GLOBAL

Educational
I~courses

Probability and Statistics

GLOBAL
-I Educational
~courses
 GLOBAL
Educational
I==========:;;Courses
PERMUTATIONS

Permutations are the different arrangements of a given number of

things by taking some or all at a time

Exa1nples

All pernurtations (or arrange1nents) that can be fonned '\\<ith the letters a~ b, c by taking
three at a time are (abc~ acb, hac, bca cab~ cba)

All pennutations (or arrangen1ents) that can he forn1ed with the letters a, b, c by taking
two at a thne are (ab~ ac, ba, be~ ca, cb)

The nun1ber of different]Jernlutations of n distinct

objects taken rat a tinze is
P(n,r) = . nl
· (n- r)!

GLOBAL
Educational
I===========:;;Courses
Combinations

Each of the different groups or selections formed by taking

some or all of a number of objects is called a combination

Exa.1nples

Suppose we 'vant to select t\VO out of three girls P, Q, R. Thent possible eornbinations are
PQ, QR and RP. (Note that PQ and QP represent the sarne selection.)

Suppose '\v"e vvant to select three out of three girls P, Q, R. Then, only possible co1nbination
isPQR
The nutnber of difier·ent cornbinations of n distinct
objects taken rat a tirne is
C ( n~r) = P(n, r) = . n! .
· , r! [r! (n - r ) ! ]

nCr and (n,. .)·. are alten1ative notations i~r Ct(n•. r)

..
\ '
 GLOBAL
IEducational
~~co urses

. d.' .·a,1g11. en n;eers. are e .,· ·~,.bl.e· tOr

S. 1x ~ ~· t . . . . •.
pro1notton '
to .Pay
grade G8, b11t o:nly four spots are available~ Hovv many
different conibiilatiorm of promoted a.ngineers are
possible?

(A) 4 1 6!
C,(6,4) = t\'·(.' n-
.1.t • ·r. )·,·~. - t•t(6)
1t. -4 !
)9
(B) 6 6x5x4x3x2xl
-· 4x3x2xlx2x1
(C) 15 = 15

(D) 20 Tf1e answer is {C)"

GLOBAL
-I Educational
==============Courses

• h' t
A
4~11 1·d·e11tllcatao11
"'fi ·..., . •cocte
l . ·•
1ns \V!t :. . : ·ee !:etters.
'l . Tl
, 1e
possible letters are A, B, C1 D, and E. If no-n.t~ ,of tlJ,e
letter{i are ttsed mor·e than once, h:cJw ma.n)~ different
ways can tlu;l letters lJe ar:range,d. to Inatke .a code?

(A) 10
(B) 20
P( 5.3) = n.! = 5! _ 5 x 4 x 3 x 2 x l
(C) L!Q · (n- r)! (5-3)! 2x1
(D) 60 =·60

Tlte answer Is (D).

 GLOBAL
Permutations -I Educational
~~courses

of Different object types

Tl1e n11n1bei of clitleretlt p,ernzzttatio11s of 1'1 objects tc1ken

11 c1t c1 ti111e,, giv"re11 tl1at 1'li are oftY]Je i, where i == 1, 2, ...... ,
k a11d l:n.l == n~- is
P (,,·n ; r1b . ••• , l'lk .·)· == ·.·.·.· , n! .,. . ,,
. l'l'b
l'lt ~, 112 1
•••• J1k~'

GLOBAL
Educational
I=======~Courses

An Ul:Tl ,c ontaitns 18 marbles ·& otal: 4 , blae](. r:r:J,a rbles,"; 2 red,

1narl:>les, ancl 7 yello"'r m&"'bles. ArrangerrJet:r ts of 13 ma.r'-
bles are .rna.cte.. l'vlost n;ea:t~ly, 1:1 0"\V Inany unlqu,e ¥v:a.:ys c:::1n
t he 13 ·m arbles be: ordered (ruTanged')?
( A .) a.oo,
Tl1e tnarb1e colors represent different. types of objects~
(B) 1:2 0.0 ,rbe number elf· pe.rn1utations of the n1arbles taken 13 at
(C) l4,t 000 a tin1c is
(D~) 26?000 . n! _ 13!
P(13; Ll, 2, 7) = ' •·. ' . · ·...•. , - r.n
nt 1n2 ..•. n~; . 4.2.7.
13 X 12 X 11 X 10 X 9 X 8 X 7
x6x5x4x3x2xl
·- t--
! X3 --------
:x 2xlx2xl
x7x6x5x4x3x2xl
= 25;740 (26 ~000)

The answer is (D).

 GLOBAL
-I Educational
~=====Courses

GLOBAL
Educational
I============:!Courses
Basic definition

• Sample space
• Random experiment
• Random Variable
• Random Variables
• Event and type of event
• probability
 GLOBAL
Educational
I~courses
Sample space

Sample space :collection of possible output

of an experiment

taWI/iiliGLOBAL
• ~Educa~~~~~
Random expenment

Random experiment :experiment whose outcomes

are unpredictable
 ~~GLOBAL
~Educational

Random Variables courses

Discrete random variable

Continuous random variable

GLOBAL
Educational
I~===Courses
Event

Event :the out come of random experiment is

known as experiment mathematically event is
subset of sample space
 Event types
Complementary event (£):contain all out come
of sample space that is not in E
Equally likely events : events that have the same
probability
Dependent events : the probability of one event
depend on This occurs of the other event
Mutually Exclusive: can't happen at the same time .
•

QR~~GLOBAL
~Educational

LAWS OF PRQBABILIT J 1
Courses

Probability is the likelihood that an event will

occur under a set of given conditions. The
probability of an event occurring has a value
between 0 and 1. An "impossible event" would
have a probability of 0; a "certain event" would
have a probability of 1.

No of Ways Event A can occur n(A)

P(A) = =-
Total Number of Possible Outcomes n(S)

where n(A) is the number of ways in which Event A can occur and n(S) is the total number of
possible outcomes
 Example
Tw,o fair coins are tossed. What is the probability
of gettin,g one heads and one tails?

A die is roUed, find the pro;bability that the

number obtained is greater than 4
 GLOBAL
Educational
I============Courses
Total probability

Property 2 .. La'\Y of Total Probability

P (A + B ) == P (A) + P(B) - P(A$ B ), \Vll ,e re
P(A + B ) == tl1e probability tl1at eitb.er A or B occttr al or1e or
tl1at botl1 occilr together
P(~4 ) - the IJrobability that ~4 occ;IJrs
P (B ) - the probability that B occttrs
P(~4 , B) - tl1e probability tl1at both: A ar1d B o cc11r
sitilliltaneolisly

GLOBAL

-I Educational
==========Courses

A clecl< of te11 cllildren,'s carcls co,tltains 'tllt)ee fisl1 carcls}

t\VO ,dog carcls,, <:\tlcl five c,at :carets. ,_l1at is tl1e l)rofJ,~
al:>ilit)' of clra\ving eit,}Ier t\ cat~ ca\rcl or a dog CEtrct fi~OI11 a
fttll clecl{?
(A) 1/10
(B) 2/10
(C) 5/10
(D) 7/10
 GLOBAL
Educational
I===========Courses

ri'l1e t\'~lO events a.l~e llll ttllally exclttsive, S<.l the Jil"Obabil-
it_y of botl1 11ai)P€lli:tlg , P(A B) t is zero .. 'l"'tl1e total prol)-
al)ility of clra\ving eitl1er ct cat card or a clog card is

P(A +B)= P(A) + P(B )- P(A) B) = fa+ fa- 0

=7/10

GLOBAL
Educational
I=============Courses
joi nt probability

Propert)r 3. LalY of C~ olttpottnd or ,J oint Prob,ability

If t1eitl1er P(A) 11or P (B) is zero,
P (A} B) == P(A)P(B I A) == P(B)P(A 'I B)) \vl1ere
P(B IA) == tl1e prol1ability that B occurs given tl1e tact that A
l1as OCClll1·ed
P(A I B) == tl1e JJrobability that A oec11rs given the fact that B
l1as o,ccttll"ecl
If eitl1er P(A) or P(B) is zero, the11 P(A, B) == 0*
 GLOBAL

-I Educational
============Courses

A })l:tg COilt,a iilS se\retl or~'ll1ge r.>-alls, eigllt ,g reerl 1Jalls, ailCl
t~\VO \Vl1ite balls., 'I·",~.ro balls are clravvn fro111 ·tl1e l:>ttg
'~itl1out re}Jla,c lllg eitl1er of tl1err1. 1\!los·t 11earljl ~ ''"ll~tt is
th.e probability ttla't tl1e first l:)all clra\\7n is \Vllite a11d. tl1e
secoru:l l:>all clraw·11 is orange?
(A) 0J)36
(B) 0 .052
(C) 0'.10
(D) 0.53

~~GLOBAL
. . . .. . . . . . .~ Educational
"'r here is a t;ot,a l of 17 balls. ~I'b.e:r.~e are 2 '\Vl1ite, ball Courses
probability of pi<~king a white 1Jall as the first ball is
.
f>(rl).· = -1'27·.

