Ch 6 Test Review
Random Variables
Use the following for questions
1 - 2:
In order to set premiums at profitable levels. insurance companies must estimate how much they
will have to pay in claims on cars of each make and model. based on the value of the car and
110\\' much damage it sustains in accidents. Let Cbe a random variable that represents the cost of
a randomly selected car of one model to the insurance company. The probability distribution of
C is given below.
'l:2.. 1. The expected value of C is
U
(a) $155.
(b) $595.
AA'i.=
\)
(c) $875.
(d) $645.
(e) $495.
(6X,,~+ (6ro)C4l0~-t 00(0)(.22) + (?OCDXt»Z2") = Sq5
2. Which of the following is the best interpretation of expected value? (In the choices below.
"Exp(C)" represents expected value you found in question 1).
(a). If the company insures 10 cars of this model, they know they will incur 10 x Exp(C) in
costs.
(b) The maximum cost to the company for insuring this car model is Exp( C) per car.
(c) The company must insure at least Exp( C) of these cars to make a profit.
(d) If the company insures a large number of the-e cars. they can expect the cost per car to
average approximately Exp(C).
(e) If the company insures a large number of these cars, they can expect the variability in
~e~C~~il:;fEPhr~)(C)'
nril"VVvov
C
df' rn~t.vn:..¥1
~~~.
CCLV'5
lo.,.,~
t\~~
.
~~~.
3. A raiidomly chosen subject arnves for a stud of exercise ailO fitness. Consider these
statements.
',~~
I. After 10 minutes on an exercise bicycle, you ask the subject to rate his or her effort 011
the Rate of Perceived Exertion (RPE) scale. RPE ranges in whole-number steps frolll~
(no exertion at all) to 20 (maximum exertion).
L:vY ~ nUCtt Il. You measure V02'.the maximum volume of oxygen consumed per minute during
_
exercise. V02 is generally b~veen 2.5 liters per nlinute and 6 liters per minute.
D\50-J~. Ill. You measure the maximum heart rate (beats )er minute) .
.. "-
f\l).l\ti
S
The statemenrts) that describe a discrete random variable are
(a) I.
(b) II.
(c) I. III.
(d) I. II. III.
(e) None of the statements describe a discrete random variable.
~ In the town of Tower Hill. the lltlll~ber of cell phones in a household
the following; distribution:
17 with
I
P(X~ '2..j:= \-PCO) - r(~)
W
peW)
0.1
0.1
0.25
0.3
0.2
z:
0.05
1_(.\ +.\)= ...t
that a randomly-selected household has at least two cell Pl~.· es is
(b) 0.25.
(c) 0.55.
(d) 0.70.
~0.80.
The probability
(a) 0.20.
A ~
h
is a random variable W
A random variable
Yhas the following distribution:
r
,5::=3c-t2 C
pry)
Ae
value of the constant Cis: \
~0.10.
(b) 0.15.
'~ ~~
(d) 0.25.
(c) 0.20.
c~_ (
(e) 0.75.
A marketing survey compiled data on the number of personal computers in households. If X
household, and we omit the rare cases of
more than 5 computers. then Xhas the following distribution:
P(~ 2'J...)= \ -=-(y:J'Y
~
= the number of computers in a randomly-selected
I
.
X
P(X)
PC'/..
=
~
..
What IS the
randomly chosen household has at least tw~ personal c~uters?
(a) 0.19
(b) 0.20
(c) 0._9
. ~9.39
B 7)
A random variable Xhas
I
x~
P(X)
a prob~:::~li~ distribution
~
m
1~(·2.4+ 31\
1-.-l.9l~..-3G(
that a
(e) 0.61
as follows:
~~
~,
\
20- ..05
.~~~
Where k is a positive constant. The probability P(X <: 2.0) is equal to
(a) 0.90.
~.25.
(c) 0.65.
(d) 0.15.
c~
probability
X::;..f)
PC'/..~Dor \)
= ~~ t 2k
'5 ~
(e) 1.00.
5(JY5)~.
A business evaluates a proposed venture as follows. It stands to make a profit 0[S10.000
with probability 3/20. to make a profit of$5000 with probability 9/20. to break even with
probability 5/20. and to lose $5000 with probability 3/20. The expected profit in dollars is
(a) 1500.
(b) O.
(c) 3000.
(d) 3250.
(e)
-1500.
IOIccc(Jlt5)-1- t5C:c0C.45) + 0 ('2-5)+ C-5pCX5)(J5)
A rock concert producer
has scheduled
z:
3000
an outdoor concert. If it is warm that day, she expects
-I) to make a $20.000 profit. If it is cool that £1ay. she expects to make a $5000 profit. If it is
very cold that day. she expects to suffer a $12.000 loss. Based upon historical records. the
weather office has estimated the chances of a \Va1111 day to be 0.60: the chances of a cool day
to be 0.25. What is the producer's expected profit?
