for
"th
e nu
ll h
ypot
hesi
s"ju
is th
e po
pula
tion
mea
n tr
ead
thic
knes
s fo
r al
l tir
es o
f thi
s ty
pe.
The
aut
o m
anuf
actu
rer m
ay s
uspe
ct t
hat t
here
has
bee
n a
chan
ge in
the
mea
n th
ickn
ess
of th
e ti
re t
read
, so
the
y de
cide
to c
heck
sev
eral
of
the
tire
s. T
his
lead
s th
e au
to c
ompa
ny to
dev
elop
wha
t is
cal
led
anal
tern
ativ
e hy
poth
esis
. T
he
alte
rnat
ive
hypo
thes
is
is
a co
mpe
ting
hypo
thes
is a
nd c
ould
be
wri
tten
in
one
of th
e fo
llow
ing
thre
e w
ays:
Ha:
p* 0
.312
5 in
. or
fj. <
0.3
125
in.
orH
> 0
.312
5 in
.he
re,
Ha
stan
ds f
or "
the
alte
rnat
ive
hypo
thes
is"
Bec
ause
th
e au
to
man
ufac
ture
r su
spec
ts
that
th
e m
ean
trea
dth
ickn
ess
has
chan
ged,
but
doe
s no
t ha
ve a
spe
cifi
c di
rect
ion
in m
ind,
they
wou
ld u
se j
u ^
0.31
25 i
n. a
s th
e al
tern
ativ
e hy
poth
esis
.N
o m
atte
r w
hich
al
tern
ativ
e hy
poth
esis
th
e co
mpa
ny u
ses,
th
ehy
poth
esis
tes
ting
pro
cedu
re o
nly
allo
ws
us t
o fa
vor
this
alte
rnat
ive
ifth
ere
is s
tron
g ev
iden
ce a
gain
st t
he n
ull
hypo
thes
is.
Thi
s ev
iden
cew
ould
com
e fr
om s
ampl
e da
ta.
We
wou
ld e
valu
ate
wha
t w
e se
e in
the
the
sam
ple
to d
eter
min
e if
the
sam
ple
mea
n ti
re t
read
is
just
too
far
from
wha
t th
e nu
ll h
ypot
hesi
s sp
ecif
ies
to b
e ex
plai
ned
by ju
st c
hanc
edi
ffer
ence
s fr
om s
ampl
e to
sam
ple.
Thi
s sa
me
reas
onin
g is
use
d in
all
hypo
thes
is t
ests
con
side
red
in th
e A
P S
tatis
tics
cour
se.
The
nul
l hyp
othe
sis
is u
sual
ly w
ritte
n as
H0:
som
e po
pula
tion
cha
ract
eris
tic =
the
hyp
othe
size
d va
lue
and
the
alte
rnat
ive
hypo
thes
is i
s w
ritt
en a
s on
e of
the
follo
win
g:H
a: s
ome
popu
lati
on c
hara
cter
istic
*
the
hyp
othe
size
d va
lue
Ha:
som
e po
pula
tion
cha
ract
eris
tic
< t
he h
ypot
hesi
zed
valu
eH
a: s
ome
popu
lati
on c
hara
cter
istic
>
the
hyp
othe
size
d va
lue
EXAM
PLE
The
mar
keti
ng m
anag
er f
or a
n on
line
com
pute
r ga
me
stor
eta
rget
s th
e co
mpa
ny a
dver
tisi
ng
tow
ard
mal
es b
ecau
se h
e be
lieve
sth
at 7
5% o
f th
e co
mpa
ny's
pur
chas
es a
re m
ade
by m
en.
The
sal
esm
anag
er c
laim
s th
at t
he p
ropo
rtio
n of
pur
chas
es m
ade
by f
emal
es h
asin
crea
sed.
He
belie
ves
that
the
pro
port
ion
of p
urch
ases
mad
e by
men
is n
ow l
ess
than
wha
t th
e m
arke
ting
man
ager
bel
ieve
s. W
hat
null
and
alte
rnat
ive
hypo
thes
es
wou
ld b
e us
ed t
o te
st
the
sale
s m
anag
er's
clai
m?
The
app
ropr
iate
hyp
othe
ses
are
show
n be
low
. (N
otic
e th
at w
ew
ill n
ow u
se p
to
repr
esen
t th
e pa
ram
eter
, si
nce
the
hypo
thes
es a
reab
out a
pro
port
ion.
)
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 215
Ha:p
ma
les<
0.7
5
The
nu
ll an
d al
tern
ativ
e hy
poth
eses
ar
e w
ritt
en
in
term
s of
popu
lati
on c
hara
cter
isti
cs. I
n th
is e
xam
ple,
the
alte
rnat
ive
is w
ritte
n as
"les
s th
an"
sinc
e th
e sa
les
man
ager
's c
laim
is
that
the
pro
port
ion
ofpu
rcha
ses
mad
e by
mal
es i
s le
ss t
han
wha
t th
e m
arke
ting
man
ager
belie
ves,
0.7
5.
ERRO
RS IN
HYP
OTHE
SIS
TEST
ING
(Intr
oduc
tion
to S
tatis
tics
& D
ata
Ana
lysi
s 3r
d e
d. p
ages
531
-534
/4th
ed. p
ages
582
-586
)
Now
tha
t w
e ha
ve a
n un
ders
tand
ing
of h
ow t
o ge
nera
te t
he n
ull
and
alte
rnat
ive
hypo
thes
es,
a te
st p
roce
dure
will
be
used
to
deci
de i
f w
esh
ould
rej
ect
the
null
hypo
thes
is.
Tes
t pr
oced
ures
ar
e co
nsid
ered
in
the
next
sec
tion.
Onc
e a
deci
sion
is
mad
e af
ter
the
test
pro
cedu
re i
spe
rfor
med
, th
ere
is a
cha
nce
that
the
fin
al d
ecis
ion
is w
rong
. In
oth
erw
ords
, an
err
or c
ould
hav
e be
en m
ade.
The
re a
re t
wo
poss
ible
typ
esof
err
ors
and
they
are
cal
led
a Ty
pe I
err
or a
nd a
Typ
e II
err
or.
Eith
er t
ypes
of
erro
r m
ay o
ccur
whe
n m
akin
g a
deci
sion
eith
er t
ore
ject
or
to f
ail
to r
ejec
t th
e nu
ll hy
poth
esis
. Fo
r ex
ampl
e, i
n th
e tir
etr
ead
prob
lem
, if
a d
ecis
ion
is m
ade
to r
ejec
t th
e nu
ll hy
poth
esis
, th
isco
uld
be a
wro
ng d
ecis
ion
that
wou
ld c
ause
the
aut
o m
anuf
actu
rer
toco
nclu
de t
hat
the
tire
s di
d no
t m
eet
spec
ific
atio
ns.
How
ever
, if
the
deci
sion
was
to
fail
to r
ejec
t th
e nu
ll hy
poth
esis
, th
is c
ould
als
o be
wro
ng a
nd th
e au
to m
anuf
actu
rer
coul
d en
d up
usi
ng t
ires
that
do
not
mee
t sp
ecif
icat
ions
. In
eith
er c
ase,
the
re i
s a
poss
ible
err
or e
xist
s th
atis
pot
entia
lly d
amag
ing
in s
ome
way
.A
Typ
e I
erro
r is
mad
e if
we
reje
ct th
e nu
ll hy
poth
esis
and
the
nul
lhy
poth
esis
is
actu
ally
tru
e. A
ltho
ugh
the
hypo
thes
is t
est,
base
d on
prob
abili
ty,
supp
orts
the
dec
isio
n, w
e ar
e le
d to
an
inco
rrec
t in
fere
nce
abou
t th
e po
pula
tion
. T
his
wou
ld a
mou
nt t
o ha
ving
str
ong
enou
ghev
iden
ce to
con
clud
e th
at t
he t
ires
do
not
have
a m
ean
trea
d th
ickn
ess
of 0
.312
5 in
. The
com
pany
wou
ld d
ecid
e to
ret
urn
the
tire
s, c
ausi
ng t
heti
re
man
ufac
ture
r to
lo
se
mon
ey.
If
the
tires
ac
tual
ly
mee
tsp
ecif
icat
ions
, th
e ti
re m
anuf
actu
rer
lost
mon
ey d
ue t
o th
e de
cisi
oner
ror. A T
ype
II e
rror
is
mad
e if
we
fail
to r
ejec
t the
nul
l and
in
real
ity t
henu
ll hy
poth
esis
is
not
true
and
sho
uld
have
bee
n re
ject
ed.
Thi
s ty
pe o
fer
ror
wou
ld a
mou
nt to
not
hav
ing
enou
gh e
vide
nce
to s
ay th
e tir
es d
idno
t m
eet
spec
ific
atio
ns.
In
this
in
stan
ce,
the
com
pany
w
ould
unkn
owin
gly
use
thes
e ti
res.
Thi
s er
ror
coul
d m
ean
that
cus
tom
ers
rece
ive
cars
with
fau
lty t
ires
and
thi
s co
uld
caus
e a
law
suit
for
the
com
pany
and
pot
enti
ally
eve
n th
e ri
sk o
f los
s of
life
for
the
cus
tom
er.
Avo
idin
g th
ese
erro
rs i
s, o
f co
urse
, de
sira
ble.
How
ever
/it
isn'
tal
way
s po
ssib
le to
avo
id m
akin
g an
err
or b
ecau
se w
e m
ake
deci
sion
sba
sed
on l
ooki
ng o
nly
at a
sam
ple.
Wha
t w
e ca
n do
is
to t
ry t
o ke
epth
e ch
ance
of t
hese
err
ors
as s
mal
l as
poss
ible
.W
e us
e th
e sy
mbo
l a
to d
enot
e th
e pr
obab
ilit
y of
a T
ype
I er
ror
and
the
sym
bol
/? t
o de
note
the
pro
babi
lity
of
a T
ype
II e
rror
. We
can
cont
rol
the
valu
e of
a
by t
he s
igni
fica
nce
leve
l w
e se
lect
for
the
tes
t.
