Final Review Part B

Final Review Part B - 
 
 
 1. The
lifespan...

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Unformatted text preview: 
 
 
 1. The
lifespan 
of
a 
qu een 
ant
follows
th e
followin g
d istribution :
 
 


‐(4/5) x+1
 0≤x<1
 f(x)
=





c

 
 1≤x≤3
 
 
 



0
 
 x<0,
x>3
 
 (a) Find
the
function 
f(x) =c
tha t
is 
dis tribu ted 
1
≤
x
≤
3
(note:
c
is
a 
cons tant).
 
 
 
 
 
 
 (b) What
is
th e
a vera ge
lifespan
of
a
qu een
an t?
 
 
 
 
 (c) If
after
one
year,
th e
qu een 
ant
is
s till
a live,
what
is
th e
probability
it
will
still
b e
alive
for
a t
leas t
 another
five
months ?
 
 
 
 
 
 
 
 
 (d) Wh en
on e
queen 
ant
d ies,
anoth er
qu een
a nt
immedia tely
ta kes
its
p lace,
as
an y
given
ant
hill
will
 alwa ys
b e
ru led
b y
a 
qu een
ant.
If
an 
ant
h ill
goes
through
30
different
qu een
ants ,
what
is
th e
 probability
that
this 
ant
h ill
is
at
least
28
years
and 
fou r
mon ths
old?

 
 
 
 
 
 
 
 
 
 
 MATH
218
SI
Final
Review
(Spring
2009)
 Part
B
 2. Slick
Motor
Tire
In c.
us es
a
tire
mold
which
occasionally,
and
randomly,
produ ces 
out‐of‐round
tires.
In 
fact
 20%
of
the
tires 
wh ich 
it
produ ces
are
ou t‐of‐round .
Suppose
Slick
ta kes
a 
random
samp le
of
10
tires
 produced
b y
this
tire
mold.
 
 (a) Find
the
exp ected
nu mb er
of
out‐of‐round
tires
a mon g
th es e
10.
 
 
 (b) Find
the
probab ility
tha t
two
or
more
of
th es e
10
tires 
are
ou t‐of‐round .
 
 
 
 
 
 
 (c) Suppose
now
tha t
th e
Fraud
Motor
Company
bu ys
120
of
th e
tires
produced
b y
this 
tire
mold .
Find
 the
approximate
probability
that
b etween 
20
and
30
(inclusive)
of
th es e
tires
are
ou t‐of‐round .
 
 
 
 
 
 
 
 
 
 3. John
regularly
pla ys
a 
target
shootin g
comp etition 
in 
his
yard .
He
is
not
very
good
a t
th e
ga me
and
h is
s core
 has
th e
following
probability
dis tribu tion
 
 Score
 0










1











2










3
 Prob.
 0.1






0.4








0.3







0.2
 
 (a) Find
the
exp ected
valu e
and
standard 
d eviation
of
Joh n’s
score.
 
 
 
 
 
 (b) Suppose
he
pla ys
th e
game
twice.
Find
th e
probability
that
h is
total
is
a t
leas t
5.
 
 
 
 
 (c) Suppose
he
pla ys
th e
game
50
times .
Find
the
probab ility
tha t
h is
total
is
a t
leas t
75.
 
 
 
 
 
 4. Happy
Hun gry
Hamburger
restaurant
cooks 
an
a verage
of
1800
ha mburgers
pa tties 
a
week,
with
a
s tandard
 deviation 
of
200
ha mburger
patties.

Assu me
tha t
the
numb er
of
hamburger
patties
grilled 
follows
a
n orma l
 distrib ution.
A
loca l
report
sta tes 
that
n ext
week’s
d emand
for
hamburger
patties
will
b e
2200.
 


 (a) What
is
th e
probab ility
that
Happ y
Hungry
Ha mburger
will
not
be
able
to
meet
d emand?
 
 
 
 
 (b) This
week,
Happ y
Hungry
Hamb urger
grilled 
an
a moun t
of
ha mburgers 
that
ran ks
in 
th e
top
5%
of
 weeks
of
a mount
of
ha mburgers
grilled .
How
man y
pa tties
d id
Happ y
Hungry
Ha mburgers
grill,
a t
 leas t?

 
 
 
 
 
 (c) What
is
th e
probab ility
that
th e
a vera ge
nu mb er
of
ha mburger
patties 
grilled
each 
week
durin g
a
15
 week
period 
is
a t
leas t
1830?