After I'icl<lng a 1tvl1ite l:>all first, tJter,e ~-u."'e 16 balls

t:en1ah1ing) 7 of' \vhich. a1~e ora1:1ge. Th,e p·ro·b.abilit~y of
J>icking O.Il orange ball seco.n d give:n tlJ..a t t'1. \<'\;r}lit.e ba.Jl
\v.as chosen first is

P(BIA)= Z 6

l"rheJ)rol:>abilit~:*t of )Jicl<itlg a "v·hi te ball first a ,11.d an

orange t)all seconcl is

P(;:t, B)= P(.tl)P( ,BJA)

= 2)('
(.17 7)
'. .: ·161'

The ans,w er is (B)..

 GLOBAL
Educational
I============Courses
If three students vvork on acertaintnath question, student Ahas aprobability of success of 0.5,
student B~ 0.4 and student C, 0.3. If they vvork independently, the probability that no one \Vorks
the question successfillly is:

A. 0.12
B. 0.21
C. 0.25
D. 0.32

Multiplying the cotnplitnentary probabilities:

(1 - 0.5)x(1 - 0.4)x(1 - 0.3) = 0.21

GLOBAL
Educational
I~====Courses

Afinn rents cars fro1n three rental agencies: 60% fron1 agency D, 20% iron1 agency E, and the
rest frmn agency F. If 12% of the cars fl-on1 Dhave bad tires~ 4°/o frotn Ehave bad tires~ and 10°/o
fro1n Fhave bad tires, vvhat is the probability that acar that is rented \¥ill have bad tires?

A. 0.02
B. 0.10
C. 0.20
D. 0.24
 GLOBAL
-I Educational
~co urses

Let B == eve11t of l1a\li11g bad tires

P(D) == 0.6 P(E) == 0.2 P(F) == Ow2

(probabilities of re11tir1g fro111 age11cies)

P(BID) == 0.12 P(BIE) == 0.04 P(BIF) == 0,*10

P(B) == P(D) P(BjD) + P(E) P(B IE) + P(F) P(BIF)

== (0.6)(0.12) + (0.7)(0.04) + (0.2)(0.1 ·0) ·== 0.10

~~GLOBAL
~Educational

Condition al probability courses

...·\ . ··· , . ..., ). P ( B an A)

P(BJ 1A = - ··. (A)
 GLOBAL
-I Educational
~===Courses

A 111eclical patie1Tt .exhfl:Jits a sy111ptotll tl1c11t occl1rs Ilatll-

raJljr l :O % of tl1e tin1e i11 t:!.ll peo.r,le. Tl1e sy1·n 1'torn, is also
exJ1iiJitecl 1Jjf ~\ll})atie· Jits \~lllo ):Ja,re a. particttlar clisease.
' l1e i~:lciclez1ce of tl1at~ pa~rt,icttb:tr disease a ,lllOJ.lg all pe·<)-
]Jle is 0.00()2%. . l1at is tl1e prolJability of tl1e patie11t
.11av1ng . ·t J)arvlC
·.... ·.:·~·: . . :. t.~..·Ila :'·"~·· · t 1.&
'l· .r. ·d·.Jseas
• ~: et'l
:-·
{A) 0.0{)2%
(B) 0. 01%
(C) 0.3%
(D) 4%

GLOBAL

Bayes theorem
-I Educational
~====Courses

Ba~?·es• Tl1.eore1:t1.

P (BJIA ) = :(B~)P (AIBJ) .

L P (r1 IB, ) P(Bi)
i = 1

wl1ere P ( /:L .t) is the tJra·b abiiity of eve11tt .r1J "\¥itl1i11. tl'le
J:>OJJUiation of A
P(B.t)is tl1e probability of ev·e 11t B 1 "\>Vithir1 the
populatio:n of B
 GLOBAL
Educational
I~====Courses

A firm rents cars from three rental agencies: 60% from

agency D, 20% from agency E, and the rest from agency
F. If 12% of the cars from D have bad tires, 4% from E
have bad tires, and 10% from F have bad tires, given
that there is a bad tires, what is the probability that
was supplied by agency E :

GLOBAL

-I Educational
~===Courses

Let B = eve11t of 11a\ri11g bacl tires

"-"

P(D) = 0.6 P(E) = 0.2 P(F) = 0.2

(probabilities ofret'ltitlg fro11:1 age11eies)

P(BID) = 0.12 P(BIE) = 0.04 P(BIF) = 0.10

. P(E)P(B :E)
P(E : B) = P(E)P(B : E) + P(F)P(B : F) + P(D)P(B : D) = O.OB
 GLOBAL
- IEducational
========Courses

~~GLOBAL
Measures of central tendenciE~Educa~~Jr!~J

. h . dispersions
A r1t met1c mean
- Xl+X2 +... +Xn
X =Jl =__;:.______;; __ _____;,;_
n

Mode
The mode is the observed value that occurs most frequently.
Median
The median is the point in the distribution that partitions
the tota l set of observations into two parts conta ining
equa l numbers of observations

\Vhen the discrete data are rearranged in increasing order and n

'-~ ' *-

is odd~ the tnedian is the value of the ( n !1

· )lh•
item
\Vhen n is even, the tnedian is the average of the
n)' th and ·(n + .l )th. lte:rns.
(2 ~
2
 GLOBAL
Educational
I============Courses

Root-Mean-Square

The sample root-mean-square value = j(l!n)L.Jf;

The scuuple range R is the largest sarnple value 1ninus the

smallest san1ple value.

GLOBAL
Educational
I=============Courses

,.Tl1e ''.rater le·vel 011 a ta11l<. in: a c~heinic<;tl IJla.:rtt is t1'le:2t-

sureci ever:y· 6 l1ours,. , e ta.tll< l1t1s a deiJ'tll of 6 111 .. Tl1e
water le'vels 011 tl1e t~tl'tl< on a certai11 clay '~"ere fotli1(l to
'b e 2 . 5 In, 4.2 111} 5.6 ll'l ltl~cl 3.3 In. '¥11~\.t is Ill;O St nearly
tl1e root-IIle~t,n-scattare value of \vatet level for tl1at cha.:yr~l
(A) 2.0 n1
(B) 3*3 1:11
(C) £1.1 rx1
(D) 5.8 n1
 GLOBAL
-I Educational
~========Co urses

r
X
. rnts =

~:5~6(:~21~:3.3 m)
25
(i) ( ( . 2)

== IJ . Q7 111 (tl.l lll)

Th'e answer Is (C).

100 r~'\,tlclon1 sallll=>l<:~s '\V:e re ·,t.f.lk:etJ. frc>t:ll a l~'trg<;! poJ::>tllatiorL

1\. }:)al"'t icuJat· n·u:rl~Jarical cll.a.ract-eris·t ic of sa..t~l.J:)le. ci ite1111S
'-'vas :tlJC:aSlu~ed... 'rhe resul't..:s ·of ·t lle l:t'leasure:rne:tlts "ver<:.l .a s
ftl llo"vs ..
• 45 tlleasttrellle:n, t~s '""'ere l.:>et,.,vet''!:n () . 859 ~l;.l'1l·Cl Cl . 9(l0~

0.901 '\vas ol:>serve<:l Otl.ce ..

• 0~902 '\Vk\S ol).serv.e cl t~l'lree tirn.es ..
• 0~90:3 '~'"as c>1:>s<:~rv<~<:l '"\Vice~
0.90·4 ··v~ras ol:JserveiL·l £o·u.r ti1~nt~.

• '-!5 n::t<~.asttrell1.~1llts '\l\rere }:)e:t,YeC:tl 0 .. 905 at~d 0.9:58 ..

Tl'le stnallest 'raltle '\ Vas 0 . 8,59~- at1cl tl'le le:\,r gest v.alt.lt~ '\\ras
0.95·8 . 'l~l1e Sl.ln~t of all 10~0 l::ru::!~lst:tre:nlellts "'ras 91.170.
Excer)t t;J~ose :no·t ect,_ Il<> lnea.S Llt:"em;CllJb.s C>C<~l.l.:t":recl nlt:)re
t 'h a.xt t\vice,.
'\~llta:t, are tl1e (a.) rrte<.:'t-rt, (t'l-) lXlOdk"!, a.1~1<:l (c) r.r tec::lia11 <)f t~ lle
1,..l"1 ·. lll:d'lo ~;<... S' .,. s .,~e·
... - ~<::::.-.$~,.. 1'.. . ~, ..&.:..
.;·,.-.r-·r· ... ~:~
.a. ..:&,.L- "'~ ·tw.-.:;,:.ll·
~ · JL . :I..
~ ':: .,.1.....•·c:·'l:··:•
'--,_.;< ff.:.. ·t·),~t.;'#
.,. -'-,....:-',.,,.
\....,.... .~, ·•.'!!' r.tn~.}J'r ~'"!
1
1\..:.:~.
:{J .. ;, ''ti •
1

(A) 0~908; 0 . 902; 0~902

(I:~)() . ~}08;. Ow90Li; 0.9'03

(CJ) 0.912; 0 . 902; 0 . 90'2
CD') 0.912; (J.90.tl,; 0.903
 GLOBAL
- I
Educational
~~ courses

(a) . i tbe arithn1etic n1e.a n is

....,,.. ~- c··l /· ) ~ '\?' . -

/ \ . - , . 7.:L , ~,.r\.t~
( ..
1
]()Q)' <··91: · ·o
··· · ·17 ··) .-_, o·.;;;l( n.t. 1,. 7·.·
·.. (().~)12)
I'= l '

(b) ~l.hr~ tnode is tho value; 'that. occlrrs 1nost freqtten·t ly.
~T'he value of 0.904 o·c curred four C;itlles, an.cl 110 ot;her
tlleastu·E~11:1Cllts rcp<mt·ed l:ttore tbatl ft"Jttr t~in'tes .. O.f)04 is
t~ he tn<>de ..