216
* C
hapt
ers
Typ
e II
err
or i
s m
ore
prob
lem
atic
as
it is
som
ethi
ng t
hat
we
can'
tco
ntro
l ea
sily
. The
val
ues
of a
an
d /?
ar
e re
late
d — th
e sm
alle
r w
em
ake
a , t
he l
arge
r /3
beco
mes
, all
othe
r th
ings
bei
ng e
qual
. For
thi
sre
ason
, w
e ge
nera
lly c
hoos
e a
sign
ific
ance
lev
el a
th
at i
s th
e la
rges
tva
lue
that
is
cons
ider
ed a
n ac
cept
able
ris
k of
Typ
e I
erro
r. T
his
will
help
con
trol
for
the
err
ors
by k
eepi
ng a
sm
all a
s w
ell
as c
ontr
ollin
g a
LARG
E-SA
MPLE
HYP
OTHE
SIS
TEST
S FO
R A P
ROPO
RTIO
N(I
ntr
od
uct
ion
to
Sta
tistic
s &
Dat
a A
naly
sis
3rd
ed
. pag
es 5
37-5
48/4
th e
d. p
ages
589
-599
)
Nex
t, w
e ta
ke t
he h
ypot
hese
s w
e de
velo
ped
and
syst
emat
ical
ly t
est
them
to
deci
de w
heth
er
or n
ot t
o re
ject
the
nul
l hy
poth
esis
. T
his
proc
ess
is k
now
n as
a te
st p
roce
dure
and
the
sam
e ba
sic
proc
edur
e is
used
in
the
man
y di
ffer
ent
hypo
thes
is t
ests
. H
owev
er,
depe
ndin
g on
the
type
of
data
tha
t w
e ha
ve a
nd t
he q
uest
ion
of i
nter
est,
ther
e ar
edi
ffer
ent
hypo
thes
is t
ests
. The
fir
st t
est
we
cons
ider
is
a la
rge-
sam
ple
hypo
thes
is t
est f
or a
pop
ulat
ion
prop
orti
on.
In t
his
case
, w
e ar
e lo
okin
g at
cat
egor
ical
dat
a th
at c
ome
from
asi
ngle
sam
ple,
suc
h as
the
dat
a on
the
pro
port
ion
of c
usto
mer
s w
hoar
e m
ale.
In
this
situ
atio
n, t
he
data
con
sist
s of
obs
erva
tions
on
aca
tego
rica
l var
iabl
e w
ith tw
o po
ssib
le v
alue
s — m
ale
or f
emal
e.Ju
st a
s w
as t
he c
ase
with
a c
onfi
denc
e in
terv
al,
the
hypo
thes
is t
est
is b
ased
on
the
pro
pert
ies
of t
he s
ampl
ing
dist
ribu
tion
of p
, th
esa
mpl
e pr
opor
tion.
Rec
all t
hat
p is
the
sam
ple
prop
ortio
n ba
sed
on a
rand
om s
ampl
e,
1-
/< =
P p(l
-p)-
n
3.
Whe
n n
is
larg
e en
ough
, th
e sa
mpl
ing
dist
ribu
tion
of
p is
appr
oxim
atel
y no
rmal
.It
is
impo
rtan
t to
ch
eck
to m
ake
sure
the
sa
mpl
e si
ze i
s la
rge
enou
gh b
efor
e ca
rryi
ng o
ut a
one
-pro
port
ion
hypo
thes
is te
st. T
o ve
rify
that
the
sam
ple
size
is la
rge
enou
gh, c
heck
to m
ake
sure
tha
t
np>
10
Onc
e w
e ha
ve v
erif
ied
that
the
sam
ple
size
is la
rge
enou
gh a
nd t
hat
the
sam
ple
is a
rand
om s
ampl
e, w
e ca
n pr
ocee
d w
ith th
e te
st.
Usi
ng th
e pr
oper
ties
.of th
e sa
mpl
ing
dist
ribu
tion
of p
, w
e ca
n fo
rma
z te
st s
tatis
tic:
z-
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 217
whe
re p
= p
opul
atio
n pr
opor
tion
.In
th
e ex
ampl
e w
here
w
e w
ante
d to
te
st
H0
: p =
0.7
5 ve
rsus
Ha
:p<
0.7
5,
we
cons
ider
whe
ther
the
sam
ple
prop
orti
on i
s en
ough
smal
ler
than
the
hyp
othe
size
d pr
opor
tion
of
0.7
5 th
at t
he
diff
eren
ceca
n't
be e
xpla
ined
just
by
sam
plin
g va
riab
ilit
y. T
o do
this
, w
e ca
lcul
ate
the
valu
e of
the
z st
atis
tic u
sing
0.7
5 (f
rom
the
nul
l hyp
othe
sis)
as
the
p-0
.75
0.75
(1-0
.75)
valu
e fo
r p: z-
;u./t
HV
nIf
the
null
hyp
othe
sis
is tr
ue, t
hen
this
z s
tatis
tic w
ill h
ave
a st
anda
rdno
rmal
dis
trib
utio
n.
If t
he v
alue
of
the
z st
atis
tic i
s so
met
hing
tha
tw
ould
be
"une
xpec
ted"
for
a s
tand
ard
norm
al v
aria
ble,
we
rega
rd t
his
as e
vide
nce
that
the
nul
l hyp
othe
sis
shou
ld b
e re
ject
ed.
EXAM
PLE
Sup
pose
tha
t th
e sa
les
man
ager
in
the
onlin
e co
mpu
ter
gam
ecu
stom
er e
xam
ple
sele
cts
a ra
ndom
sam
ple
of 4
23 p
revi
ous
cust
omer
san
d fi
nds
that
29
8 w
ere
mal
es.
The
sa
mpl
e pr
opor
tion
is
th
en29
8=
0.70
. W
e ca
n se
e th
at 0
.70
is s
mal
ler
than
0.7
5, b
ut i
s it
sm
all
enou
gh t
o co
nvin
ce u
s th
at c
hanc
e di
ffer
ence
s fr
om s
ampl
e to
sam
ple
coul
d no
t ac
coun
t fo
r th
is
diff
eren
ce?
Thi
s is
the
qu
estio
n th
at i
san
swer
ed b
y a
hypo
thes
is t
est.
Fir
st, l
et's
che
ck th
e as
sum
ptio
ns n
eede
d. T
he s
ampl
e w
as a
ran
dom
sam
ple
of c
usto
mer
s, s
o th
at c
ondi
tion
is m
et.
The
sec
ond
cond
ition
is
that
the
sam
ple
size
is la
rge
enou
gh,
so w
e ch
eck
423(
0.75
) =
317
.25
> 1
0
423(
1 -
0.75
) = 1
05.7
5 >
10
Nex
t, w
e ca
lcul
ate
the
valu
e of
the
z te
st s
tatis
tic.
298
p =
= 0
.704
so
we
can
now
sub
stit
ute
into
our
tes
t sta
tist
ic42
3 0.70
4-0.
75z
= -0.
75(1
-0.7
5)42
3B
ecau
se t
he c
ondi
tions
wer
e m
et,
we
know
tha
t if
H0 w
ere
true
, z
has
an a
ppro
xim
atel
y no
rmal
dis
trib
utio
n. U
sing
wha
t w
e kn
ow a
bout
the
norm
al d
istr
ibut
ions
, we
know
tha
t ge
ttin
g a
z s
core
mor
e th
an +
2or
les
s th
an -
2 do
es n
ot o
ccur
ver
y of
ten.
In
fact
, w
e ca
n co
mpu
te th
epr
obab
ilit
y of
obs
ervi
ng
a z
valu
e as
sm
all
as -
2.16
giv
en t
hat
the
dist
ribu
tion
is s
tand
ard
norm
al:
P(z
< -2
.16
if H
0 is
tru
e)=
area
to
the
left
of
- 2.1
6=
0.01
54
In o
ther
wor
ds,
a sa
mpl
e pr
opor
tion
as
sm
all
as o
r sm
alle
r th
an0.
704
wou
ld h
appe
n on
ly a
bout
1.5
% o
f th
e tim
e if
the
po
pula
tion
prop
orti
on i
s re
ally
0.7
5. B
ased
on
this
sam
ple,
it
does
see
m t
hat
the
null
hypo
thes
is i
s un
likel
y to
be
corr
ect,
so w
e w
ould
rej
ect H
0.
The
prob
abil
ity
just
com
pute
d is
cal
led
a P
-val
ue.
218
Cha
pter
9
Tw
o ke
y pa
rts
of a
hyp
othe
sis
test
are
the
tes
t st
atis
tic
and
the
P-
valu
e. Tes
t st
atis
tic—
a va
lue
com
pute
d fr
om t
he s
ampl
e da
ta t
hat
is u
sed
to m
ake
a de
cisi
on t
o ei
ther
rej
ect
H0
or f
ail
to r
ejec
t H0
P-v
alue
—th
e pr
obab
ility
of
gett
ing
a te
st s
tati
stic
val
ue a
t le
ast
asex
trem
e as
w
hat
was
ob
serv
ed
for
the
sam
ple,
if
th
e nu
llhy
poth
esis
is
actu
ally
true
If th
e P-
valu
e is
sm
all,
we
reje
ct th
e nu
ll h
ypot
hesi
s. H
ow s
mal
l doe
sth
e P
-val
ue h
ave
to b
e in
ord
er t
o re
ject
H0
? T
his
will
dep
end
on h
owla
rge
a ri
sk o
f a
Typ
e I
erro
r w
e ar
e w
illin
g to
ass
ume.
Thi
s le
vel o
f ris
kis
pre
set b
y th
e re
sear
cher
and
is
know
n as
the
sign
ific
ance
leve
l of
the
test
. It i
s de
note
d b
y a
. For
exa
mpl
e, i
f we
set
a =
0.0
5 r t
his
mea
ns w
ekn
ow t
hat
a re
sult
as
extr
eme
as w
hat
we
saw
in
the
sam
ple
coul
dha
ppen
as
ofte
n as
5%
of
the
time
if t
he n
ull
is t
rue,
but
we
can
live
with
a T
ype
I er
ror
occu
rrin
g fo
r ab
out
5 ou
t of
100
of
all
poss
ible
rand
om s
ampl
es.
Onc
e yo
u ha
ve
calc
ulat
ed
the
valu
e of
the
te
st
stat
istic
an
d th
eas
soci
ated
P-v
alue
, co
mpa
re t
he P
-val
ue to
the
valu
e of
a .