 
 
 
 
 
 
 
 (d) A
statistics
professor
mad e
th e
following
statement
on 
th e
MATH
218
website:
“Th e
probability
tha t
 Happy
Hun gry
Hamburgers
will
cook
at
least
39,000
pa tties
in
the
n ext


$#@!


weeks 
will
b e
a 
sma ll
 chance
of
9.51%”
Unfortunately
th ere
was
a 
typo
in
th e
s enten ce.
How
man y
weeks
sh ould 
that
 statement
have
read?
 
 
 
 
 
 
 
 
 
 5. Let
X
b e
the
tra veling
time
from
h ome
to
work
for
a
ra ndomly
chos en
office
worker
in
Los
Angeles .
Assume
 that
X
has 
a
normal
distribution .
A
samp le
of
6
observa tions
from
this 
popu lation
gave
th e
following
times
 (in
minutes):
 35

20

25

15

22

21
 
 (a) Find
point
es timates
for
th e
p opula tion 
mean
μ
an d
population
varian ce
σ2.
 
 
 
 (b) Find
a
p oin t
estimate
for
the
standard
d eviation
of
th e
sample
mean
X.
 
 
 (c) Find
a
95%
confid ence
in terval
for
μ.
 
 
 
 
 
 
 
 (d) Suppose
that
you
are
told 
that
σ
=
8.
Us ing
this 
additional
informa tion 
find 
a
95%
confid ence
interval
 for
μ.
 
 
 
 
 
 
 (e) Rela ted 
to
part
d) ,
how
large
should 
th e
sa mple
b e
in 
order
to
estima te
μ
to
with in
4
un its
with
95%
 confid ence?

 
 
 
 
 
 6. A
youn g
bas eball
fan
n otices
tha t
professional
baseba ll
pla yers 
are
either
right‐hand ed 
or
left‐hand ed.
 
 (a) This
youn g
bas eball
fan 
would 
like
to
determin e
a
90%
confid ence
in terva l
for
th e
tru e
proportion
of
 baseball
p layers 
wh o
are
left‐hand ed
us ing
a
samp le
of
75
p layers .
What
would 
this 
young
bas eball
 fan
estimate?
 
 
 
 
 
 (b) Suppose
that
in 
a
samp le
of
75
profess iona l
bas eball
p layers ,
24
are
left‐hand ed.
Determin e
the
n ew
 90%
confid ence
interval
for
p.
 
 
 
 
 
 (c) Bas ed
on 
th e
preliminary
sa mple
in 
part
b) ,
d etermin e
how
man y
more
professional
baseba ll
pla yers
 are
n eed ed
to
estimate
p
to
with in
+/‐0.05?
 
 
 
 7. The
American 
Ch ees e
Compan y
s ells
pa ckages
of
ch eese.
Th es e
packa ges
are
sta mp ed
with
an
exp iration
 date;
thos e
packages
n ot
sold
b y
th e
expira tion 
date
are
d iscard ed .

The
compan y
cla ims
that
on
a verage,
 the
ch eese
can
be
kep t
for
at
least
14
da ys
past
th e
expiration
date
before
goin g
bad .

Th e
govern ment
 believes
that
ch eese
cannot
b e
kept
14
days 
past
its
expiration.

To
test
th is
claim,
governmen t
inspectors 
 decid ed
to
keep 
30
packages
a nd
record 
how
long
past
its 
exp iration
da te
it
ta kes
each 
package
to
go
 bad.

Govern ment
insp ectors
found 
that
th e
a verage
time
it
took
th e
30
pa ckages
to
go
bad 
was 
13
 days.

Th e
standard
d evia tion 
σ
is
known 
to
be
2.1
da ys.
 
 (a) Formula te
th e
null
and
a lternative
h ypoth es es
for
th is
test.
 
 
 
 (b) Which
tes t
sta tistic
should 
b e
us ed
to
tes t
th es e
h ypothes es?

Evaluate
its 
numerica l
valu e.
 
 
 
 (c) Find
the
P‐va lue.
 
 
 
 (d) Bas ed
on 
th e
P‐valu e,
at
wh ich 
significance
level(s)
should
th e
insp ectors
reject
th e
n ull
 hypoth es is?

Circle
all
tha t
app ly.
 