(c) ~rilE~ 111ediau. is

tl'1e value at "the lnldJlOitlt of ar1 <lrdered
(sorted) set of n1easttre111ents . There \Vere 100 Ineasm·e-
I1l8Ilts~ so; tl1e In.i:ddJe of tl1e ord:ered set occ.ttra betwee:rt
tl1e 50th aJ1cl 51st Inet1St:treme~:lts~ Sitlc·e these tne:asure-
Irleilts are l)otl1 0.903, tl1e t\ver . e of tl1e t'vo is 0.903.
The·.answer Is (D)~

GLOBAL
Educational
I======~ Courses
Weighted Arithmetic Mean

Tl1e vveighted arithn,1 etic n,1ea11 is

L W·X
X 1-V = .z.: ·'llVi.· z, where
. .
 ~~~a~~~~ional
~ --courses
A course I1as tottr exc.llllS tt1at co1111)rise 'tl1e e11tire gracle
f(xt the course,. E,a.cl1 ex~tin is '\Veigltted . A stt1clet1t's
scores 011 a11 fotir exan1s and tl1e , veigl1t for eacl1 exa111
<::1:re as given.

exa111 stttcler1t score v.reiglrt

80% l
95% 2
72% 2
95%

'Vl1~tt is n1c)st; lleltrly tl1e stttdent 's fi11al gracie i11 tlte
c0 tl·t··s·e
.· -?
~
4

(A) 82%
(B) 85%
(C) 8.7%
(D) 89%t

GLOBAL
-I Educat.ional
============Courses

Tl1e sttlf]etlt S finaJ gt~a,cle is tilE! 'veigl1terl aritllm,etic

1

t11eat1 of tll;e i1lcli·vicltta.l exa111 scores~

(1)(80%) (2)(95 ) -1- (2)(72%) + (5)(95%)

-t~
2 ~+~ 5
= 88~9% (89%)

The answer ,Is (D),.

 GLOBAL

Geometric Mean
-I Educational
~==== Co urses

The sample geo1netric mean = nj A!X2 A:'} ... Xn

\\'l1at is nlos-t nearly t'he geornetr:ic rnean of tl1e followu1g

dat.a. set?

(A) 0 ~ 79
san1ple geon1etric n1ean =\IX1X2/'(a .. ·. )tn
(B) 0.81
(C) 0.'98 ~ (0.820)(1. )(2.22)
(D) 0 ~ 96 X (0.190}(LOO)

= 0.925 (0!93)

GLOBAL

standard deviation -I Educational

============Courses

Standard deviation measures the spread of a data distribution. The

more spread out a data distribution is, the greater its standard
deviation.

(j population = ~(1/N) L (A[ - t-1 Y

Tl1e satnple stctnclard deviation is

s = ~ I[ll (n. - .
1)], ~
n

1
( - )2
X;- X
 GLOBAL
IEducational
~====Courses

A cat colot1y li"\ri11g ir1 a. SlTlall to'v11 l1as a. total popttla-

ticJ.tl of severt :c ats . 'I,tle ages of tl1e cats a,r e <-lS sllO'Wll+
a,gtl lllttn]Jer

7 yr l
8 yr 1
10 yr 2
12 yr 1
1a yr 2
\t\l l1at is IIlost Jle<:\rlj tJ1e stail<t;;1.r<l <:levi~"ttdorl of tl1c:~ a. ge of
tl1e ·Ca,t I)OJJt11£ttioil~?
(A) 1.7 yr
(B) 2~() )'~r
(C) 2~2 j'r
(D) 2~~4 y"r

GLOBAL
Educational
u I============Courses
/J~ = (l/71e) 2: x i
:i = 1

= (!)· ··· +.··• ·• (1).·( 8. yr} ·.+ (. 2~(1..O. yr) ).·

(··· (1)(7 +yr)(1)(12
7 ·.. yr) + (2)(13 yr) :
= 1(}.4 yr
""

(7 ?
y:~~· -- lO~LJ J'r)~ + (8 :;rr - 10.4 yr)2
· ?
-J- (2){1() yr ·- - l()~tl yr)-
(~) + {12 )rr -· · 10 . 4:. )tr) 2
yr- lO.~t yr)
2
+ (2)(1.3
(2.2 yr)

The a~nswer is (C)#

Sar.r1ples o :f <.l.lt1·Ininu:rn:.oo.a1J.o:ir el1a:r1nels '\:i\rer·e. t·t~s:teci fo~r
stiffness~ The follo"V\.r.i ng distrfb:tl.ti.o n.. of results '\>v·e re
o bt;ai::ri·ed.

stiffness freql21ency;o-
2ot.180 2:3·
2>..~l40 35
'2 :4 .00 ·4 0
23.·60 33.
23·20 21

If ·thfJ mea·n of the · sru:nples is 2402., \Vhat is the approx-

imate st~andard deviatJo11 of the pop·u latio11 from wl1ich
tl1e san1ples are takenf?
{i\) 48.2
(B) ·1 9.7
(C) 50.6
(D) 50 .8

GLOBAL

variance
• -I Educational
======~Courses

population variance is defined by

= (I !N*Xj - fl) + (~ - fl)2 + ... + (X:v - fl)2 ]

2 2
6
N ·2
= (1 /LV) ~ C-\f - !1)
i= l
The saniJJ/e variance is
s 2 = [ll(n -1 )] i: (x;- X}
i= 1 .

Tl1e sa1nple coefficient of variation == CV == sf X

 GLOBAL
' I:'l1e nuJ:ltl:>er of san1,ples is Educational
I======Courses

'I'he sa;n~tple s,ta,nclard cle,viation, B, is the lll1,biaset.l esti-

n1J~l.tor of tlte J:lOIYttlation stailclar<] <lC'\'ia.t1o·r l, u.

s= /{1/(n -l)]~(Xi- X) 2

(2.3)(2480 -- 24()2f2
_._, (35)(24.t:t Q - · 2402) 2
_;1- (~10)(2400 - 2 -4 02) 2
+ (!33) (2360 - 2 4l02) 2 '
-!- (21)(2320 - 24,02) 2
=:· 50 .. 82 (50.8)

The .a .n swer Is· (D) ..

GLOBAL
-I Educational
======~ Courses
 GLOBAL
Educational
I============Courses
PROBABILITY FUNCTIONS

probability distribution function is some

function that may be used to define a
particular probability distribution. Depending
upon which text is consulted, the term may refer
to:
a cumu lative distribution function,
a probab ility mass function, and/or
a probabi lity density functio

GLOBAL
Educational
Probability Mass Function I===========Courses

T11ediscrete probability of a11y single,eveut, X=xi, occurring

is defined as P(xi) \\d1ile theprobabili~1 lnass fi1nctionof tl1f (1.·~~'1""""'------,
random variable Xis defined bvJ 1!:1$ •

f(xJ =P(}( =xJ~ k=t 2~ ···~ n

l~ f(:x) ~ 0

2~ ~f(x) - l
:X
 r, no. of heads Probability~ p{r
0 ..l.
0.3 -
32
..?;. p(r)
32
2 .!!! 0 ..2 -
32
3 to
32
4 s ...

,
0.1
32
5 1
32
Total
0 l
t I
0 2 3 4 s
Number of heads, r

GLOBAL
Educational
I~~courses
Probability Density Function

If)( is continuous, the probabili~v densi~vfunction, f, is

defined such that
b
P(a < X < b)= Jj(x)dx r De..nsfty Function

where the function j(x) has the properties

1. f(x) ~ 0
2. ~-~ f(x)dx "" I
- :.'C
 .[ti GLOBAL
Educational
============Courses

Cumulative distribution function

The cuJnulative distribution function~ F; of a disc.rete randotn
variable ..c¥ that has a probability distribution described by P(xj
is defined as
111

11 ( :r:m) == L . P (xk) == P (A."" ·< X 111) , 111 == :L, 2~ ..... ., 11

k=l

If_¥ is co11tinuous., the cu1nulative distribution fitnction~ F; is

defined by
,'\"'

FCx) == _( .f\t) cit

"\Vhich in1plies tl1at F(a) is the probability tl1at ~4""< a.

GLOBAL

-I Educational
~====Courses

l~or tl1<!
proba,b iHty density function slto\1vn~ ''tht:l.t is
the probability of tbe randon1 variable x· beb1g le.';)s
than 1/3?

f(xl :

1 X

(A) 0.11
(13) 0~22

(<;) 0.25
(D) 0.3~)
 GLOBAL
-I Educational
~~courses

1
~['he ])robabHity that; /~l is eqllal to t~lle area under
::ro: <
1
th.e, Cllrve bet\veen 0 and /3 . Frorn J~,q .. 5 .30,

l/ 3 1/ 3

F{O < x < i) = j f(x)dx = j 4x dx = 2x2~~/3

0 0

= (2)(i)2- 0
::;:::; 0~222 (0.22)

The answer Is { B) ..

A probability density function is given by:

f(x) = 0 for x < 1
f(x) =1.2zs for 1 < x < 5
X

f(x) =0 for x > 5