IfP
-val
ue <
a,
H0
shou
ld b
e re
ject
ed.
P-v
alue
> a
, H
0 sh
ould
be
not b
e re
ject
ed.
Whe
n w
riti
ng a
sol
utio
n to
a h
ypot
hesi
s te
st p
robl
em,
mak
e su
re t
oex
plic
itly
com
pare
the
P-v
alue
to
a T
his
show
s yo
u kn
ow h
ow t
olin
k th
e P-
valu
e an
d a
in m
akin
g yo
ur d
ecis
ion.
The
n al
way
s st
ate
your
con
clus
ion
in th
e co
ntex
t of
the
prob
lem
.
Fina
lly, w
e no
te t
hat t
here
are
thr
ee d
iffe
rent
com
puta
tion
s of
the
P-
valu
e to
co
nsid
er.
The
com
puta
tion
ch
osen
will
dep
end
on w
hich
ineq
ualit
y (<
, >,
or *
) ap
pear
s in
the
alt
erna
tive
hyp
othe
sis.
Her
e ar
eth
e th
ree
poss
ible
sit
uati
ons
that
can
occ
ur.
Upp
er-t
aile
d t
est:
Ha
-. p >
hyp
othe
size
dva
lue
Low
er-t
aile
d t
est:
Ha
-. p <
hyp
othe
size
dva
lue
Tw
o-ta
iled
tes
t:H
a : p
•£ h
ypot
hesi
zed
valu
e
P-v
alu
e =
area
in
uppe
rta
il.
P-v
alu
e =
area
in
low
erta
il
P-v
alue
= su
m o
f ar
eatw
o ta
ils
calc
ulat
ed Z
calc
ulat
ed Z
calc
ulat
ed z
, -Z
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 219
AP
Tip
' -
. •-
•'".
•
• •
f '
- •
' • .
''
••
'
Usi
ng
inco
rrec
t no
tati
on
can
low
er
your
sc
ore
on f
ree-
resp
onse
ques
tion
s. R
emem
ber t
o ke
ep th
e di
ffer
ent
"p"
nota
tion
s st
raig
ht.
P-v
alue
Thi
s is
a p
roba
bili
ty
com
pute
d fr
om
the
valu
e of
the
tes
tst
atis
tic.
All
hyp
othe
sis
test
s in
volv
e th
e us
e of
a P
-val
ue t
o m
ake
ade
cisi
on.
p T
his
is th
e no
tati
on u
sed
to d
enot
e th
e po
pula
tion
pro
port
ion
whe
nw
e ha
ve c
ateg
oric
al d
ata.
p R
ead
as p
-hat
, is
used
to
deno
te th
e sa
mpl
e pr
opor
tion
.
HYPO
THES
IS TE
STS
FOR
A M
EAN
(In
tro
du
ctio
n t
o S
tatis
tics
& D
ata
Ana
lysi
s 3r
d e
d. p
ages
550
-55S
/4th
ed. p
ages
602
-61
0)
Now
tha
t w
e ha
ve a
han
dle
on a
hyp
othe
sis
test
for
pro
port
ions
, le
t'sco
nsid
er t
ests
bas
ed o
n nu
mer
ical
dat
a. W
ith
num
eric
al d
ata,
we
are
usua
lly i
nter
este
d in
mak
ing
infe
renc
es a
bout
a p
opul
atio
n m
ean.
In
Cha
pter
8 o
n co
nfid
ence
int
erva
ls,
ther
e w
ere
two
type
s of
int
erva
lsfo
r es
timat
ing
mea
ns,
a z
inte
rval
and
a
t in
terv
al.
Whi
ch o
f th
ese
inte
rval
s is
use
d in
a p
arti
cula
r si
tuat
ion
is d
eter
min
ed b
y w
heth
er w
ekn
ow t
he p
opul
atio
n st
anda
rd d
evia
tion
(a) o
r w
heth
er w
e ha
ve o
nly
the
sam
ple
stan
dard
dev
iatio
n (s
j. I
n ei
ther
cas
e, w
e fo
und
that
if
n is
larg
e en
ough
or
if
we
know
th
e po
pula
tion
is
ro
ughl
y no
rmal
lydi
stri
bute
d, t
hen
eith
er
z =x
- u
- —
whe
n w
e kn
ow a
a
has
appr
oxim
atel
y a
stan
dard
nor
mal
dis
trib
utio
n or
y _
ii
t = -
— ,
whe
n w
e D
ON
'T k
now a
has
a t d
istr
ibut
ion
with
df
= n -
1In
eit
her
case
, th
e fo
rms
of t
he h
ypot
hese
s of
int
eres
t lo
ok v
ery
muc
h lik
e th
ose
of th
e pr
opor
tion
s te
st.
The
dif
fere
nce
is th
at n
ow t
hehy
poth
eses
are
in
term
s of
the
popu
lati
on m
ean,
so
the
null
hyp
othe
sis
isH
0 :
n =
hyp
othe
size
d va
lue
and
the
alte
rnat
ive
hypo
thes
is i
s on
e of
the
follo
win
g:H
a : n
*• h
ypot
hesi
zed
valu
eH
a: /
/< h
ypot
hesi
zed
valu
eH
a: n
> h
ypot
hesi
zed
valu
e
220
*
Cha
pter
9
The
tes
t st
atis
tic i
s ba
sed
on e
ither
the
z o
r th
e t
stat
istic
sho
wn
abov
e, d
epen
ding
on
whe
ther
or
not
we
know
a.
If a
is k
now
nx
- hy
poth
esiz
ed v
alue
z =
•<7 Vn
"P
-val
ue:
com
pute
d as
are
a un
der
the
z cu
rve
If a
is N
OT
know
n_x-
hypo
thes
ized
val
ue
A_
Vn"
P-va
lue:
com
pute
d as
are
a un
der
the
t cu
rve
with
df
= n -1
It i
s ve
ry r
are
that
the
pop
ulat
ion
stan
dard
dev
iatio
n is
kno
wn,
so
we
will
foc
us m
ainl
y on
the
t s
tatis
tic.
Rev
isiti
ng th
e ea
rlie
r tir
e tr
ead
scen
ario
, w
e ca
n w
rite
the
hyp
othe
ses,
che
ck a
ssum
ptio
ns,
calc
ulat
eth
e te
st s
tatis
tic a
nd P
-val
ue,
and
give
a c
oncl
usio
n in
con
text
.
EXAM
PLE
Tru
stw
orth
y T
ires
sel
ls th
eir
lead
ing
high
per
form
ance
tire
toa
car
man
ufac
ture
r. T
he c
ar m
anuf
actu
rer
requ
ires
tha
t th
e tir
es h
ave
a m
ean
trea
d th
ickn
ess
of 0
.312
5. T
he c
ar m
anuf
actu
rer
thin
ks t
ires
rece
ived
fro
m T
rust
wor
thy
may
not
be
mee
ting
this
req
uire
men
t an
dth
at t
he m
ean
trea
d m
ay b
e gr
eate
r th
an 0
.312
5, b
ecau
se t
hey
have
been
fin
ding
tha
t in
som
e ca
rs t
he t
ires
are
hitt
ing
part
s of
the
whe
elw
ell a
rea
on b
umps
, hi a
ran
dom
sam
ple
of 3
2 ti
res,
they
fou
nd t
he t
ire
trea
ds t
o ha
ve a
mea
n of
0.3
625
in.
and
a st
anda
rd d
evia
tion
of 0
.094
in. S
ince
bot
h th
ese
valu
es c
ame
from
the
sam
ple,
we
use
the
nota
tion
for
sam
ple
stat
istic
s:
x =
0.33
60 a
nd s
x =
0.0
94.
For
this
situ
atio
n, l
et's
look
at t
he c
ompo
nent
s th
at a
re c
omm
on to
all
hyp
othe
sis
test
s.
HYPO
THES
ESH
0: f
i =
0.3
125
Ha:
jj. >
0.3
125
(sin
ce th
e co
mpa
ny s
uspe
cts
the
mea
ntr
ead
is g
reat
er t
han
the
requ
irem
ent
of 0
.312
5)se
t the
sig
nifi
canc
e le
vel (
a) a
t 0.0
5.
ASSU
MPT
IONS
The
ass
umpt
ions
tha
t m
ust
be s
atis
fied
are
the
sam
e as
thos
e w
ith c
onfi
denc
e in
terv
als.
We
mus
t ha
ve a
ran
dom
sam
ple
and
the
sam
ple
size
mus
t be
lar
geor
th
e po
pula
tion
di
stri
buti
on
of
trea
d th
ickn
ess
mus
t be
appr
oxim
atel
y no
rmal
. T
he p
robl
em
stat
es
that
the
sa
mpl
e w
as a
rand
om s
ampl
e, s
o th
at c
ondi
tion
is m
et. T
he s
ampl
e si
ze is
larg
e (n
isgr
eate
r th
an 3
0). B
ecau
se b
oth
cond
ition
s ar
e m
et,
it is
rea
sona
ble
topr
ocee
d w
ith t
he t
est.
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
221
TEST
STA
TISTIC
x -
hyp
othe
size
d va
lue
t —
0.33
60-0
.312
50.
094
= 1.
414
nw
ith
df =
n -1
= 3
2-1
-
. .
-=
31P
- v
alue
= 0
.08
Not
e th
at b
ecau
se t
he i
nequ
alit
y in
the
nul
l hy
poth
esis
was
>,
the
P-
valu
e is
the
area
to
the
right
of 1
.41
und
er t
he t
cur
ve w
ith
df =
31.
CONC
LUSI
ON
In th
is e
xam
ple,
wit
h a
= 0
.05
we
see
that
the
P-v
alue
is n
otle
ss t
han
a. W
e fa
il to
rej
ect
the
null
hyp
othe
sis
and
conc
lude
that
ther
e is
not
con
vinc
ing
evid
ence
that
the
mea
n tr
ead
thic
knes
s of
tire
sis
gre
ater
tha
n 0.
3125
in.
Whe
n ca
rryi
ng o
ut a
tes
t fo
r m
eans
, w
heth
er o
r no
t yo
u kn
ow t
hest
anda
rd d
evia
tion
of
the
popu
lati
on is
wha
t de
term
ines
if y
ou s
houl
dus
e a
z te
st o
r t
test
. O
nly
use
the
z te
st w
hen
you
are
sure
the
stan
dard
dev
iati
on g
iven
is
from
the
pop
ulat
ion.
