Bai Giangbusiness Statistics Hai Phong Dang Soan (1)

8/11/2019 Bai Giangbusiness Statistics Hai Phong Dang Soan (1)

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BUSINESS STATISTICS

Assoc .Prof. Nguyen Huu Bao

Department of Mathematics !"U

BUSINESS STATISTICS Course sy##a$us

Instructor: Assoc.Prof .Nguyen Huu Bao - Department of Mathematics

E - mail: NghBao@Wru. eu. !n" phone: #$%#'&&(

%ffice: Department of Mathematics"W)*%+, ayson Dong a - Hanoi Cooperator: Mmath. Nguyen !an Dac Phone : #$/,,#,,#$

Prere&uisite: Basic Mathematics s0ills

Te't$oo(: Essentials of Moern Business 1tatistics 2ith Microsoft E3cel. Anerson

12eeney an William .


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)aptop* 4ou 2ill 5e e3pecte to use your laptop in class an to complete the

Home2or0 5y using Microsoft E3cel soft2are. 1ome 6uestions on the mi - term

an final e3ams 2ill re6uire using 7aptop. Bring your laptop to class e!ery ay.

Course content: his course 2ill introuce you to all the important 5usiness-

relate topics in applie statistics in one semester. his semester 2e 2ill learn a5out

ata ac6uisition an analysis" ta5ular" graphical an numerical methos 2ill use of

escripti!e statistics" pro5a5ly istri5utions" statistical inference an regression

analysis. We 2ill use the statistical capa5ilities of Microsoft E3cel to reuce your

calculations

Tests*hese 2ill 5e three ests uring the course. he est 2ill 5e in at least '#

minus an each 2ill 5e score on the 5asic of %## points.

Home+or(: Home2or0 from the te3t5oo0 2ill 5e assigne an score regularly.

here 2ill usually 5e home2or0 assignments ue on uesay each 2ee0. 7ate

home2or0 2ill not 5e accepte. Missing home2or0 2ill 5e score -( points.

,ra-ing Stan-ar-s*


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Tests * / 0 122 points 3 /22 points

Home+or( Assignments / 0 45 points each 3 65 points

Contri$ution to Team !or( 5 points

C#ass Atten-ance an- Participation 42 points

Tota# Possi$#e 722 points

/82 9 722 points : A ; /42 9 /5< points : B ; 4=2 9 /1< points : C

472 9 46< points : D ; Be#o+ 472 points : > >ai# 3

,ra-ing Stan-ar-s in C#ass Atten-ance an- Participation *

- Each a5sent time : - , points

- 8ame late: - ( points - Each missing Home2or0: - ( points - Presentation: 9 , points - 8orrection the e3amples at class: 9 , points

Course 9 Sche-u#e


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Day Course Content

+-%# ntrouce the course. 8hapter %: Data an 1tatistics . 8hapter ( : Bar chaan

Pie chat

/-%# 8hapter &: Descripti!e statistics. Numerical Measures

%%-%# eam Wor0 : Practice Microsoft E3cel

%(-%# 8hapter ;: ntrouction to Pro5a5ility

1ome e3amples calculating Pro5a5ility of a e!ent A : P


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(%-%# est for Population Mean

((-%# 8hapter %(: 7est s6uare metho . 1imple 7inear )egression

E3cel>s )egression ool

(,-%# eam Wor0 : Dra2ing the 1imple 7inear )egression 7ine

('-%# )e!ie2 for est &

(+-%# Test / nform the 8lass a5out Attenance an Participation

1core


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CHAPTE" 1 DATA AND STATISTICS

1.1 DATA* Data are the facts an the figures " analy?e an summari?e forpresentation an interpretation. All the ata collecte in a particular stuy arereferre as the ata set for the stuy

E#ements are the entities on 2hich ata are collecte

A ?aria$#e is characteristic of in interest for elements

Data sources* Datacan 5e o5taine from e3isting sources or from sur!eys ane3perimental stuies esigne to collect ne2 ata


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1.4 Statistica# Stu-ies* 1ometime the ata neee for a particular application are nota!aila5le through e3isting sources. n these cases" the ata can often 5e o5taine 5yconucting a statistica# stu-y.

1tatistical stuies can 5e classifie as either e3perimental or o5ser!ational

1./ Descripti@e Statisticshere are t2o omains in 1tatistical 1tuies: Descripti!e 1tatistics an heoretical

Most of the statistical information in ne2spapers" maga?ines" company reports another pu5lications consists of ata that are summari?e an presente in a form that is

easy to reaer to unerstan. 1uch summaries of ata 2hich may 5e ta5ular" graphicalor numerical are referre to as Descripti@e Statistics

1.7 Statistica# Inference

The popu#ation an- the samp#e*


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Many situations re6uire information a5out a large group of elements < ini!iuals"companies" !oters" househols" proucts" customersBut" 5ecause of time" cost" an other consierations" ata can 5e collecte from only asmall portion of the group

he large group of elements in a particular stuy is calle the Popu#ation an thesmaller group is calle the samp#e

Some e'amp#es in c#ass

1.5 Statistica# Ana#ysis Using Microsoft E'ce#

% ntrouce Microsoft E3cel

( Basic perations 2ith Microsoft E3cel& Bar chart 2ith Microsoft E3cel


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CHAPTE" 4

DESC"ITI?E STATISTICS TABU)A" AND ,"APHICA)

P"ESENTATI%NS

4.1 Summariing Categorica# Data

>re&uency Distri$ution

A >re&uency Distri$ution is a ta5ular summary of ata sho2ing the num5ers ool to 8onstruct a Bar 8hart an Pie 8hart of BranPurchase