$5'OOO
$11.450
.P'I- z: 2O,Q)O
OCO(,
'r-(-50CO
l,to)+- 5
E(i):::
$13.000
~
(d) $13.250
(e) $15.050
=-
l\ ,460
2'5)
Xt \5)
26
X and Yare independent random variables. and a and b are constants.
following statements is true?
(a)
<7.\,+y
=
<7x
Which one of the
+ <7y
@.var(x-Y)=
'ar(X)+
Var(Y)
(c) Var{a+bX)=bVar(X)
(d)
<7x_y
= <7x
(e) Var ( X
- <7T
+ Y) = ~r-v-al-'(X-2 )-+-,-,.m-'(-y-) 2
*/~'
Let the random variable X represent the profit made on a randomly selected day by a certain
.-.). store, Assume that Xis Normal with mean $360 and standard deviation $50. What is P(X>
00)':
0.2119
(b) 0.2881
(c) 0.5319
(d) 0.7881
(e) 0.8450
111e variance
1) I~
Cal
OfS\Un~f~?r~2
(
+ (<7yl
OX + av. but only ifXalld Yare independent.
(o:xi + (Clyl but only if X and Yare independent.
e None of these,
A/4)
==
y~~~
Z
0)(+ 0\'.
(b) (ay,i
em
\:a~s(:nd
r?-
p tz 7 •~)
+ 0-2
~ .2\l'1
=: U
~
~
uJh.e.X\ od.c::ll()(). GY BLt'ot:to.cb n:
~"i~
ty\Q V ()d'
0 --
aX\CQ..,D
...--.
......\.J ,,_a
o....'f'e._U-U-=--_(}.X()\
__
Let the random variable Zrepresent the weight of male black bears before they begin
hibernation. Research has shown that Xis approximately Normally distributed with a mean
~~o pounds and a standard deviation of 50 pounds. What is P(X> 325 pounds)?
~.0668
(b) 0.2514
(c) 0.7486
(d) 0.8531
(e) 0.9332
r(:7323)= iJ(~?8A;:O)= P(Z IL6)~O.~~
The probability of getting exactly 4 eights in those 12 rolls is
Let the random variable Xl'epresent the amount of money Dan makes doing lawn care in a
randomly selected week in the summer. Assume that X is Normal with mean $240 and
standard deviation $60. TIle probability is approximately 0.6 that. in a randomly selected
week. Dan will make less than
Z: =0 \ (0 'X- ,2.5 5-C ~ (
(a) $144
(b) $216
(c) '255
(d) $30 :\
(e) $360
®
~
n)
C
2,40
If A
then
= result
f.iA
of the following is true?
(a) JID =l.vD =1.71+2.29
@lID = 1. vD = .J1.712
(d) JiD = 1, vD
J
9J
JlD
-=-
2.6':5
1?D \~
216 .3ba~~
= .J1.712
~2-
2..
- 2.292
=l.o-D =,./1.71+2.29
1l- ::: C5'~
~t
+ 2.292
()J5-A-
'2.
vU~ :::;;".r»--1t
+(),4
:: ~I)
(y~ +- O~
2
{
e:
i ('2 .'2Aj2.+{( .IQZ
.
~~
A vending machine operator has determined that the number of candy bars sold per week by
a certain machine is a random variable with mean 125 and standard deviation 7. His profit
011 each bar sold is $0.25. and it costs him $5.00 per day to maintain the machine and rent the
space for it. What are the mean and standard deviation for Y = the profit he earns from this
machine in a randomly-selected
week?
,{)
I 2.
0- :z
(a) Mean = 31.25. Standard deviation-$3.25
.
.c.a
(b) Mean = 31.25. Standard deviation $1.25
"
pro, rtCo'S
=
(c) Mean = 31.25. Standard deviation $1.75
~~fean
= 26.25. Standard deviation $1.25
®\1eall=26.25.Stalldarddeviation$~.75
'
AJ D -=: »(3 -A-
~.~
(b) JiD=LvD=1.71-2.29
(e)
+ LuC(.2tJ)
of a single roll of a six-sided die and B = result of a single roll of an s-sided die.
= 3.5. v.{ = 1.7 1.JIB = 4.5. and VB = 2.29. If D = the difference B -A. then which
5
+
I
'I =- /2.5 X --- 5,06
..J.)y
~()<25{t2S)-5 ~
=:8 (~2?-5
=- 2~.:L5
~;.5
·
It
,
I
I
)
\
25e. 280
(j;)The weights of adult men are approximately Normally distributed with a mean of 190
pounds and a standard deviation of 30 pounds. A::: tq 0
a- .;::30
,0
,l9- Jt:tD22D
~
(a) If you randomly select three men. what are the mean and standard de'dation of the stun of
theirwei21 ts?
~n
~=
db ()Umcf;rriR.ana
IqD + 190 -t-/QO
z::
570
(TT
=- of30:2. +"30
'2.