Not
e th
at t
he s
teps
in
a te
st a
bout
a p
opul
atio
n m
ean
are
the
sam
eas
for
the
tes
t fo
r a
popu
lati
on p
ropo
rtio
n. W
hat
dist
ingu
ishe
s th
e tw
ote
sts
is t
he
type
of
dat
a (n
umer
ical
for
mea
ns a
nd
cate
gori
cal
for
prop
orti
ons)
, th
e sp
ecif
ic a
ssum
ptio
ns t
hat
mus
t be
che
cked
, an
d th
ete
st s
tati
stic
use
d fo
r th
e te
st.
SAM
PLE
PROB
LEM
1
A
wel
l-kn
own
bra
nd
of
pain
re
lief
ta
blet
s is
adve
rtis
ed
to
begi
n re
lief
w
ithi
n 24
min
utes
. T
o te
st t
his
clai
m,
ara
ndom
sa
mpl
e of
18
sub
ject
s su
ffer
ing
from
th
e sa
me
type
s of
head
ache
pai
n re
cord
whe
n th
ey f
irst
not
ice
reli
ef a
fter
tak
ing
the
pain
relie
f ta
blet
. The
dat
a ga
ther
ed f
rom
thi
s st
udy
are
show
n.
Tim
e to
Pain
Relie
f (in
min
ute
s)
25 27
24 24
27 23
23 26
25 28
22 25
25 24
25 26
24 23
Doe
s th
e sa
mpl
e su
gges
t th
e m
ean
pai
n re
lief
tim
e is
lon
ger
than
the
adve
rtis
ed t
ime?
Tes
t th
e ap
prop
riat
e hy
poth
eses
usi
ng a
0.0
5si
gnif
ican
ce le
vel.
SOLU
TION
TO
PRO
BLEM
1 T
o so
lve
this
pro
blem
, we
carr
y ou
t a
hypo
thes
iste
st. A
ll fo
ur p
arts
of
a hy
poth
esis
sho
uld
be c
ompl
ete
as s
how
n.
22
2 *
Ch
ap
ters
HYPO
THES
IS
With
]i r
epre
sent
ing
the
popu
latio
n m
ean
time
to r
elie
f,
Ha:/
/>2
4
a =
0.0
5 (g
iven
]
ASSU
MPTIO
NSR
ando
m s
ampl
e: T
he p
robl
em s
tate
s th
e 18
sub
ject
s ar
e a
rand
omsa
mpl
e.L
arge
sam
ple
or n
orm
al p
opul
atio
n di
stri
buti
on:
Sin
ce t
here
wer
eon
ly 1
8 su
bjec
ts,
we
need
to
be w
illin
g to
ass
ume
that
the
dist
ribu
tion
of r
elie
f tim
es i
n th
e po
pula
tion
is a
ppro
xim
atel
y no
rmal
. A
dot
plot
of
the
sam
ple
relie
f tim
es
is
show
n be
low
. B
ecau
se t
he
dotp
lot
isap
prox
imat
ely
sym
met
ric
and
ther
e ar
e no
out
liers
, it
is r
easo
nabl
e to
thin
k th
at th
e po
pula
tion
dis
trib
utio
n is
app
roxi
mat
ely
norm
al.
pain
rel
ief
o o
o o
oo
o e
o
22
23
24
25
26
27
28
29
30ti
me
TEST
STA
TIST
IC A
t t
est
will
be
used
sin
ce w
e do
n't
know
the
popu
lati
onst
anda
rd d
evia
tion.
x =
24.
78,
sx =
1.5
9,
df =
17
Vis
P-v
alue
= 0
.027
CONC
LUSI
ON S
ince
0.0
27 <
0.0
5, w
e re
ject
the
null
hyp
othe
sis
in f
avor
of
the
alte
rnat
ive
hypo
thes
is. T
here
is
conv
inci
ng e
vide
nce
that
the
mea
ntim
e to
pai
n re
lief
is g
reat
er t
han
24 m
inut
es.
As
a fi
nal
note
, sh
ould
you
be
in t
he u
nusu
al s
itua
tion
whe
re t
hepo
pula
tion
sta
ndar
d de
viat
ion
(CT)
, is
giv
en,
you
wou
ld u
se t
he z
test
stat
istic
and
the
ass
ocia
tion
P-v
alue
wou
ld b
e de
term
ined
usi
ng t
hest
anda
rd n
orm
al d
istr
ibut
ion.
Oth
erw
ise,
the
pro
cess
wou
ld b
e th
esa
me
as th
e pr
oces
s il
lust
rate
d in
the
exam
ple
abov
e.
Hyp
othe
sis
Test
ing U
sing
a S
ingl
e S
ampl
e *
223
AP
Tip
For
a
stan
dard
hy
poth
esis
tes
t, yo
u w
ill b
e re
quir
ed
to p
rovi
deco
mpl
ete
answ
ers
to f
our
key
part
s in
eve
ry t
est.
Get
in
the
habi
t of
thin
king
in
term
s of
the
se f
our
piec
es.
If y
ou c
hunk
the
inf
orm
atio
nun
der
each
of
thes
e ca
tego
ries
, yo
u w
ill h
ave
a be
tter
ch
ance
of
rem
embe
ring
all
asp
ects
of
any
hypo
thes
is t
est.
The
fou
r pi
eces
are
Stat
e th
e H
ypot
hese
s an
d
defi
ne
any
sym
bols
us
ed
in
the
hyp
oth
eses
.
Ho
and
Ha
(cor
rect
ly w
ritt
en)
a le
vel
Iden
tify
the
tes
t p
roce
du
re b
y na
me
or b
y fo
rmul
a. C
heck
All
Ass
umpt
ions
(or
Con
diti
ons)
The
ass
umpt
ions
var
y by
tes
t, b
ut y
ou s
houl
d al
way
s st
ate
AN
DC
HE
CK
the
assu
mpt
ion
appr
opri
ate
for
the
test
you
are
per
form
ing.
Tes
t St
atis
tic
Sho
w y
our z
or
t cal
cula
tion
Giv
e th
e P
-val
ue a
ssoc
iate
d w
ith th
e va
lue
of th
e te
st s
tatis
tic
Con
clus
ion
(be
su
re y
ou s
tate
it i
n c
onte
xt t
oo)
P <
a,
H0
shou
ld b
e re
ject
ed.
P >
a,
H0
shou
ld n
ot b
e re
ject
ed.
POW
ER A
ND P
ROBA
BILIT
Y OF
TYPE
II E
RROR
(Int
rodu
ctio
n to
Sta
tistic
s &
Data
Ana
lysi
s 3r
d e
d. p
ages
562
-567
/4th
ed.
pag
es 6
13-6
21)
Whi
le t
he A
P c
urri
culu
m d
oes
not
requ
ire
you
to c
alcu
late
pow
er, y
ouar
e ex
pect
ed to
kno
w th
e fa
ctor
s th
at a
ffec
t th
e po
wer
of
a te
st. T
here
are
thre
e fa
ctor
s th
at a
re g
ener
ally
con
side
red
whe
n th
inki
ng a
bout
pow
er.
Fir
st,
Incr
easi
ng «
will
rai
se t
he p
ower
. A
ltho
ugh
this
see
ms
like
a fa
st f
ix, i
t is
dan
gero
us b
ecau
se t
he p
roba
bili
ty o
f a
Typ
e I
erro
rw
ill a
lso
incr
ease
. Ano
ther
way
to r
aise
pow
er i
s si
mpl
y to
incr
ease
the
sam
ple
size
, alt
houg
h th
is i
sn't
alw
ays
prac
tica
l. T
he o
ther
thi
ngs
that
affe
ct p
ower
are
n't
real
ly th
ings
tha
t w
e ca
n co
ntro
l, bu
t it
is h
elpf
ul t
okn
ow t
hat
they
do
affe
ct
pow
er.
The
var
iabi
lity
in
the
popu
lati
onaf
fect
s po
wer
, w
ith
pow
er b
eing
gre
ater
whe
n th
e va
riab
ilit
y in
the
popu
lati
on i
s sm
all.
Als
o, th
e di
ffer
ence
bet
wee
n th
e ac
tual
val
ue o
f th
epo
pula
tion
cha
ract
eris
tic
and
the
hypo
thes
ized
val
ue a
ffec
ts
pow
er.
The
larg
er th
e di
ffer
ence
, th
e gr
eate
r th
e po
wer
of t
he t
est.
SAM
PLE
PROB
LEM
2 C
onsi
der
the
earl
ier
scen
ario
of
the
onli
ne c
ompu
ter
gam
e cu
stom
ers.
The
sal
es m
anag
er w
as i
nter
este
d in
dec
idin
g if
the
224 *
Cha
pter
9
prop
orti
on o
f m
ales
was
les
s th
an t
he 0
.75
clai
med
by
the
mar
keti
ngm
anag
er. T
he a
ppro
pria
te h
ypot
hese
s fo
r th
is s
itua
tion
wer
e:
H0:p
= 0
.75
Ha:p
<0
.75
(a)
Iden
tify
the
Typ
e I
erro
r in
thi
s sc
enar
io a
nd p
rovi
de a
pos
sibl
eco
nseq
uenc
e of
this
err
or;
(b)
Iden
tify
the
Typ
e II
err
or i
n th
is s
cena
rio
and
prov
ide
a po
ssib
leco
nseq
uenc
e of
this
err
or.
(c)
Wha
t ca
n be
don
e to
inc
reas
e th
e po
wer
of t
his
test
?
SOLU
TION
TO PR
OBLE
M 2
(a)
A
Typ
e I
erro
r w
ould
re
sult
if
it
w
ere
conc
lude
d th
at
the
prop
orti
on
of c
usto
mer
s w
ho a
re m
ale
is l
ess
than
0.7
5, w
hen
infa
ct t
his
prop
orti
on i
s 0.
75. A
pos
sibl
e co
nseq
uenc
e of
thi
s er
ror
wou
ld b
e th
at t
he c
ompa
ny m
ight
cha
nge
its
stra
tegy
of
targ
etin
gm
ales
in it
s ad
vert
isin
g, w
hich
mig
ht r
esul
t in
a de
crea
se in
sal
es.