Find the cumulative distribution function of X.
For~,\: 1 < 1: F (x1 ) = ] 0 dx = 0
X!

For 1 < '"'l;t < 5,: F(x1 ) = 0+ JL25x- 2 dx

I

For 5 < x 1 < oo:

1 5 Xt

=J0 dx+ Jl.25x- dx + J0 dx

2

0 I 3

= 0+ (-1.25)[ x- 1 + 0 f
=( ~1.25)(.!5 -1 )·
=1
Then to sun1n1arize, the cumulative distribution function of Xis:.
0 for x1 < 1

1.25 ( 1- ~~ ) for 0 <x1 <5

and l for x1 >5

GLOBAL
Educational
I~====Co urses
 GLOBAL
-I Educational
=============Courses

Expected Values for discrete random

Tl1e ex1Jectecl \.raltie of X is defit1ed as

Tl1e \rariance of X is clefi11.ed as

2
6 = V[X] =
i ·•·· .. ···. . n:
~
.. . .
(xk- ...
11)2J(xk)
.
k ~ 1
T 'h e standard de·viatiot~ is give11. l>y
cs == /v[L¥]
Tl1e coe.f ficie11t of,rariatiotl is defit~ed as cr/ J-L.

Expected Values for continuous

random variable
The 111ea11 or expected valtte of tl1e ra11dom ·variable X is 110\¥
defu1ed as
00

~ == E[X] == f ~t:f(x)ctr

-oo

6 == jv[x]
Tl1e coefficient of\rariatio11 is defi11ed as crl f.-L.
 GLOBAL
Educational
I=======~ Courses
~P.l1e rJr(ll:lal:>ilit~, ciisl~. x~ibuti<..)t:l of tl1<~ l1t.:tt'l:tber of calls,_ ..X,
tJ:1 at a . Ct.lstcllT.leT' st:~rvice a .g E'n t~ rect;~ivc~s each. llOl:tr is
sl"H)Wl~l ..

:c J(a:)
0 0.00
2 0.04-
·4 0.05
6 0~10
8 0.85
10 (L46

'"'ll;;:t;t is tt1ost lle<-:tl·· ly t~ l1-e a'\,er\;;\,ge nlll'lll:>er of pb.<>lle caJls

tl1at ..£\ Ctlsi;o:rn<:~r· se.r vice age:nt a;..;::m ;) ects t.,o l"'<!Oeive i1:t a:11
h<)lrr?
(A-) 5
(B) 7
(C) 8
([>) 9

GLOBAL

-I Educational
==========Courses

n
IL = E[X] = "',xkf(xk)
A~= I

= (0)(0.00) + (2)(0·~04) + (4)(0~05)

+ (6)(0~10) 1~ (8)(0 .3·5) + (10)(0~46)
= 8.28 (8)

The·answer Is (C),.
 Combinations of Random Variables
y = a I XI + a2X2 + . ·- + Cin.A:;-~

Th.e ex1jectecl ·val1-1e of l 7 is:

IJ-y = E(Y·) == atE( Xi) + c12E( A2.) + ... + Cln E( ~1 )
If tl'le randot'1'1 variat-,les are statistically irzdepende11t, tlJ.etl tl1e
·v·ariat'lce of .Y is:
a~== T-7 (Y) == a~V(Xt:) + ct~V(X2) + ... + a;v(./¥,1)
2 2 2 2 2 2
= 0 1 0'1 + C:l2 (52 + ~ ·· + Cln (Jn

Also, th.e stan.dard de·v·iatio11. of Y is:

(JJ. == g
Wh.e11 Y = ./{X1 , X 2 , .. , ~.1 ) at'ld ~are it"1de1Jendet1.t, tl1.e stat'ldard
de,riatiot"l of 17 is expressed as:

GLOBAL

- I
A. gas station sells three grades of gasoline; regular, extra and super. These are priced at $21.20
Educationa
==========Course

£21.35 and $21.50 per gallon, respectively. Let X1, X2, and XJ denote the an1ounts of thest
grades purchased (gallons) on a particular day. The Xi are independent \¥ith p-1 == 1000, p2 == 50(
1nd ~t3 == 300~ 01 ==100, cr2 = 80 and 03 =50. The standard deviation is nearly:

A. 222
B. 322
C. 422
D. 505
 GLOBAL

-I Educationa
~====Course

The revenue fron1 sales is:

Y = 21.2X1 +21.3SX2 +21.5X3

E(Y) = 21.2!11 +21.35tL 2 +21.5tl3

V(Y) = (21.2) 2a[ + (21.35) 2af + (21.5) 2a-} =104,025

ay = ~ 104,025 = 322.53

GLOBAL

-I Educationa
==========Course
 GLOBAL

inomial Distribution
- I Educationa
~====Course

P(x) is th_e prol"">ability th_at x Sl.1ccesses -vvill occ1.1r in. n trials.

Ifp = }=>robabilit)r ofs"Llccess ar1d q =probability offail"Llre =
1 - p, tl1e11

.Pn (.. ~JC ) == C.., ( 17" .JC ).. px qn. - x = ( 11'

. ) ]:J X q 11- X ,.
x! · rz - ,.,x- !
-vv1'1ere
_x- = 0, 1, 2, ... , n
C(n, --~) = tl1e nl.ltTlber of cot1'1bi11.atiotl.S
1'1, p = 1=>arat1.1.eters
0.-~

Tl1.e variat1ce is given by tl1e forn'l:

.,
.
cr- = npq O.t

o 1 2 3 4 s & 7 a a 10 11 12

Mean=np

GLOBAL

'
-I Educationa
~~course

Four fair coins are tossed at once. Wl1at is tl1.e

probability of obtaining three heads and one tail?
(A) 1/4 (0.25)
(B) 3/8 (0.375)
(C) 1/2 (0.50)
(D) 3/4 (0.75)
 GLOBAL
Educationa
I~===Course

.. Tqe binomial probability funetio11 can be used to

deter,nine tl"le probabilit:y of three h.eads i11 four trials.
p · P(heads) = 0.5
q = P(n.ot heads)= 1 - 0~5 = 0.5
n = n.u.m .b er of tria.Is = 4
x= n1..1mber of successes= 3

From the binomial function)

P n(a;)
·
= n~'
x!(n.-x)l
. p·T: qn..-x

41 ) · ( . )3 (0 5)4.-3
= ( 3!(4- 3)[ 0 •5 . . $

= 0.25 (1/4)

The answer is (A)..

GLOBAL
-I Educationa
~~· Course

What is the approximate probability of exactly two

_ _ J

people in a group of seven having a birthday on

April15?
(A) 1.2 X 10-18
(B) 2.4 X 10-17
(C) 7.4 x w- 6
(D) 1.6 X 10-4
 GLOBAL
-I Educations
=============Course

U s e the binoxnial prob ability func tiox1 to calculate

g

the p robability that two o f the sev en s a mples '\Vill h.ave

been b o rn on April 15 ~ x = 2 ) and the sa:rnple size,. n, is 7.
The probability that a pers o n ' v ill l1.ave bee11. bort~ on
A pril 15 is 1/365.. Tl~erefore, the probability of ''suc-
ces s ,"' p, is 1/365, and the p r obability of c:'failure,''
q = 1 - p, is 364/365~

P n(x) = . I( n! ) p"'qn- :.
· X. n - X!

~ 2)t) Cs!s) G~:)

7 2 7 2
P 7( 2 ) = (2!(7 -

1 ) ·(364)
2 5
- ( 2 l) ( 365 .365 .
1.555 x lo-.(1

The a n s w e r i s (D).

Cumulative Binomial probabilities

Cum .nl:athte Bi.n o·:m.i:n] Probabi1it£e-!l P(.Y.~ _,-)

1()5000 0.4000 (L3000 0 .2000 0.1000 0 .0500 (1_0100

2 0 0 .:8100 0.6400 0.2500 0 . !1.600 0.0900 0J)400 0.\HOO 0.002..5 0 . 0001
l 0.9900 0 . 9600 0 .9 1 00 0.7500 0 .6400 0 ..5100 0.3600 0.1900 0J)9'75 0 . 0199
3 0 0.7290 0 .51.20 0.3430 0.2160 (1.1250 0J)640 0.0270 0.0080 0 . 0010 0.0001 0 . 00'{)0
1 0.9720 ()_8960 0..7840 0.6480 0.5000 0 ..3520 0.2160 0.1040 0 .028{} O.OG73 0.0003
2 0.9990 0.9920 0 .9730 0.9.360 0 .8150 0.7:840 0.6570 0.4SSG 0 . 2710 0 . 142.6 0.0297
4 (} 0 ~6.561 OA096 CL240Ji. 0.1296- QJ)625 0.02S:6 CUOOSl O.OOHJ. 0 .0001 o.oooo. 0.00(10
l 0 .9477 0 .8192 0.6517 0.4752 0.3>125 0 .1792 0 .0'.837 0 .0272 0 _0()37 0 .0(}05 0 .0000.
2 0.9963 0 .9?2.8: 0.9163 O.:S:20S 0.6875 05248 0.34:83 0 .1808: 0..0523 0 . 014-0 ()_()006
3 0.9999' 0.9984 0.9919 0.9744 0.9375 0 .8:704 0 .7599 0..5904 0.3439 0 . 1855 0 . 0394
"
...;! 0 0 .5905 0.3277 0 . 168:! 0.0778 0.0313 0 .0102 0.0024 0.0003 CHJOOO 0.0000 0 . 0000
1 0.9185 0.7373 0 .52S2 0.3370 0.1875 0 .0870 Ct030S 0J)067 0.0005 0.0000 0.0000