T~2
~ 6 LQ(p
CJT- -=: 5/, q{P
(b) An elevator in a small apartment building has a maximum weight capacity of 700
pounds, If three (randomly-selected) adult men g'et on the elevator. w iat IS t ie
probability that they exceed the maximum capacity:
perllOO'
')
®
-z:
P&' > ,00-619')::pr:.
oJ,q(o )
~
2.5D)
\:= 6.001.9 2.I
r-
'-~"/
The Internal Revenue Service estimates that 8% of (Ill taxpayers filling out long
mistakes.
e
p = 0~
fOll11S
make
(a) An IRS employee starts to randomly select forms=-one at a time-to check for mistakes.
: What is the probability that the [ust fOlm wit
es is the 7th one she checks?
0
r (It= ~)1t-pXP pC 101')
z:
cyoW\pJf1(.O~\-i)
6lOY'rttv' \ C
"
(.CJ2)(.DZ)
.+-J'
c=:t0,04 'l'5 \
<
"
(b) The same IRS employee announces at lunch one day that she checked 25 forms this
morning and didn't find mistakes on any of them, Is this surprising enough so that her
supervisor should worry about whether she is missing enol's? Explain,
PenO
yY\isto.~
in 26)::: (1- ,oif'l~~o-<\-~44~J
(pC )~ Q-,o~rr)(O'lf)~
no ,-fuyms Wlm
eVVOfs w.t
26 'is CJose to
\ l'(\ ~ ~-rn\s
13 not lAi\com~
exlOUZ)Yl -ftJy t1L{ ~~f\j\~
1YU. ~
woc~.
cr
C# ~~
let
Use the following for questions-a and
W
1-0
It has been estimated that about 30<)/0 of frozen chickens are contaminated with enough
salmonella bacteria to cause illness if improperly cooked. Chickens are delivered to grocery
stores in crates of 24. Assume the chickens are independently selected for inclusion in the crate.
..
C J~
The probability
(a) 0.0424
2.S)
D
C
p::Ol,3
.
.
.
b nom cO ~ (2.4 o,~,4-) c.~~~q
(JCX >4)
-:= \ "i
I
:=:
The mean and standard deviation of the number of contamina red chickens in a crate are
(a)
~1=~:
(b) P
.JJ.x
= 2.24
c
= I: o = 2.68
~~;~::
~~2~:
:'~:4
(e)
.21)
r)-::::l4
that a certain crate has ore than -4 contaminated chickens IS
(b) 0.0686
c
.8889
(d) 0.9313
(e) 0.95i6
~t
=
7.2:
(j
vLl'f:
= 5.04
::=:
OX ==--..r~ (\
np
NiI_P~
I I \,-,,\--1
z:
C7A-L e)
Z
1/2-
.J)
c; ~~(24)C3re')
z
2 -2.4-
Which of the following random variables is geometric?
(a} The number of phone calls received ill a one-hour period
~ The number of times I have to roll a six-sided die to get two 5s.
0 StACCe"?'X.~
~Tbe
number o.f digits I will read beginning at a randomly selected starting point in a table
" :\.
of random digit~.2~ltil I find a i.
fO\Y"YLf\::) (5UCC12~-S/fc::t"d.J
(d) The number of IS in a row ot 40 random dlgItS·ft~
i -lr(\~
(e) All four of the above.are geometric random variables.
., -. QQ.(X)fc\
rr'\a.\"S>
-rw
....r
"B-
-tt:-cf
S SlACC£"$S
-pnit:t;Zbl~\~
.s o...y~
5lA.'1~
Free Response
The Census Bureau reports that 27% of California residents were boru outside the Untied
States. Suppose that you choose four Califomiai . mde'pell et. of each other and at random.
Let F = the number of f~gll-bom
people in a ran on y-selected group of four.
(\Om I
p:::
(a) Find P(F = 1).
"16\
.2-, (\ ::;.4
Q\
P(F~~= [ ~Jc.:;t1)'C./3)3,z bhiOyY1frlf'(4
J
.21 J
i) =·D.42.
(b) Find P(F21).
.-
P (f- - I) =- \ - P(!==~ = 1i!3h \ -
PInOy¥)
r.rl f'C4, ,21,0
J\=;.---.lit"
(c) Find and interpret the expected value ofF.
C j=
P= 4l.21) ~ 1-0~
Y\U.YVl,l?e¥
(d)
F~~~tlt~
of w~0Y\
st~
~t-Erce.
of~'Y17U{J
oc~
r U)\ cle,vl~lJ .f;n D-- OCAmO Lt 0 P
~
~----------------------
('e~\Ctents "\()Q
bOV"Vl
0; -:::
'lnptl-p) ~~-t(oll)(~-=
OJM)(,ODt
'S ot:;ouJ.-
~\fe{oqf
'l11L )o~rUJ\
v ~
of 4-
J. o~
ss \hi ~~\-ed
Y\L,lyV\'y)ex
4 (Jdcl
0 "1s'~~.
HS
oP
~
fi:N'ei~Y1-~Yl
~
_
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