(b)
A T
ype
II e
rror
in
this
sit
uati
on w
ould
occ
ur b
y co
nclu
ding
ther
ew
asn'
t en
ough
evi
denc
e to
say
tha
t th
e pr
opor
tion
of
cust
omer
sw
ho a
re m
ale
is l
ess
than
0.7
5 w
hen
this
pro
port
ion
real
ly is
less
than
0.7
5. I
n th
is c
ase,
the
com
pany
wou
ld p
roba
bly
cont
inue
to
targ
et m
ales
in
its a
dver
tisi
ng,
whi
ch m
ight
res
ult
in a
los
s of
pote
ntia
l sa
les
to f
emal
e cu
stom
ers.
(c)
One
way
to
incr
ease
pow
er w
ould
be
to i
ncre
ase
the
sign
ific
ance
leve
l of
the
tes
t. H
owev
er,
this
will
als
o in
crea
se t
he c
hanc
e of
aT
ype
I er
ror.
Als
o, in
crea
sing
sam
ple
size
will
incr
ease
pow
er.
INTE
RPRE
TATIO
N OF
RES
ULTS
IN H
YPOT
HESI
S TE
STIN
G(In
trod
uctio
n to
Sta
tistic
s &
Dat
a An
alys
is 3
rd e
d. p
ages
571
-574
/4th
ed
pag
es 6
23-6
25)
Onc
e da
ta
has
been
ga
ther
ed
and
an a
ppro
pria
te
hypo
thes
is
test
carr
ied
out,
the
find
ings
are
typ
ical
ly s
hare
d w
ith o
ther
s in
tere
sted
in
the
outc
ome.
In
com
mun
icat
ing
resu
lts
in jo
urna
ls a
nd n
ewsp
aper
s, i
tis
not
com
mon
to
prov
ide
the
sam
e le
vel o
f de
tail
that
you
wou
ld w
ant
to p
rovi
de i
n a
solu
tion
to a
hyp
othe
sis
test
ing
ques
tion
on
the
AP
exam
. S
ome
of t
he
impo
rtan
t th
ings
to
co
nclu
de w
hen
repo
rtin
gre
sult
s ar
e:
• H
ypot
hese
s: I
n e
ither
sym
bols
or
wor
ds,
you
nee
d to
cle
arly
sta
tebo
th t
he n
ull h
ypot
hesi
s an
d th
e al
tern
ativ
e hy
poth
esis
.K
Tes
t P
roce
dure
s: C
lear
ly s
tate
wha
t te
st y
ou u
sed
(lar
ge s
ampl
e z
test
for
pro
port
ions
, an
d so
on)
and
men
tion
any
ass
umpt
ions
that
are
nece
ssar
y in
ord
er f
or th
is te
st t
o be
app
ropr
iate
. ;
• T
est
Sta
tist
ic: B
e su
re t
o re
port
the
val
ue o
f the
test
sta
tist
ic a
s w
ell
as t
he a
ssoc
iate
d P
-val
ue. T
his
will
all
ow th
e re
ader
s to
kno
w if
they
wou
ld d
raw
the
sam
e co
nclu
sion
giv
en th
e sa
mpl
e da
ta.
B C
oncl
usio
n in
Con
text
: B
e ce
rtai
n yo
u ha
ve p
rovi
ded
a con
clus
ion
in t
erm
s of
the
ori
gina
lly
pose
d re
sear
ch q
uest
ion.
Thi
s ne
eds
to
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 225
incl
ude
a co
mpa
riso
n of
the
P-va
lue
to a
. Sta
ting
that
you
rej
ect t
heH
0 is
not
suff
icie
nt.
In m
any
case
s, th
e re
port
ed r
esul
ts o
nly
incl
ude
a st
atem
ent
such
as
P-va
lue
< 0
.05.
Thi
s is
com
mon
and
tel
ls t
he r
eade
r th
at t
he r
esul
ts o
fth
e te
st
yiel
ded
a P
-val
ue s
mal
ler
than
0.
05,
henc
e st
atis
tical
lysi
gnif
ican
t. Jo
urna
ls m
ay a
lso
use
a st
anda
rd m
etho
d of
cod
ing.
* -
sign
ific
ant,
wou
ld m
ean
thei
r P
-val
ue w
as <
0.05
, **
= v
ery
sign
ific
ant,
mea
ns P
-val
ue <
0.0
1.A
s yo
u re
view
pub
lish
ed r
epor
ts,
be s
ure
to l
ook
for
the
four
key
com
pone
nts
you
wou
ld r
epor
t an
d as
k yo
urse
lf s
ome
ques
tions
abo
utth
ese
piec
es. W
hat
wer
e th
e hy
poth
eses
the
y te
sted
? D
id th
ey u
se a
nap
prop
riat
e te
st f
or t
hese
? W
hat
was
the
asso
ciat
ed P
-val
ue a
nd w
hat
sign
ific
ance
le
vel
was
us
ed?
Als
o, w
ere
the
conc
lusi
ons
reac
hed
cons
iste
nt w
ith th
e re
sult
s of
the
test
?
Tip
In w
riti
ng a
con
clus
ion
for
a hy
poth
esis
te
st,
rem
embe
r yo
u ar
eal
way
s ei
ther
rej
ectin
g or
fai
ling
to r
ejec
t th
e nu
ll hy
poth
esis
. T
his
mea
ns y
ou e
ither
hav
e co
nvin
cing
evi
denc
e in
fav
or o
f th
e al
tern
ate
hypo
thes
is o
r th
at t
here
is
not
enou
gh e
vide
nce.
You
nev
er s
ay t
hat
you
acce
pt t
he n
ull
hypo
thes
is b
ecau
se t
hat
impl
ies
stro
ng e
vide
nce
for
the
null
hypo
thes
is.
HYPO
THES
IS TE
STIN
G US
ING
A SI
NGLE
SAMP
LE: S
TUDE
NTOB
JECT
IVES
FOR
THE
AP E
XAM
• Y
ou w
ill b
e ab
le t
o w
rite
the
nul
l an
d al
tern
ativ
e hy
poth
esis
for
ate
st a
bout
a p
opul
atio
n m
ean
or a
pop
ulat
ion
prop
orti
on.
« Y
ou w
ill b
e ab
le to
des
crib
e T
ype
I and
Typ
e II
err
ors
in c
onte
xt.
• Y
ou w
ill b
e ab
le t
o de
scri
be a
pos
sibl
e co
nseq
uenc
e of
eac
h ty
peer
ror
in c
onte
xt.
• Y
ou w
ill b
e ab
le to
car
ry o
ut a
test
of h
ypot
hese
s ab
out
a po
pula
tion
mea
n.•
You
will
be
able
to c
arry
out
a te
st o
f hyp
othe
ses
abou
t a p
opul
atio
npr
opor
tion
.•
You
will
be
able
to
inte
rpre
t th
e re
sult
of
a hy
poth
esis
tes
t in
cont
ext.
22
6 *
Ch
ap
ter9
MULT
IPLE
-CHO
ICE Q
UEST
IONS
1.
A p
sych
olog
ist r
epor
ts th
at th
e re
sult
of a
hyp
othe
sis
test
was
stat
isti
call
y si
gnif
ican
t at
the
0.05
leve
l. W
hich
of t
he f
ollo
win
g is
cons
iste
nt w
ith
this
sta
tem
ent?
(A)
The
P-v
alue
cal
cula
ted
was
sm
alle
r tha
n th
e si
gnif
ican
ce le
vel
of 0
.05.
(B)
The
P-v
alue
cal
cula
ted
was
lar
ger
than
the
sign
ific
ance
lev
el o
f0.0
5.(C
) T
he s
igni
fica
nce
leve
l cal
cula
ted
was
lar
ger
than
0.0
5.(D
) T
he s
igni
fica
nce
leve
l cal
cula
ted
was
sm
alle
r th
an 0
.05.
(E)
The
re w
as n
ot e
noug
h in
form
atio
n to
mak
e a
deci
sion
.
2.
A c
oncr
ete
lear
ner
is a
stu
dent
who
lea
rns
best
whe
n va
riou
s ty
pes
of h
ands
-on
or m
anip
ulat
ive
activ
ities
are
use
d to
illu
stra
teab
stra
ct c
once
pts.
Res
earc
hers
hav
e lo
ng b
elie
ved
that
60%
of a
llst
uden
ts r
emai
n co
ncre
te le
arne
rs u
ntil
they
are
bet
wee
n 16
and
21ye
ars
of a
ge. E
ach
stud
ent
in a
ran
dom
sam
ple
of 3
2 st
uden
ts a
ge17
to 1
9 w
as e
valu
ated
, an
d it
was
fou
nd t
hat
24 o
f the
32
wer
eco
ncre
te l
earn
ers.
Wou
ld it
be
appr
opri
ate
to u
se th
e z
test
for
apo
pula
tion
pro
port
ion
to te
st to
det
erm
ine
if th
e pr
opor
tion
of
conc
rete
lea
rner
s in
this
age
gro
up is
less
than
0.6
0?(A
) Y
es. S
ince
32(
0.6)
= 1
9.2
and
32(1
- 0.
6) =
12.
8, a
nd w
e ca
npr
ocee
d w
ith t
he te
st.
(B)
Yes
. Sin
ce 3
2 is
larg
er t
han
30, t
he s
ampl
e is
suf
fici
ently
lar
gean
d w
e ca
n pr
ocee
d w
ith th
e te
st.
(C)
Yes
. Sin
ce w
e kn
ow f
rom
the
sam
ple
was
a ra
ndom
sam
ple,
we
can
proc
eed
wit
h th
e te
st.
(D)
No.
Sin
ce 3
2(0.
05) =
1.6
, we
do n
ot h
ave
a la
rge
enou
gh s
ampl
eto
pro
ceed
with
tes
t.(E
) N
o. W
hile
32
is la
rger
tha
n 30
, it i
s so
clo
se to
30
and
we
don'
tkn
ow if
the
popu
lati
on d
istr
ibut
ion
is n
orm
al.
3.
A T
ype
I er
ror
occu
rs i
n w
hich
of t
he f
ollo
win
g si
tuat
ions
?(A
) H
a is
rej
ecte
d an
d th
e nu
ll h
ypot
hese
s is
true
.(B
) H
0 is
not
rej
ecte
d an
d th
e nu
ll h
ypot
hese
s is
fal
se.