2 0..9914 0.9421 CUB6'9 0.6826. Ct5i000 03174 0.1631 .OJJ579 0 .00&6 O.OG12 0 .0.000
3 0.9995 0.99.33 0 .969'2 0.9130 CLS125 0.6:630 0.4718 0.2627 0..0815 0 . 0:216 o.oo:w
4 1.0000 0.9997 0 . 9976- 0.989£ Ct96SS 0.9222 O.S319 0.6723 0.40~5 0 .2262 CL0490
6 0 0.5314 0 .2621 0 ..1 176 0 .046 7 CL0156 ().0041 0.0007 O.OOI:H 0..0000 0.0000 0 . 0000
l 0 .8857 0 .65.54 '0 .4202 0.23:33 O. Hf.94 0J)410 0.(H09. 0.0016 0.0001 (1.0000 0 . 0000
2 0 ..9:842 0.9011 0 . 7443 R5443 0 .3438 0.11.792 0.0705 0.0170 0.00.!1.3 0.0(}01 0 .0000
3 0.9987 0 .9830 0 .9295 CL820S 0.6563 0.4557 0.2557 0.0989. 0 .0159 0 ..0022 0 .0000
4 0.9999 0 . 9984 0 . 989'1 0.9590 0.8-906 0.7661 05798 0.3446 0.1 143 0.0328 0 . 0015
5 1 .:000.0 0 ..999'9 (t99'58 0.9.9 :59 0 .9844 0.9'533 0.8824 0.737'9 ,0 .4686· 0 .2649 (1.0585
7 0 0.4783 0 .2097 0 .0824 0.0280 0.0078 0.0016 0.0002 0.0000 0 .0000 0.0000 0.0000
1 O.S503 0.5767 0..3294 0.1:586 0.0525 0 .0188 I!L0038 0.0004 0.0000 0 .0000 0.0000
2 0.9743 0 .8520 0.6471 0.4199 0.2266 0.0963 0.0288 0 ..0047 0 .0002 0.0000 0 . 0000
3 0.9973 0 ..9667 0.8740 0 .7102 05000 0 ~2:898 0.126'0 0.0333 0 . 0027 0.0002 0 .0000
4 0 ..999S 0 . 9953 0 .9712 0 . 9037 0 . 7734 0.580 1 0 .3529 0.14:80 0.0257 0.0038 0 .0000
:5 1.0000 0 .999'6 0.9962 0.981:2 0 .9375 0.8'414 C:L6706 0.4233 0 ...1497 0.0444 0.0020
l nnon 1 (101110 il QQR&. il 11 Qr'"Hl n 0 ;:;;·Hr O";tl'lll r n n.,;;7Q
" ·ll QQQ<R Ql:}")";t CH7" 07Qf11'::t
On e Fair Co in is Tossed 10 times .what is the
probability of obtaining at least 4 head ?

X \··> ;fO:ll ,a::s 0.99

10 ('}. 0.34:87 0.1074. 0.0282 110060 (LOOlO 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
l 0.7.361 0.3758 OJ.4'9:3, OJ>464 tt010'7 -o.0017 0.0001 0.0000 0.0000 <LOOOO 0.0000
2 0.9298: 0.6718 0...3828 0.16.7.3 {JJ}547 OJH23 ()_0016 0.0001 0.0000 0.0000 0.0000
3 0.9S72 0.8'791 0.6496 Q.3&23 tU1l9 0.0548 O.J1106 0..0009 0.0000 0;0000 0.0000
4 0.99'84 (L9672 O.S4'97 0.633 1 0.3770 0.1662 0.0473 0.0064 0.0.001 (tOOOO 0.0000
5 0.9999 0.9936 0.9527 1'1.8338 0.6230 0.3669 0.1503 0J)328 Il0016 o~oooi 0.0000
6 1.0000 0.9991 0.98'94 0.9452 0 ..8281 0.6177 ().3504 OJ209 (}.0128 0.0010 0..0000
1 1.0000 0.9999 0.9984 0.9877 0.9453 fL8327 (U:H72 03222 0.0702 CHHl5 o.oocn
8 1.0000 U:tDOO 0.9999 t19983 {L9893 0.9536 0.8507 0. ..6242 0.26:39 0J}B61 0.0043
9 1.0000 1.0000 1.0000 0.9999 0..9990 0..9.940 0.9718 0.892.6 0.6.513 0.4013 OJJ956
- - -·- ·--
 Poisson Distribution
1./e-l
J(x;l)= I 0.2

X.
0 ;1
Mean=Variance= ).._. =np

Poisson Distribution

~ Discrete
~ Number of evertts. dier,< time
~ Errors per I,000 transaction
~ ·Calls per hour
~ Breakdowns per week
~ Lambda (/~) is the mean a-n d the
variance

A sho p sells five pieces of shirt everyday, then

what is the probability of selling three shirts
today?
 Mean value

What is the approximate probability of exactly 2

in a group of 1000 have a birthday on 15 April ?
 GLOBAL

Normal Distribution -I Educationa

====~Course

(Gaussian Distribution)

f(x) == ···) e-2

1 (X - !-!-)2
-- <r-. - , where
. 6 /2ii.
J-t th.e pop1.1lation. tl.'lean.
=
0' = the standard de·v iation of tl'le pop1.1latiot1
-CX)<x<oo

When J..L = 0 at'ld cr2 = cr = 1, tl1e distribution. is called a

stanclardized or unit nor1nal distrib1.1tion. Then

j(x) = ,;-k e- x 212

, where -oo < x < oo.

It is 11oted that Z == x - ~1 follovvs a stan.dardized n.orrn.al

()
distribl-ttiot'l fur1ctior1~
 GLOBAL

Normal Distribution
- I Educationa
==========Course

GLOBAL
Unit Normal Distribution
-I Educationa
==========Course

X fi.d~) F{-r) R(;'t) 2.R,'t) Jri-r.)

0.0 0.3989 05000 0.5000 1.(1000 0.0000
O.l 0.397() 0.5398 0.4602 0~9203 0J)79
0.2 0.3910 05793 0.4201 0.8415 0.1585
OJ OJ814 0.617:9 OJ821 0.7642 0.2358
0.4 0.3683 0..6554 0.3446 0.6892 OJ lOS

0.5 0.3521 0.6915 0.3035 0~6171 0.3S29

0.6 0.3332 0.7257 0.2743 0.5485 0.4515
0.7 0.3123 0.75&{) 0..2420 0.4&Jg 0.5161
O.S 0..2'897 0.7&81 0.2119 OA237 0.5763
0.9 0.2661 0..8159 0.1841 03681 0.6319

LO 0.2420 0.&413 0.158'7 0.3173 0.6827

u 0.2179 0.864:3 0.1357 0.2713 0.728
1.2 0.1942 0.8849 0.1151 0.2301 0.7699
lJ 0.1714 0.9032 0.09£8 OJ936 0.0064
L4 0.1497 0.9192 O.OSIJS OJ615 0.8385

1.5 0.1295 0.9332 1l066S OJ336 0.8·664

1.6 0.1109 0..9452 1).0548 o~I096 0.8904
 GLOBAL
-I Educationa
=============Course

Tlte lteigltts of st~vera1 tl10t1sancl fiftl1-gracle boys i11

Sa11ta Cla.r a CottntJ',.' aTe tne·asttrecl~ Tl1.e I:Jtea11 of tl1e
. . 1 2
l1eigl1ts is 1~20 111, a11cl 'th·e varia11ce is 25 x 10-J 111.
Approxillla,telj' lVh.a t p·erce11 ·Of tJ1ese boys i.s taller
tl1a1t 1.23 1r1{'?

(A) 27%
(B) 31%
(C) 69%
(D) 73%

~ ~ .\ ' X
A A
.\ ' __,. X

.\' fl.\") F~x) R(x} 2R~Y)

0.0 0..3989 0.5000 0.5000 LOOOO

0.1 0.3970 0.5398 0.4602 0.9203
0.2 0.3910 0.5793 0 .4207 0.8415
0.3 0.3814 0.6179 0.3821 0.7642
0.4 0.3683 0.6554 0.3446 0.6892

0.5 0.3521 0.6915 0.3085 0.6171

0.6 03332 0.7257 0.2743 0.5485
0.7 0..3123 0.7580 0.2420 0.4&39
 GLOBAL

-I Educationa
~=-Course

a= -fdi = V25 x 10'=>~ 4 rn 2 = 0.05 111

.z.. ··- ~ =J1». . . .... . 1.23 (J.()5

U -·
{T
- ·
rll - . 1 .. 20 .Ill - .
. .........
Ill
- 0·..'. ,. .B··".

at Z= 0.6 is F(Z) = 0.7257. Tl1<:: J?ercentage of l:>Oj'S

ha.vi11g heigl1t greater tl1a11 1.23 111 is

pereen.ttlg·e taller tl1art 1.23 n1 = 100%- (0.7257)(100%)

= 27.43~) (27%)

The, Bl'lSWer Is (A).

GLOBAL
-I Educationa
========~Course

2. Sa111ple8 of altt111lnun1-allo:;~ charm·e ls are testecl for

stif ess~ St;iffnes.s is ·tlO.r lllally" ciJstriblJtec:t Tl1e follo'v'i1,1g
fT·eCJtlellc:y (listrll'>lltio,l l is obta.it1ecl .
St, ·1·:. .'f~c·
~ . .J...n •;t"JjQ.Q ·
: ·
~ .v~'~ frequ.:en.cy
2480 2.3
2l!.tl0 :35
2tl00 40
2360 83
2320 21
\¥l1at is tl1e .a pproximate prol)ability tl1a:t, t11E'!' stifftle&~
of a11y gi\"011 chatl.Ilel sectio11 is less thm1 23-5 0?
(A.) 0'.,08
(B) 0.16
(C) 0.23
(D) 0.36
The arithmetic mean is an unbiased estimator of tl-~~~~~~~cioounrsae
population n1ean.
n
.)[ = (1/n)Z: .Lyi
f= t

(2480)(2,3) + (2440)(35) )•

(·.~.
152. ) (
1
= . ·.· +:. (. ..2·... ~:1·0· :·.0.·:· ).··(·
. 4··· 0·.· )···: -f- (2360)(33) :.i