 0.1%



 
 
0.5%





1%


 









5%







 




10%
 
 (e) What
is
th e
probab ility
that
th e
insp ectors
wron gly
conclud ed
that
th e
ch ees e
cannot
be
kep t
for
at
 leas t
14
days 
past
its 
exp iration
if
the
level
of
significan ce
is
s et
at
5%?
 
 
 
 8. A
finan cia l
ana lys t
with
a
major
brokera ge
hous e
sp ecializes
in
a
group 
of
technology
s tocks .
Th e
a verage
 price‐to‐earn ings 
(P/E)
ra tio
of
su ch
stocks
has
b een
35.
We
will
assume
that
th e
d istribution
of
P/E
ratios
is 
 approxima tely
normal.
The
analyst
is
interested
to
s ee
if
th e
P/E
ratio
has
changed
after
th e
recent
a ctivity
 in
th e
stock
market.
Th e
popula tion
standard 
d eviation
is
un known .
 
 (a) State
th e
ap propria te
null
and 
alterna tive
h ypothes es .
 
 
 
 (b) Choos e
a
tes t
sta tistic
and
rejection
rule
to
tes t
th e
null
h ypoth es is
at
th e
1%
s ignificance
level,
usin g
 a
sample
of
20
tech nology
stock
P/E
ratios .
 
 
 
 (c) Suppose
th e
a verage
P/E
ra tio
of
the
twenty
stocks
ch osen
is
43,
with
s 
=
15.
What
would 
b e
th e
 analys t’s
conclusion
after
p erformin g
this 
hypoth esis 
test?
 
 (d) The
P‐value
of
the
result
in 
(c)
lies
in
wh ich 
of
th e
followin g
intervals?
Circle
on e,
and
justify
your
 answer.
 (i)

 more
than 
.10

 
 (ii)

 .05
to
.10

 
 (iii) 

 .02
to
.05

 
 (iv)

 .01
to
.02
 
 (v)

 .005
to
.01

 (vi)

 less 
than
.005
 
 
 9. One
of
the
larges t
firms
ma kin g
an d
marketin g
popu lar
music
has 
chan ged 
its
manager
respons ible
for
 acquiring
new
mus ical
ta lent.
In 
th e
pas t,
a t
leas t
35%
of
recordin g
contra cts
ha ve
resulted
in
h its.
Th e
 high er
level
executives 
wish
to
find
ou t
if
th is
prop ortion
has
d ecreas ed
with
the
n ew
mana ger.
Th ey
d ecid e
 that
they
will
collect
data
on 
how
man y
of
th e
record in g
contracts
mad e
by
th e
n ew
mana ger
have
resu lted
 in
h its.
 
 (a) State
th e
ap propria te
null
and 
alterna tive
h ypothes es .
 
 
 (b) Choos e
a
tes t
sta tistic
and
rejection
rule
to
tes t
th e
null
h ypoth es is
at
th e
5%
s ignificance
level,
usin g
 a
sample
of
70
n ew
con tracts .
 
 
 
 (c) The
samp le
of
70
contracts
resulted
in
17
hits.
Wha t
ca n
you 
con clud e
about
th e
prop ortion
of
 record in g
contracts
resultin g
in
h its?
Clearly
exp lain 
your
reason ing.
 
 
 
 (d) Determin e
th e
P‐va lu e
of
th e
resu lt
in
( c).
 
 
 
 (e) Without
prejudice
to
your
ans wer
to
part
c),
wha t
wou ld
b e
a 
type
I
error
conclusion ?
 
 
 
 10. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 11. Consid er
th e
following:

 
 [This 
is
an
exam ple
of
how
things 
could
be
worded
in
a
way
that
you
have
never
s een
before,
and
that
is
why
 this 
past
final
exam ple
is
on
the
review.]
 
 (a) Let
W
b e
a
norma lly
d istributed
ran dom
variable
with 
E(W) =15
and
P(12<W<18) =0.74.

Find
the
 standard
d eviation
 
 
 
 
 
 
 (b) Assume
X
is
normally
d istributed
with
mean
7
and
variance
4.
Find 
a
valu e
c
such 
that
P(X>c) =0.42
 
 
 
 
 
 
 
 (c) Y
is
a 
normally
dis tribu ted 
random
variab le
withmean 
20
and
s tandard
d eviation 
6.
Find 
P(|Y‐ 21|>6).
 
 
 ...
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