(C)
H0 i
s re
ject
ed a
nd t
he n
ull h
ypot
hese
s is
tru
e.(D
) T
he P
-val
ue is
too
smal
l to
reje
ct th
e nu
ll h
ypot
hesi
s.(E
) T
he «
leve
l is
too
smal
l and
so
the
null
hypo
thes
is is
reje
cted
.
4.
A T
ype
II e
rror
occ
urs
in w
hich
of t
he f
ollo
win
g si
tuat
ions
?(A
) H
0 is
reje
cted
and
the
nul
l hyp
othe
ses
is tr
ue.
(B)
HO
is n
ot r
ejec
ted
and
the
null
hyp
othe
ses
is f
alse
.(C
) H
a is
not
reje
cted
and
the
nul
l hyp
othe
ses
is f
alse
.(D
) T
he P
-val
ue is
too
smal
l to
reje
ct th
e nu
ll h
ypot
hesi
s.(E
) T
he a
leve
l is
too
larg
e an
d so
the
null
hypo
thes
is is
reje
cted
.
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 227
5.
A g
rad
uat
e st
uden
t at
a p
riva
te u
nive
rsit
y w
ante
d to
stu
dy th
eam
ount
of
mon
ey th
at s
tude
nts
at h
is u
nive
rsit
y ca
rrie
d w
ith
them
.A
rec
ent
stud
y re
port
ed t
hat
the
ave
rage
am
ount
of m
oney
car
ried
by c
olle
ge s
tude
nts
is $
31.
He
deci
des
to c
olle
ct d
ata
and
carr
y ou
ta
test
to
see
if th
ere
is e
vide
nce
that
the
ave
rage
is h
ighe
r fo
rst
uden
ts a
t hi
s un
iver
sity
. Whi
ch o
f the
fol
low
ing
desc
ribe
s a
Typ
eII
err
or
in th
is c
onte
xt?
(A)
Thi
s w
ould
lead
to t
he i
ncor
rect
ide
a th
at s
tude
nts
at h
isun
iver
sity
, on
ave
rage
, sp
end
mor
e m
oney
eac
h m
onth
tha
nst
uden
ts a
t ot
her
univ
ersi
ties
.(B
) T
his
wou
ld le
ad t
o th
e in
corr
ect
idea
tha
t st
uden
ts a
t hi
sun
iver
sity
car
ry,
on a
vera
ge,
mor
e th
an $
31.
(C)
Thi
s w
ould
lead
to th
e in
corr
ect
idea
th
at s
tude
nts
at h
isun
iver
sity
car
ry, o
n a
vera
ge, l
ess
than
$31
.(D
) T
his
wou
ld l
ead
to t
he i
ncor
rect
ide
a th
at t
here
was
no
reas
onto
bel
ieve
that
stu
dent
s at
his
uni
vers
ity
carr
y, o
n av
erag
e,m
ore
than
$31
.(E
) T
his
wou
ld le
ad t
o th
e co
rrec
t id
ea t
hat
stud
ents
at
othe
rca
mpu
ses
carr
y, o
n av
erag
e, l
ess
than
$31
.
6.
An
anim
al r
ights
gro
up h
as b
een
very
sup
port
ive
of a
new
sili
con
prod
uct th
at c
aps
the
nail
s on
cat
s as
an
alte
rnat
ive
to s
urgi
call
yde
claw
ing
the
pets
. The
com
pany
who
mak
es th
e ca
ps c
laim
s th
eyla
st f
or a
n av
erag
e of
69
days
bef
ore
need
ing
to b
e re
plac
ed.
Bef
ore
publ
ical
ly e
ndor
sing
the
prod
uct,
the
ani
mal
rig
hts
grou
ppl
ans
to c
olle
ct d
ata
to s
ee if
ther
e is
con
vinc
ing
evid
ence
tha
t th
em
ean
tim
e be
fore
rep
lace
men
t is
nee
ded
is a
ctua
lly
less
tha
n w
hat
the
com
pany
cla
ims.
Whi
ch o
f the
fol
low
ing
wou
ld b
e an
appro
pri
ate
pair
of h
ypot
hese
s fo
r th
e an
imal
rig
hts
gro
up t
o te
st?
(A)
H0:
ju =
69 d
ays,
Ha:
ju
> 69
day
s(B
) H
0: V
= 6
9 da
ys,
Ha:
ju <
69 d
ays
(C)
H0: /
/ =
69
days
, H
a: n
* 6
9 da
ys(D
) H
0: x
= 6
9 da
ys, H
a: x
> 69
day
s(E
) H
0: x
= 6
9 da
ys,
Ha:
x <
69 d
ays
22
8 *
Ch
ap
ter9
7.
Neu
teri
ng d
ogs
is a
com
mon
sur
gica
l pra
ctic
e. T
he m
ean
tim
e to
reco
ver
from
the
gene
ral a
nest
heti
c us
ed is
28
hour
s. A
vete
rina
rian
bel
ieve
s th
at s
ince
cha
ngin
g to
a n
ew a
nest
hetic
, th
em
ean
reco
very
tim
e is
sho
rter
tha
n be
fore
. To
inve
stig
ate,
she
sele
cts
a ra
ndom
sam
ple
of 4
0 su
rger
ies
done
with
the
new
anes
thet
ic a
nd f
inds
that
the
mea
n re
cove
ry ti
me
was
25
hour
s an
dth
e st
anda
rd d
evia
tion
was
2.5
. She
pla
ns to
use
this
sam
ple
data
tote
st t
o se
e if
ther
e is
evi
denc
e th
at th
e re
cove
ry t
ime
is s
hort
er w
ithth
e ne
w a
nest
heti
c. W
hich
of t
he f
ollo
win
g is
the
corr
ect
test
stat
istic
for
this
stu
dy?
OK
_9
Q(A
) z
=
, w
ith d
f =
392.
5p
c;_
9P
(B)
t = =
, w
ith d
f =
39
'39
PS -
PR
(C)
t = =
^^-,
wit
h d
f =
39
V40
PS -P
R(D
) t =
0-
, w
ith d
f =
40
V40
(E)
t =
~
, w
ith
df
= 4
02.
5
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 229
A r
ecen
tly
publ
ishe
d st
udy
repo
rted
tha
t 63%
of t
he n
atio
n's
stud
ents
hav
e so
me
type
of s
truc
ture
d ho
mew
ork
stud
y tim
e. A
scho
ol s
urve
yed
each
stu
dent
in a
ran
dom
sam
ple
of 8
3 st
uden
tsw
ho a
tten
d th
e sc
hool
and
fou
nd t
hat
only
52%
repo
rted
hav
ing
ast
ruct
ured
hom
ewor
k tim
e. T
his
data
was
use
d to
car
ry o
ut a
hypo
thes
is te
st to
det
erm
ine
if th
ere
was
evi
denc
e th
at t
hepr
opor
tion
of s
tude
nts
at th
e sc
hool
who
had
str
uctu
red
hom
ewor
k tim
e w
as le
ss th
at th
e pr
opor
tion
rep
orte
d in
the
nati
onal
stu
dy. W
hich
of t
he f
ollo
win
g w
ould
be
the
test
sta
tistic
for
this
tes
t?0
.52
-0.6
3(A
) z=
, |0
.52(
0.48
)>
(B)
z =
i 83
0.5
2-0
.63
{0.5
2(0.
48)
\) z =
-'82
0.5
2-0
.50
(0.5
2(0.
48)
\) z
=83
0.5
2-0
.63
(0.6
3(0.
37)
\)
z =
-'83
0.5
2-0
.63
0.63
(0.3
7)82
9.
Bic
ycle
s pu
rcha
sed
from
a d
isco
unt
stor
e co
me
unas
sem
bled
. The
asse
mbl
y in
stru
ctio
ns t
hat
com
e w
ith t
he b
icyc
le c
laim
that
the
aver
age
asse
mbl
y tim
e is
30
min
utes
. A c
onsu
mer
gro
up h
asre
ceiv
ed c
ompl
aint
s fr
om p
eopl
e w
ho s
ay th
at th
e as
sem
bly
time
was
gre
ater
tha
n th
e tim
e cl
aim
s. T
hey
deci
de to
pur
chas
e 40
of
thes
e bi
kes
and
have
ask
ed 4
0 di
ffer
ent
peop
le to
ass
embl
e th
em.
The
con
sum
er g
roup
bel
ieve
d th
at it
was
rea
sona
ble
to r
egar
dth
ese
40 p
eopl
e as
rep
rese
ntat
ive
of th
e po
pula
tion
of p
eopl
e w
hom
ight
pur
chas
e th
is b
ike.
For
this
sam
ple,
they
fou
nd t
hat t
heas
sem
bly
times
had
a m
ean
of 3
4.2
min
utes
and
a s
tand
ard
devi
atio
n of
8.6
min
utes
. Is
ther
e co
nvin
cing
evi
denc
e th
at t
hecl
aim
ed a
vera
ge a
ssem
bly
time
is to
o lo
w a
t the
0.0
5 si
gnif
ican
cele
vel?
(A)
No,
z =
0.4
9 P-
valu
e =
0.31
2.(B
) Y
es, t
= 3
.09,
df =
39,
P-v
alue
= 0
.002
.(C
) Y
es, t
= 3
.05,
df
= 39
, P-v
alue
= 0
.004
.(D
) N
o, t
= 0
.49,
df =
39,
P-v
alue
= 0
.313
.(E
) Y
es, t
= 3.
05, d
f 40
, P-v
alue
- 0
.002
.
230
<•
C
hapt
er 9
10.
The
pro
m c
omm
itte
e is
thin
king
ab
ou
t ch
ang
ing
the
loca
tion
of
the
prom
. The
new
loc
atio
n is
mor
e ex
pens
ive
to r
ent,
and
for
the
incr
ease
d co
st t
o be
rea
sona
ble,
the
y w
ould
wan
t to
be f
airl
yce
rtai
n th
at m
ore
tha
n 46
% o
f the
sen
ior
clas
s w
ould
att
end
the
prom
. A s
urve
y of
a r
and
om
sam
ple
of 5
2 se
nior
s fo
und
that
25
wou
ld a
tten
d if
the
site
cha
nged
. Whi
ch o
f th
e fo
llow
ing
pair
s of
hypo
thes
es s
houl
d th
e p
rom
com
mit
tee
test
?(A
) H
0: ju
= 4
6%, H
a: /
/ >
46%
(B)
H0:/
/ =
46
%,H
a:>
«*
48
%(C
) H
0: p
= 0
.46,
Ha:p
> 0
.46
(D)
H0:
p =
0.4
6, H
a:p
* 0
.46
(E)
H0:
p =
0.4
8, H
a:p
< 0
.46
11.