-..- (2320){21) .·
=2402
The san1ple stan(lard deviation t,., an unbiased esthnator
of the sta11dard deviation.

r;:;;
. .· [1/(n-
s ""' VLJ./\' IJlL
.i =l
" .( x, -X)
2

(23)(2480 - 2402) 2
+ (35)(24L!Q - 2402) 2
1
(:' 152 - l ):: + :( 40)·(.· 2400- 2402) 2
4- (33)(2360- 2402) 2
+ (21)(2320 - 2.402) 2
= 50.82

A X
~ .\'
A .\"

X fl:Y) F(x) R{x)

0.0 0.3989 0.5000 0.5000
0.1 0.3970 0.5398 0.4602
0.2 0.3910 0.5793 0.4207
0 .3 0 .3814 0 ..6179 0 ..3821
OA 0.3683 0.6554 0.3446

0.5 0.3521 0 ..6915 0.3085

0.6 0.3332 0 . 7257 0.2743
0 ..7 0.3123 0.7580 0.2420
0 .8 0.2897 0.7881 0.2119
0 .9 0.2661 0.8159 0.1841

LO 0..2420 0.8413 0. 1587

. .- ....... _
...... A_.,.. ,....
.... ·---
 GLOBAL
- I Educationa
~=====Co urse

Z = 'X -· /};, = 2:;3~0 ~- 2LiQ2

u 50.82
= -1.0

Sinct:! tl1e llnit; rttJrlTl~lJ clistril:) t.tf;iot:l is, syr:nl:tl<~tricftl about

:1:>= 0 , the J:::>rol::Jability of a; bei11g in tht~ interval [ -O<.J,~ -l)

is the saxne as :c b ,ei11-g i.n -t,he i:nte~rv<:ll [+ 1, +CoO). This

corresponds to tll<:~ v~1..lue of R(x) irt r:r'al)le 5 .2.

P(,X< 2350)

Z = - 1 .. 0 0 X

..tr:>( , ,. .-· <·

" · ,.tl. ··.
'*2 ~-.> c:: : :o···)·
. ttJiO ·. · · ·-- · J:>( z- <·
·. ·. . .';,.;1 ·- · , · · .
1: ~ {·
· )·)··

= R"(l.O)
= 0.1587 (0.16)
The answer i s ( B )'.

GLOBAL

-I Educationa
~===Co u rse

6., }\ llOrrnal clistribtltiOllfl " a 1110 Of 12 a11cl a S't atl-

cl cl deviatiotl ·Of 3. If as ple is ta.ke11 fro·nl 'the r1ormal
clistribt1ti , 111ost 11ec:\rly, vvl1at is tl1e pro·babilit)' that
·t.·~ }.1e
··. · ·s.
;a
.· ·'.Illll · ·'v
· ·. ·1e . · .·_c~~··t·l,·l:~ b: 1•~- ' Ir,
> e Je ·' ' ee11 ~ ; o c:t1
· · }·>..... t-- · r
< · ·.
1Cl-~ 1.-8
11
. ..·<?•

(A) 0. :0:91
(B) 0.12
(C) O.ltl
( ) 0.16
 11\_ ~ ."\'
·'~
A A A X -X ,"\:' - :,.\' X

;1;- Jtx) F(x) R(;Y) 2R(\:) Jr{x)

0.0 0.3989 0.5000 0.5000 LOOOO 0 .0000
0.1 0.3970 05398 0.4602 0.9203 0 .. 0797
0.2 0.3910 0..5793 0.4207 0.8415 0. 1.585
0.3 0.3814 0.6179 0.3821 0.7642 0.23.58
0 .4 0.3683 0.65.54 0.3446 0.6892 0.3108

0.5 0.3521 0.6915 0.3085 0.6171 0.3829

0.6 03332 0.7257 0.2743 0.5485 0 .4515
0.7 0.3123 0.7580 0.2420 0.4839 0.5161
0.8 0.2897 0.78.81 0.2119 0.4237 0.5763
0.9 0.2661 0 .8159 0.1841 0..3681 0 .6319

LO 0.2420 0.8413 0.1587 0.3173 0 .6827

u 0.2179 0.8643 0.1357 0.2713 0.7287
1.2 0.1942 0.8849 0.1151 0.2301 0.7699
1.3 0.1714 0 .9032 0.0968 0.1936 0.8064
1.4 0.1497 0.9192 0.0808 0.1615 0 .8385

1.5 0.1295 0.9332 Q.0668 0<.1336 0.8664

L6 0.1109 0.94.52 0.0548 0.1096 0.8904
L7 0.()940 0.95.54 0.0446 0.089'1 0.9109
L8 0.0790 0.9641 0.{)359 0.071.9 0 ..9281
1.9 0 .06:56 0.9713 0J)287 0.0574 0.9426

2.0 0.0540 0.9772 0.0228 0.0455 0 .9545

GLOBAL
-I Educationa
===========Course

:t:1- J.L =: _15_

.;·. __,.,..,..1_2
zj 1
(J' 3
. . !1;2 ~
Z 2= J.1~
- ·
18 - l~ = 2...·.
rr ~i. ·
Frorn the standard norn1al ta,ble~ the probabilities are

P (Z < 1) = 0.8413
l~ (z < 2) = o.9772
rrhe probability that the outcorne will be bet'\veen 15
and 18 is

P(15 < :t < 18) = P(x < 18) - P(x < 15)
= P(Z < 2) - P(Z < l)
= 0~9772- 0.8413
= 0.1359 (0.14)

The answet· is (C).

 GLOBAL

T-distribution
-I Educationa
=====~Course

.
r(. v +2 1 ) (. '2)'- 2 v+ 1

j( t) = ( v·)·. 1 + v .·
~VTtr 2 Student's !-Distribution

vvl1ere

v == tltnnber of degrees oj{reedon1

n == san1ple size
v==n - l - - - - ------1--

[' == ga111111a ftu1ction VALUES OF tn,v

== x-·· ..~.
t
Ls~/ rn
-co< t < OCJ

-I GLOBAL
Educationa
I~=====Course

Ct.
v l'
0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005
1.000 1.376 1.963 3.078 6.314 12.7C6 31.821 63.657
2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 2
3 0.765 0.978 1.350 1.638 2.353 3.182 4.541 5.841 3
4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 4
5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 5
6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 6
7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 7
8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 8
9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 9
10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 10
11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 11
12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 12
13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 13
14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 14
15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 15
16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 16
17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 17
 CONFIDENCE INTE RVALS

Population properties such as means and

variances must usually be estimated from
samples. The sample mean, X/ and sample
standard deviation, s/ are unbiased estimators,
but they are not necessarily precisely equal to
the true population properties. For estimated
values, it is common to specify an interval
expected to contain the true population
properties
 GLOBAL
-I Educationa
==========Course

Confidence interval for the mean

(A.) Standard deviation CJ is knovvn
,/· .~ z
.(\ al2 /ii(j -< IJ <x .· +Z .iL
--= · ..ial2 /ii

(B)Standard deviation f) is not kno\:vn

.x· ~ tal'·vn;~ .· < f_t < .~f +tal?·vn~.·.·
where ta12 corresponds ton ~ 1degrees of frcedo1n.

Values: of Ze12
Confid·ence
lnffiTal
go~~ 1.2816
90% 1.6449
95% 1. %00
96C}~ 2.0537
98% 2.3263
99% 25758

GLOBAL
-I Educationa
===========Course

Assurne that the heliutn porosity (in percentage) of coal satnples taken a\vay frotn any particular
sean1 is notn1ally distributed vvith tn1e standard deviation 0.75. The 95o/o CI for the tn1e average
porosity of a cet1ain semn if the average porosity for 20 specilnens frorn the seatn vvas 4.85 is:

A. (4.52, 5.18)
B. (4.66; 5.22)
C. (4.87; 5.22)
D. (5.22, 6.18)

(1.96)(0, 75)
4.85 ± {20 = 4.85 ± 0.33 = ( 4.52, 5.18)
20
The mean length for the population of all screws be in g prod uced
by a certain factory is targeted to be 5
Assume that you don' t know what the population standard
deviation is. You draw a sample of 30 screws and ca lcu late t heir
mean length. The mean for your sample is 4.8, and the standard