Whi
ch o
f th
e fo
llow
ing
is c
lose
st t
o th
e P
-val
ue a
ssoc
iate
d w
ith
atw
o-ta
iled
t te
st w
ith
20 d
egre
es o
f fre
edom
if t
he v
alue
of t
he t
est
stat
isti
c is
2.0
?(A
) 0.
001
(B)
0.01
(C)
0.03
(D)
0.05
(E)
0.10
12.
Whi
ch o
f th
e fo
llow
ing
stat
emen
ts a
re t
rue?
I.
The
nul
l hyp
othe
sis
for
test
abo
ut a
pop
ulat
ion
prop
orti
onw
ritt
en a
s H
0:p
= h
ypot
hesi
zed
valu
e.n.
F
or t
he z
test
to b
e an
appro
pri
ate
test
for
a p
opul
atio
npro
port
ion, t
he f
ollo
win
g co
ndit
ion
mus
t be
met
: np
> 1
0 an
d
III.
The
stan
dard
dev
iati
on o
f the
stat
isti
c p
is a
^ -
^/np
Cl -
p).
(A)
I on
ly(B
) n
only
(C)
III o
nly
(D)
II a
nd I
II(E
) I,
H, a
nd I
II
13. A
loca
l gro
up c
laim
s th
at m
ore
than
60%
of t
he t
eens
dri
ving
aft
er10
p.m
. ar
e ex
ceed
ing
the
spee
d li
mit
. The
y pl
an to
col
lect
dat
a in
hope
s th
at a
hyp
othe
sis
test
will
pro
vide
con
vinc
ing
evid
ence
insu
pport
of t
heir
cla
im. W
hich
of
the
foll
owin
g is
true
abo
ut t
hehy
poth
eses
the
gro
up s
houl
d te
st?
(A)
The
nul
l hyp
othe
sis
stat
es t
hat
less
th
an 6
0% o
f the
tee
ns a
reex
ceed
ing
the
spee
d lim
it.(B
) T
he n
ull h
ypot
hesi
s st
ates
tha
t m
ore
than
60%
of t
he t
eens
are
exce
edin
g th
e sp
eed
limit.
(C)
The
alt
erna
tive
hyp
othe
sis
stat
es t
hat
less
th
an 6
0% o
f th
ete
ens
are
exce
edin
g th
e sp
eed
lim
it.
(D)
The
alt
erna
tive
hyp
othe
sis
stat
es t
hat
less
than
or
equa
l to
60%
of t
he t
eens
are
exc
eedi
ng t
he s
peed
lim
it.
(E)
The
alt
erna
tive
hyp
othe
sis
stat
es t
hat
mor
e th
an 6
0% o
f th
ete
ens
are
exce
edin
g th
e sp
eed
lim
it.
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 231
14. A
stu
dy b
y a
geol
ogic
al re
sear
ch t
eam
fou
nd t
hat
a ne
w p
iece
of
equi
pmen
t des
igne
d to
mea
sure
the
for
ces
of a
n ea
rthq
uake
is n
otef
fect
ive.
The
y ba
sed
this
con
clus
ion
on d
ata
from
a s
ampl
e of
40
piec
es o
f equ
ipm
ent a
nd t
hey
carr
ied
out
a te
st w
ith
a =
0.05
. T
hem
anuf
actu
rer
of th
e eq
uipm
ent c
laim
s th
is s
tudy
was
fla
wed
and
that
thei
r eq
uipm
ent i
s go
od. T
he re
sear
ch t
eam
is c
onsi
deri
ngca
rryi
ng o
ut a
sec
ond
stud
y w
ith th
e in
tent
ion
of in
crea
sing
the
pow
er o
f the
tes
t. W
hich
of t
he f
ollo
win
g w
ould
ens
ure
an i
ncre
ase
in th
e po
wer
of t
he t
est?
(A)
Mov
e th
e eq
uipm
ent t
o th
ree
rand
omly
cho
sen
new
loca
tion
s.(B
) C
hang
e a
= 0
.05
to a
= 0
.02.
(C)
Car
ry o
ut a
two-
side
d te
st in
stea
d of
a o
ne-s
ided
tes
t.(D
) In
crea
se th
e sa
mpl
e si
ze to
60
piec
es o
f eq
uipm
ent
bein
gte
sted
.(E
) D
ecre
ase
the
sam
ple
size
to 2
0 pi
eces
of
equi
pmen
t be
ing
test
ed.
15. S
uppo
se th
at th
e m
ean
heig
ht o
f wom
en i
n th
e U
nite
d S
tate
s is
64.5
in. w
ith
a st
anda
rd d
evia
tion
of 2
.5 in
. A c
loth
ing
desi
gner
feel
s th
at w
omen
who
use
her
pro
duct
s m
ay a
ctua
lly b
e ta
ller
onav
erag
e. S
he s
elec
ts a
ran
dom
sam
ple
of 7
0 w
omen
fro
m a
llw
omen
who
hav
e pr
evio
usly
pur
chas
ed h
er c
loth
ing.
Wha
t is
the
popu
lati
on o
f int
eres
t, a
nd w
hat
test
wou
ld th
e de
sign
er u
se to
tes
the
r cl
aim
?(A
)The
pop
ulat
ion
is a
ll w
omen
in th
e U
nite
d S
tate
s an
d th
eap
prop
riat
e te
st is
a t
test
with
df =
70.
(B) T
he p
opul
atio
n is
all
wom
en i
n th
e U
nite
d S
tate
s an
d th
eap
prop
riat
e te
st is
a z
test
with
=
2.5.
(C)T
he p
opul
atio
n is
all
wom
en w
ho h
ave
prev
ious
ly p
urch
ased
the
desi
gner
's c
loth
ing
and
the
appr
opri
ate
test
is a
t te
st w
ith d
f =70
. ^
(D) T
he p
opul
atio
n is
all
wom
en w
ho h
ave
prev
ious
ly p
urch
ased
the
desi
gner
's c
loth
ing
and
the
appr
opri
ate
test
is a
t te
st w
ithdf
=69
.(E
) The
pop
ulat
ion
is a
ll w
omen
who
hav
e pr
evio
usly
pur
chas
edth
e de
sign
er's
clo
thin
g an
d th
e ap
prop
riat
e te
st is
a z
test
with
= 2.
5.
FREE
-RES
PONS
E PR
OBLE
MS
1.
A b
rida
l gow
n in
dust
ry p
ubli
cati
on c
laim
s th
at n
atio
nwid
e th
eav
erag
e am
ount
spe
nt f
or a
wed
ding
gow
n is
$1,
012.
A lo
cal b
rida
lsh
op i
n an
urb
an c
omm
unit
y ha
s no
tice
d th
eir
mor
e ex
pens
ive
gow
ns a
re n
ot s
ellin
g w
ell.
Inst
ead,
the
bri
des
seem
to b
e se
lect
ing
only
low
er p
rice
d go
wns
or
clea
ranc
e go
wns
. The
sho
p w
onde
rs if
the
aver
age
amou
nt s
pent
for
a w
eddi
ng g
own
is le
ss t
han
$1,0
12fo
r th
eir
cust
omer
s. T
o in
vest
igat
e, t
hey
sele
cted
a r
ando
m s
ampl
eof
50
wed
ding
gow
n sa
les.
The
y fo
und
a sa
mpl
e m
ean
of $
985
and
a st
anda
rd d
evia
tion
of $
235.
Is th
ere
conv
inci
ng e
vide
nce
that
the
ave
rage
am
ount
spe
nt o
na
wed
ding
gow
n at
this
sho
p is
less
tha
n th
e na
tion
al f
igur
e? T
est
the
rele
vant
hyp
othe
ses
usin
g a
0.05
sign
ific
ance
lev
el.
232
* C
hapt
er 9
2.
A lo
cal s
choo
l dis
tric
t be
lieve
s th
at th
e pr
opor
tion
of s
enio
rs w
hoar
e ab
sent
from
sch
ool o
n th
e la
st d
ay o
f sch
ool m
ay b
e in
crea
sing
.O
ver
the
past
5 y
ears
, 39
% o
f the
sen
iors
hav
e m
isse
d th
e la
st d
ay.
Thi
s ye
ar, t
he s
choo
l dis
tric
t is
cons
ider
ing
a ne
w r
ewar
d pr
ogra
msp
onso
red
by lo
cal b
usin
esse
s w
here
sen
iors
who
wer
e at
sch
ool
on th
e la
st d
ay w
ould
be
ente
red
in a
dra
win
g fo
r an
iPad
. To
see
ifth
is p
rogr
am m
ight
red
uce
the
prop
orti
on o
f sen
iors
who
mis
ssc
hool
on
the
last
day
, a r
ando
m s
ampl
e of
398
seni
ors
from
the
scho
ol d
istr
ict w
as s
urve
yed.
Eac
h st
uden
t in
the
sam
ple
was
ask
edif
they
pla
nned
to a
tten
d on
the
last
day
of c
lass
giv
en th
epo
ssib
ility
of w
inni
ng a
n iP
ad. O
nly
129
of th
e 39
8 se
nior
sin
dica
ted
that
they
wou
ld m
iss
the
last
day
of s
choo
l. T
he sc
hool
dist
rict
wou
ld li
ke to
kno
w if
ther
e is
con
vinc
ing
evid
ence
that
the
new
pro
gram
wou
ld r
educ
e th
e nu
mbe
r of
sen
iors
abs
ent
on th
ela
st d
ay o
f sc
hool
.(a
) W
hat
hypo
thes
es s
houl
d th
e sc
hool
dis
tric
t te
st?
(b)
Iden
tify
the
appr
opri
ate
test
and
ver
ify
that
any
con
diti
ons
need
ed f
or th
e te
st a
re m
et.
(c)
Des
crib
e T
ype
I and
Typ
e II
err
ors
in th
e co
ntex
t of
this
prob
lem
.
An
sw
ers
MUL
TIPL
E-CH
OICE
QUE
STIO
NS
1.
A.
Whe
n a
rese
arch
er
says
th
e re
sults
w
ere
stat
istic
ally
sign
ific
ant,
it m
eans
the
P-v
alue
was
less
than
the
set
sig
nifi
canc
ele
vel
(Int
rodu
ctio
n to
Sta
tist
ics &
Dat
a A
naly
sis
3rd
ed. p
ages
57
1-57
4/4t
h ed
. pag
es 6
23-6
25).
2.
A.
In a
one
-sam
ple
prop
orti
ons
test
, on
e co
nditi
on t
hat
is n
eede
dis
np
> 10
and
n(l
- p
) >
10. N
otic
e th
ese
both
use
p a
nd n
ot
pfr
om t
he s
ampl
e (I
ntro
duct
ion
to S
tatis
tics
& D
ata
Ana
lysi
s 3r
d ed
.pa
ges
537-
548/
4th
ed. p
ages
589
-599
).
3.
C. B
y de
fini
tion,
a T
ype
I er
ror
occu
rs w
hen
the
null
hypo
thes
is is
reje
cted
whe
n it
sho
uld
not
be r
ejec
ted.
Thi
s m
ight
hap
pen
whe
nth
e P-
valu
e <
sign
ific
ance
lev
el (
Intr
oduc
tion
to S
tati
stic
s &
Dat
aA
naly
sis
3rd
ed. p
ages
531
-534
/4th
ed.
pag
es 5
82-5
86).
4.
B.
A T
ype
II e
rror
will
occ
ur a
nytim
e yo
u fa
il to
rej
ect
the
null
whe
n in
fac
t the
nul
l is
fals
e. T
his
mig
ht o
ccur
if th
e P-
valu
e is
not
smal
ler
than
a
(Int
rodu
ctio
n to
Sta
tistic
s &
Dat
a A
naly
sis
3rd
ed.
page
s 53
1-53
4/4t
h ed
. pag
es 5
82-5
86).
5.
D.
If a
Typ
e II
err
or w
as m
ade,
thi
s m
eans
he
faile
d to
rej
ect
the
null
hypo
thes
is.
In t
his
case
, th
at w
ould
am
ount
to
sayi
ng t
here
was
not
con
vinc
ing
evid
ence
that
the
mea
n fo
r hi
s un
iver
sity
was
grea
ter
than
$31
whe
n in
fac
t th
is i
s in
corr
ect a
nd th
e m
ean
real
lyis
gre
ater
tha
n $3
1 (I
ntro
duct
ion
to S
tatis
tics
& D
ata
Ana
lysi
s 3r
ded
. pag
es 5
31-5
34/4
th e
d. p
ages
582
-586
).
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 233
6.
B.
Sin
ce t
he g
rou
p is
con
cern
ed o
nly
if th
e ca
ps l
ast
less
th
an 6
9da
ys o
n av
erag
e, t
he a
lter
nati
ve h
ypot
hesi
s w
ould
be
u <
69
days
(Int
rodu
ctio
n to
St
atis
tics
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
525-
529/
4th
ed. p
ages
578
-581
).
7.
C.
Sin
ce w
e don't
know
a,
it m
ust
be
a t t
est.
Als
o, w
hile
df
= 3
9,n
= 4
0 is
the
val
ue t
hat
is u
sed
in t
he t
est
stat
isti
c ca
lcul
atio
n(I
ntro
duct
ion
to
Stat
isti
cs
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
550-
558/
4th
ed. p
ages
602
-610
).
8.
D.
In
calc
ulat
ing
the
test
st
atis
tic,
th
e de
nom
inat
or
uses
th
ehy
poth
esiz
ed v
alue
and
sam
ple
size
of
83 (
Intr
oduc
tion
to S
tati
stic
s&
Dat
a A
naly
sis
3rd
ed. p
ages
537
-548
/4th
ed.
pag
es 5
89-5
99).
9.
B.
A
t te
st
wit
h 39
de
gree
s of
fr
eedo
m
wou
ld
be
used
(Int
rodu
ctio
n to
St
atis
tics
&
Dat
a A
naly
sis
3rd
ed.
page
s 55
0-55
8/4t
h ed
. pag
es 6
02-6
10).
10.
C.
Thi
s is
a t
est
abou
t a
popu
lati
on p
ropo
rtio
n. T
he q
uest
ion
ofin
tere
st
is w
heth
er t
he p
opul
atio
n pr
opor
tion
p i
s gr
eate
r th
an0.
46 (
Intr
oduc
tion
to S
tati
stic
s &
Dat
a A
naly
sis
3rd
ed.
page
s 52
5-52
9/4t
h ed
. pag
es 5
78-5
81).
11.
D.
The
P-v
alue
is
appr
oxim
atel
y 0.
06,
whi
ch i
s cl
oses
t to
0.0
5(I
ntro
duct
ion
to
Stat
isti
cs
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
550-
558/
4th
ed. p
ages
602
-610
).
12.
B.
Not
ice
that
ch
oice
I
is w
ritt
en
usin
g p
inst
ead
of p
. A
llhy
poth
eses
ar
e st
ated
in
ter
ms
of t
he p
opul
atio
n va
lue,
whi
chw
ould
be
p. C
hoic
e II
I is
the
sta
ndar
d de
viat
ion
for
a bi
nom
ial
ran
do
m
vari
able
. T
he
stan
dard
de
viat
ion
of
p
is
'n
(Int
rodu
ctio
n to
St
atis
tics
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
537-
548/
4th
ed. p
ages
589
-599
).
13.
E.
Ans
wer
cho
ices
A,
B,
and
D c
an b
e el
imin
ated
sin
ce t
he n
ull
hypo
thes
is m
ust
incl
ude
the
equa
l ca
se.
Bec
ause
the
grou
p w
ants
to s
how
sup
port
for
the
n cl
aim
th
at m
ore
than
60%
are
spe
edin
g,th
e al
tern
ativ
e hy
poth
esis
w
ould
be
p >
0.6
0 (I
ntro
duct
ion
toSt
atis
tics
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
525-
529/
4th
ed.
page
s57
8-58
1).
14.
D.
The
eas
iest
way
s to
inc
reas
e po
wer
are
eit
her
to i
ncre
ase
the
sam
ple
size
or
use
a la
rger
si
gnif
ican
ce l
evel
(I
ntro
duct
ion
toSt
atis
tics
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
562-
567/
4th
ed.
page
s61
3-62
1).
15.
D.
The
des
igne
r is
int
eres
ted
in t
he p
opul
atio
n of
wom
en w
hoha
ve
prev
ious
ly
pu
rch
ased
th
e de
sign
er's
cl
othi
ng
and
the
stan
dar
d de
viat
ion
of t
his
popu
lati
on i
s no
t kn
own
(Int
rodu
ctio
nto
Sta
tist
ics
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
550-
558/
4th
ed.
page
s60
2-61
0).
234
* C
hapt
er9
FREE
-RES
PONS
E PR
OBLE
MS
1.
Hyp
oth
esis
Ha:/
/<$
10
12
a =
0.0
5
Ass
um
pti
on
sT
he p
robl
em s
tate
s th
is w
as a
ran
dom
sam
ple
of w
eddi
ng g
own
sale
s.
Sin
ce 5
0 >
30,
the
sam
ple
is la
rge
enou
gh f
or th
e on
e sa
mpl
e t t
est
to b
e ap
prop
riat
e.
Tes
t S
tati
stic
x =
$98
5
sx =
$23
5
n =
50
985 -1
012
p =
0.2
1
Con
clu
sion
Sin
ce t
he F
-val
ue o
f 0.
21 i
s no
t sm
alle
r th
an 0
.05,
the
re i
s no
tco
nvin
cing
evi
denc
e th
at t
he a
vera
ge a
mou
nt s
pent
on
a w
eddi
nggo
wn
at th
is s
hop
is le
ss t
han
the
nati
onal
ave
rage
of
$1,0
12
(Int
rodu
ctio
n to
St
atis
tics
&
Dat
a A
naly
sis
3rd
ed.
page
s 55
0-55
8/4t
h ed
. pag
es 6
02-6
10).
2.
(a)
H0
•. p
= 0
.39
whe
re p
= p
ropo
rtio
n of
sen
iors
ski
ppin
gH
a -. p
< 0
.39
(b)
The
app
rop
riat
e te
st i
s a
1-sa
mpJ
e z
test
wher
e
p-p
P(l
-p)
The
con
diti
ons
of th
is t
est
are
(1) r
ando
m s
ampl
e an
d (2
) lar
gesa
mpl
e. T
he p
robl
em s
tate
s th
at t
he s
ampl
e w
as a
ran
dom
sam
ple
of s
enio
rs.
Che
ckin
g th
e co
nditi
ons
for
larg
e sa
mpl
e:(3
98X
0.39
) =
155
, (3
98X
0.61
) =
243
. S
ince
bot
h n
p>
10
and
n(l -
p) >
10,
the
sam
ple
size
is
larg
e en
ough
.
(c)
A T
ype
I er
ror
wou
ld b
e th
at t
he i
Pad
dra
win
g w
ould
not
actu
ally
red
uce
the
prop
orti
on o
f se
nior
s w
ho m
iss
the
last
day
of s
choo
l, bu
t th
e sc
hool
dis
tric
t th
inks
tha
t th
at i
t w
illan
d im
plem
ents
the
draw
ing.
Hyp
othe
sis
Tes
ting
Usi
ng a
Sin
gle
Sam
ple
* 235
A T
ype
II e
rro
r w
ould
be
that
the
iP
ad d
raw
ing
wou
ld i
n fa
ctre
duce
the
pro
port
ion
of s
enio
rs w
ho m
iss
the
last
day
of
scho
ol,
but
the
scho
ol d
istr
ict
is n
ot c
onvi
nced
of
this
and
does
not
im
plem
ent t
he d
raw
ing.
(Int
rodu
ctio
n to
Sta
tist
ics
& D
ata
Ana
lysi
s 3r
d ed
. pa
ges
531-
534/
4th
ed. p
ages
582
-586
).