deviation of your sample (s) is 0.4 centimeters .

What is the 95% confidence interval for the population mean ?

Round your answer to two decimal places

a
v 1<"
0 •.2.5 0..20 0.1:5 0~1 0 0 .0 5 0.(}25 0 .0 1 0.005

29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 29

30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 30
00 0.674 0.842 1.036 1.282 1.645 1.960 2.326 2.576 00
The fonT1ula for the confidence inte rval fo r one
oopu lation ;m ean, using tt'l e t -distri bution, is
x+t s
- n-1 y'Ti
In thi s case, the samp le mean,
x
s 4.8; the samp le sta ndard deviation, s, is 0.4; the
sample s ize. n, is 30; and the d egrees of freedom, n
- 1, fs 29 . That means tn _ 1 = 2 .05.
i'>Jovv, p lug i n the numbers:
~ s
x+t,_l y'Ti

=4.8-+-2. 05 ~
=4.8+ 0al497
=4 . 6503 to 4.9497
Rounded to tvvo decimal p laces, the ansv11er is 4 .65
to 4.95 ..

~~GLOBAL
Confidence Interval for the Diff~Educa~~~~
Between Two Means

where lar2 corresponds to n1+lfJ. - 2degrees of freedom.

 GLOBAL
- I Educationa
~======Course

10·0 resistors }Jroclt.lcecl by COillp<:lilY A atl(I 150 resistors

]Jrocltlcecl b'}' colillJallY B are testecl to fit1cl t~l1eir limits
})efore })trrrli:tlg O'tlt .. rflle t ·e St restllts 8}10\\' tl1at the COll1-
l:l<:t,n:y A resistors l:tave fl Il1e<"Ul rati11g of 2 W l1efo1:(~
l:Jtll~Ilin.g 0 '1 1t1, 'vit.l1 a star1d1trci rlevia·tiot1 of 0.25 \;\ll;
a11d tl1e cornJ>t:Ul:Y1' B resistors l1ave a 3; \V tlle·~lll ra't i.r tg
})efore ])tlrnitlg Ollt, \Vitll a starl(lard cleviatiotl of
0.30 '\t\T2 , v\t}Ja,t are th.e 95% conficletlCe li111it~s for tbe
cliffeTellCt:~ l)e·t"vee11 tl1e t;,vo, n1e~a.t1s for ·tl1e CtllllJlRllY A
resistors a.r1cl cOl'lliJ~lilY B resistors (i . e., A- · B)?

(J\) - 1.1 \f\'; -1 ,o vv-

(13) -1 .1 \iV ; - 0~93 vv
(C) - -1 .1 "'' ; -- 0.90 "v
,. \ '. - 0
(1)) - ....··t .,. . o·.·.·~.,._ \'Xr
~- ..-·· •. 9>9
.. ·.,:, .. '1- t
:l~ v

GLOBAL
- I Educationa
============Course

(0.25 vV2 ) 2 . (o.3o \V2) 2

- 1.9600-·.. --id(} . . . .._ . .... 150

= - 1 ~.0686 v\' (-1.1 \V)

UCI.~(JJ;1 - ,.~.2) -~· X1 - X2 + Zo/ 2 Jui + <>]. ·.•.

rt1 n2.

= '2 ,¥ - 3\¥

+ L9oooJ.(o.2~o~V2? + (o.s~5:2)2
= -0.9314 W (-0*93 W)

Tile answer is .(B).

 Confidence Intervals for the Variance
of a Normal Distribution

(n l)s 2 < 62 < (n l)s 2

2 2
Xa/2 .n 1 xl - a/2, n 1
'

GLOBAL

x 2 -distrbution
-I Educationa
==========:Course

CRITICAL VALUES OF X 2 DISTRlBlTI10N

/f/(l

~
{)

Degrut "'ffreedom
r2
• a.n

x'w.~ x1.wn 1 X, ..~.., X~:900 x~. l!)} .'!.'"~.If!'

1
X .ffw X 1 ilU,
X •m x t.!t!n

0.0000393 0.000157! 0.0009821 0.003.9321 0.0157908 2.70554 3.84146 5.02389 6 .6~490 7.87944
0.0100251 0.020100 0.0506356 OJ02587 0210720 4.60517 5.99147 7.37776 9.21.0)4 10.596!)
0.0717212 0.1148'n 0.215795 0.351846 0.584375 6.25139 7.S1473 934840 11.3449 12..S3SI
0.2{16990 0.297110 0.484419 0.710721 1.{163623 7.77944 9.48773 11.1433 13.1767 14.8602
0.41174{) 0554300 0.831211 !.145476 1.61031 9.23635 1U>7fr5 12.8325 15.0863 16.74%
(}_675727 0.872085 1.237347 1.63539 2.104B 10.6446 12..5916 14.4494 16.8119 1S.547ti
().989265 1.239043 L68S'S7 2.16735 2..Sl3ll 12.0170 14J)671 16.0128 18.4753 20.2777
s 1.34#19 1.646482 2.17973 2.73264 3.48954 13.3616 15.507'3 17.5346 20.090:2 2L!f550
9 1.734926 2. 08791.2 2.70039 3.325ll 4.16816 14.6837 16.9190 19.0228 21.6660 235893
iO 2.15585 2.55821 3.24697 3.94030 4.86518 15.9871 18.3070 2.0.4831 23.209.3 251882
ll 2.60311 3.05347 3.8 1575 4.574Sl 557779 17.2750 19.67:51 21.9200 24.7250 26.7559
12 3.07382 3..5705~ 4.40379 5.22603 630380 1&5494 21.0261 23.3367 26.2170 28.2995
l3 3565{)3 4.10691 5.00874 S.S9lS6 7.04150 19.Sll9 22..36.21 24.7356 27..6883 29.8194
14 4.07~& 466043 5.62872 6.57063 7..78951 21.1)642 21.6S4S 26.1190 29.1413 31.3193
1.5 4.60094- 5.12935 626214 7.26094 8..5~75 21.3072 24.9958 27.48&4 30.5779 32.3013
i6 5.14224 s.n:m 6.90766 7.96154 9.31223 2.3.5418 26.2.962 28.8454 31.9999 34.2672
17 5.69724 6..40776 7564!8 8.67176 10.0852 24.7 690 27.5871 30J.9l0 33.4087 35.7185
IS 626481 i .01491 8.23075 $1.39046 10..8649 2:5.9894 28.8693 31.5264 34.8053 37.1564
19 6.34398 7.63273 8.90655 10.1170 1L6509 27.2036 30.1435 32.8.523 36.1908 38.5822
20 H33S6 8.26040 9.59CiS3 10.S50S 12.4426 1SA120 31.4104 34.!6.96 31..5562 39.9968
21 8.03366 S.S9720 10.28193 11.5913 132396 29-.6151 32.67{)5 35.4789 38.9.m 41.4010
22 8.64272 9.54149 10.9823 12.3380 !4.0415 30.8.133 33.9244 36.7$07 40.2894 42.795~
2.3 9.26042 10.19567 11.6SS5 13.0005 14.8479 32.G{l69 35.1725 38.0757 41.63&4 44JSB
24 9.88623 10.S564 12.4011 13.8484 !.5.6587 33.1963 36.4151 59.364'1 42 .9798 45.5585
25 10.5197 11.5240 13.1]97 14.6114 16..4734 34.3816 37.6525 40:.64ti 4-ql~4! 46.9278
:i ~ ',.,-_.' I· (
26 11.1603 12.1981 l3.S4J9 15.3791 17.2919 35.5631 38.8S52 41.9232 45:&U1 ' 4s'. 2s~
 In a study on cholesterol levels a sample of 12 men
and women was chosen. The plasma cholesterol
levels (mmoi/L) of the subjects were as follows: 680,
6.4, 7.0, 5.8, 6.0, 5.8, 5.9, 6.7, 6.1, 6.5, 6.3, and
5.80 We assume that these 12 subjects constitute a
simple random sample of a population of similar
subjects. We wish to estimate the variance of the
plasma cholesterol levels with a 95 percent
confidence interval.

~r .sro
2
X .~.:o
2 2
Degrees of freedom X
2
.99'5 .JC.sr1s x2.!l00 x 2.1oo X
2
G50 X .0ls
1 0.0000393 0.0001571 0.0009821 0.0039321 0.0157908 2.70554 3.84146 5.02389
2 0.0100251 0.0201007 0.0506356 0.102587 0.210720 4.60517 5.99147 7.37776
3 0.0717212 0.114832 0.215795 0.35 1846 0.584375 6.25139 7.81473 9.3 4840
4 0.206990 0.297110 0.484419 0.710721 1.063623 7.77944 9.48773 11.1433
5 0.411740 0.554300 0.831211 1.145476 1.61031 9.23635 11.0705 12.8325
6 0.675727 0.872085 1.237347 L63539 2.20413 10..6446 12.5916 14.4494
7 0.989265 1.239043 1.68987 2.16735 2.83311 12.0170 14.0671 16.0128
8 1.344419 1.646482 2.17973 2.73264 3.48954 13.3616 15.5073 17.5346
9 1.734926 2.087912 2.70039 3.32511 4.16816 14.6837 16.9190 19.0228
10 2.15585 2.55821 3.24697 3.94030 4.86518 15.9871 18.3070 20.4831
11 2.60321 3.05347 3.81575 4.57481 5.57779 17.2750 19.6751 21.9200
12 3.07382 3.57056 4.40379 5.22603 6.30380 18.5494 21.0261 23.3 367
13 3.56503 4.10691 5. 00874 5.891 86 7.04150 19.8119 22.3621 24.7356
14 4.07468 4.66043 5. 62872 657063 7.78953 21.0642 23.6848 26.1 190
15 4J50094 5.22935 6.26214 7.26094 8.54675 22.3072 24.9958 27.4884
16 5..14224 5.81221 6.90766 7.96164 9.31223 23.5418 26.2962 28.8454
17 5.69724 6.40776 7.56418 8.67176 10.0852 24.7690 27.5871 30.1910
18 6.26481 7.01491 8.23075 9.39046 10.8649 25.9894 28.8693 31.5264
19 6.84398 7.63273 8.90655 10.1170 11.6509 27.2036 30.1435 32.8523
• \ lalue of s2
s = _3918680978

• \talues of X2 frotn table

2
X.fJ75 = 21.920

,2
X.ros
= 3.816

• Calculation of the con±idence interval

(n-1)s 2 2 (n-1)s 2
2 < (j < ,2
X1-(flf2) ~.flf2

11 (. 3918 --) 2 11 (.3918 --)

- - -- < o < -- - -
21.920 3.816
. 19 664 9- < a 2 < 1. 129 59 8 --
.4435 < (j < 1. 0628

GLOBAL
-I Educationa
====:=====Course
 GLOBAL

Least Squares
-I Educationa
~cou rse

The least square regression line is the line that

makes the square of distance of vertical data
point from the line as small as possible

1 2 3 4 s 6

GLOBAL

least Squares
- I Educationa
~course

"'
_j)== a+ bx, vVll.ei·e
"'
b == SXJI~\-.X
a == V - !Jx -
S.x._v == .L.
n
z==l
( n
XtYi - (1/n) _L
z == l
Xt
)( n
.
_L
z==l
Yi )

n
S-"Kx == .L xf - (
(lin) · L _xi
n )2
;==1 i == l

y = (1/n)C±l Yi)
x = (1/n) ( iLn x 1)·.
1
 GLOBAL
- I Educationa
~·Course

The least squares method is used to plot a straight line

through the data points (1,6), {2, 7), (3,11), ad
(5 1 13). The slope of the line is most ne ly
(A) 0.87
(B) 1.7
( ) 1.9
(D) 2. 0

- ~GLOBAL
Educationa
. Course

.~
.L-J··'"'t · · l·
.t't,. • -- ..·. . --r"' .~
1 · "' - 1r ·· 3···. -L
.... ~~· ..ur; -..........· '}1
·. •.. •·..

L Yi= 6 + '7 '+ ll -1- 13 = 37

L :c7 = 2
(1) ·+. (2)2 + (3) + (5)
2 2
= 39
L xiYi = (1)(6) -1- (2)(7) _;1- (3)(11) -l- (5)(13) = 118

Sxu = i = n; XiYi- (1/n) ·(·· t Xi)·· (\. ~

.:
. L

l
. n Yi). . . . 1,=1 t= l

= 118- (!) (11)(37)

= 16.25
 GLOBAL
-I Educationa
~~course

= 8.75

~ . ·. · .· 16.25
b = Sxy/S!Iix = 8..7r:.0
= 1.857 (1 ~9 )

The answer Is (C)~

GLOBAL
-I Educationa
=======~Course

Four data points have been observed as follo\vs:

1 Xi Yi
1 2.0 5.1
2 1.5 4.2
3 3.6 7.5
4 5.7 10.4

Using linear least-square regression~ the equation that best fits this data is:

A. y=2.3 +1.5x
B. y = 2.3 +2.1x
C. y=l.5 +2.1x
D. y= 1.5 + 1.5x
 GLOBAL
-I Educationa
==============Course

Sample Correlation Coefficient and

Coefficient of Determination
Sa1nple Correlation Co·e fficient (R) Hnd Coefficient of
Determination (R~:
R= S>-y
~~Yx~vy

n 2 .. (' n)·2
S:ry = .I: Yi - (1/n) . .I:Yi
1= 1 ,t =l

Negc.ttive Sn1a1l or No Positive

Associatio:n Associ at ion Association

1. = -1 r = O r= +1

•• • •

y
r=0.998

•• ••
•••
•••

y
•
..
• •
•
. •........
• •• ' ••
••
•• •• •
• ••• r=0.278

X X

1. Large Positive Correlation 4. Small Positive Correlation

 • r=--tl061

, .....
• •
•
•• • •
• ....•••••
• •
• •
• • ••
. y

•
r=..;0.817 • • • • •
• i• •• I
X X
2. Large Negative Correlation 5. Mi.nimaLNegadve Correlation

•
• • . .. .
••••
•
r=-0.374

y
•
• •
., . •• •
•• ••
.....
•
• • •
. •. •
y
• •
• r=0.487
•• at

X
• •
X
3. Modest Positive Correlation 6. Mo4est Negative Correlation
 GLOBAL
Educationa
I~=====Co urse

1 he least sc1uares ltlet110(l is used to plot a straigl1t lir1e

througl1 t-l1e clata points (5, -5)~ (3~ ,- 2), (2 3), and
(-1, 7). rfl1e COl'l~elatiOil ,coeffi.cie:11t is lUOSt n~e~\rly
(lt) -0.97
(B) -0.92
(C) -0.88
(D) -0.80

GLOBAL
- I Educationa
~===Course

L·xi == 5 + 3 -f·- 2 + (--1) = 9

L::v. = (-5,) -1- ( - 2) '4- 3 -1- 7 = 3
L x1 = (5) 2 + (3)2 -1- {2) 2 + (--1) 2 = 39
L v~ = (-· 5) 2 + (,. . . .,. 2)'2 ,1- (3)~
2
~t~ (7) 2 = , 87
}2xiVi = , (5)(-5) -t- (3)(-2) + (2)(3) -1- (--· 1)(7) == -32
 GLOBAL

-I Educationa
~===Course

l:xt?li- (1/n)(!:xi) (!:yJ

--r======================~~~====
· ·. (L:xr- (1/n)(2:X~) ) (2:v~- (1/n)(!:vi/)
2

-32 - (!) (9)(3)

J(
-· ~========~===========
39- (!) (9) 87- (!) (3)
2
) (
2
)

= ·-0· ~972 ( -0.97)

Tlt:e answer fs· (A).

Residual

Residual
ei =Yi - .Y =Yi - (a + i)xi)

60 70 65 .411 4.589

70 65 71.849 -6.849

80 70 78.288 -8.288

85 95 81.507 13.493

95 85 87.945 -2.945
 + • • • + • •
• •••
• ,. ' " ·~
+
5 I ill
,,, ~ l • u

• * ..... ·+
• ·5....__ _ _ _ ___, ·5 + •
Random pattern Non -randorn: U-shaped Non-random: Inverted U

GLOBAL

- I
Educationa
..........- course

Standard Error of Estimate, Confidence Interval for Intercept,

Confidence Interval for Slope
Standard Et"ror of Estimate (s;):
s-e., == S.xx
_., - s)!V ·- s;,
"'':.·' == lvfSE, wl1ere
-I'\;'
~
(Y- Y') 2
Sx,.; (rz - 2) a .w - N

>2

Syy = .~
= l
Yil
- (1/n) ( .. ~Yi)·.
I= l

C ·o nfidenc·e Inter"\ral for Intercept (a):

d +
- t.xJ2~n- 2
A ( n1...· + s)'.'" 2.·- ') ·..J.Vl.~...JD
A '/./'.· .0...'1:;'
.X.,"\: ,

Conficlettce Intet~al fo1.. Slope (b):

... 1¥-JVJSE
b+ l'aJ2.,n - 2 . .·· S
.:t.X:
 Standard Error of Estimate

A B

In aregression line, the _ _ the standard 'errorof the estimate is, the more accurate thepredictionsare

fJ larger

0 smaller

0 The standard .error ofthe estimateisnot related to the accuracy of the predictions.
The graph below represents a regression Hne predicting Yfrom X. This graph shows the error of prediction for each of
the actual Y values. Use this information to compute the standard error of the estimate in this sample.

y ..

In the context of regressio n ana lysis, \Vhtch of the follmving statements are true?

I. Vt/hen the sum of the residua ls. is g reater than zero, the data set is nonlinear.
II. A random pattern of residua :ls supports a linear model.
III. A random pattern of residuals supports a non -linear model.

(A) i only
(B) U only
(C} m only
(D} I and II
(E) I and III

The correct answer is (B). A random pattern of residuats supports a linear model; a non-random pattern
supports a non-linear model. The sum of the residuals is afways zero. whether the data set is linear or
nonlinear.