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4.4 Summariing uantitati@e Data

>re&uency Distri$ution

he three steps necessary to efine the classes a fre6uency istri5ution 2ith6uantitati!e Data are: %. Determine the num5er of non-o!erlapping classes (. Determine the 2ith of each class &. Determine the class limits

E'amp#e: *sing E3cel>s PFAB7E )EP) to 8onstruct a Cre6uencyDistri5utionA ot place a5o!e the a3is

Dot p#ot* A hori?ontal a3is sho2s the range for the ata. Each ata !alue isrepresente 5y


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Histogram* his is a common graphical presentation ata 2hich can 5e prepare forata pre!iously summari?e in either a fre6uency" fre6uency or percent fre6uencyistri5ution


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NUME"ICA) MEASU"ES

/.1 Measures of )ocation

Meann statistics formulas it is customary to enote the !alue of !aria5le 3 for the firsto5ser!ation 5y 3% " for the secon o5ser!ation 5y 3( . Cor a sample 2ith no5ser!ations" the formulas for the sample mean as follo2s

ix

xn

=

N8E:%. he mean is the a!erage !alues(. f the Data are for a sample " the mean is enote 5y x

An if the ata are for the Population" the mean is enote 5y

Some e'amp#es at c#ass

Me-ian* he Meian is the !alue in the mile 2hen the ata are arrange inascening orer

N8E:%. Cor an o num5er of o5ser!ation" the Meian is the mile !alue


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(. Cor an e!en num5er of o5ser!ation " the Meian is the a!erage of the t2o mile!alues


Mo-e* he moe is the !alue that occurs 2ith greatest fre6uency

Using E'ce# to compute the Mean; Me-ian an- Mo-e

Some e'amp#es at C#ass

/.4 Measures of ?aria$i#ity

"ange* he simplest of !aria5ility is )ange

"angeG 7argest !alue 1mallest !alue

?ariance he Fariance is a measure of !aria5ility that utili?es all the ata. f the atafor a Population" the a!erage of the s6uare e!iation is calle the PopulationFariance an is enote 5y ( . f the ata for a sample" 2e shall call 1ampling

Fariance an enote 1(

:


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(

(< =ix x

s

n

=

(

(< =ix

N

= :

Stan-ar- De@iation*

Stan-ar- De@iation*

Population stanar e!iation: G (

1ample stanar e!iation: s G (s

Using E'ce# to compute the Samp#e ?ariance an- Samp#e stan-ar- -e@iation:

Some e'amp#es at C#ass

Coefficient of ?ariation

Coefficient of ?ariationtan

< %##=s dard deviation

Mean


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/./ Measures of Association $et+een 4 ?aria$#es

Co@ariance* Cor a sample of si?e n 2ith the o5ser!ation (- Dimension


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Using E'ce# to compute the Co@ariance an- the Corre#ation Coefficient


CHAPTE" 7

INT"%DUCTI%N T% P"%BABI)IT

Pro$a$i#ity is a numerica# measure of #i(e#ihoo- that an e@ent +i## occur


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7.1 E'periments; Counting "u#es

E'periment is a process that generates 2ell-efine outcomes

. E'periment E'perimenta# %utcomes

oss a coin Heat" ail 1elect a part for inspection Defecti!e" Non-efecti!e )oll a ie %"("&";","'

Samp#e space for an e3periment is the set of all e3perimental outcomes

E'amp#e

Counting Techni&ues


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1 Mu#tip#ication "u#e: Assume an operation can 5e escri5e as se6uence of 0steps an

he num5er of 2ays of completing step % is n%an he num5er of 2ays of completing step ( is n(for each 2ay of completing step %

" an he num5er of 2ays of completing step & is n&for each 2ay of completing step (" an so forth .he total num5er of 2ays of completing the operation is

n G n%. n( n04 The num$er of permutation of n ifferent elements is nJ 2here nJ G

%.(.&.n/ The num$er ofpermutationsof su5sets of r elements selecte a set n ifferentelements is

J.< %=.< (=...< %=

< =J

n

r

nP n n n n r

n r= =

7 The num$er of com$inations " su5sets of si?e r that can 5e selecte from a set

of n elements is enoten

r

orn

rC

J

J< =J

n

r

r nC

n r n r

= =


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E'amp#e* Crom a es0 of ,; cars ra2 & at ranom. Ho2 many ifferent 2aysra2ingK

E'amp#e* A 5atch of %;# semiconuctors chips is inspecte 5y choosing asample of fi!e chips. Assume %# of the chips o not conform to customer

re6uirementsa Ho2 many ifferent samples are possi5le K

5 Ho2 many samples of fi!e contain e3actly one nonconforming chipK

E&ua##y )i(e#y %utcomes*he Pro5a5ility of an outcome can 5e interprete as oursu5Lecti!e pro5a5ility or -egree of $e#ief" that the outcome 2ill occur. Whene!er asample space consists N possi5le outcomes that are e6ually li0ely" the pro5a5ility ofeach outcome is %N

7.4 Pro$a$i#ityof a e@ent

>or a -iscrete samp#e space; the pro$a$i#ity of an e@ent E ; -enote $y PE3 ;

e&ua#s the sum of pro$a$i#ities of the outcomes in E.

>or e@ery A in samp#e space; PA3 : m

n !here m is the num$er of the

outcomes ha@ing e@ent A to appear an- n is the sum of possi$#e outcomes .


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Assigning Pro$a$i#ity

Pro$a$i#ity is a num5er that is assigne to each mem5er of a collection of e!entsfrom a ranom e3periment that satisfies the follo2ing properties: % P


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E'amp#e 4* >rom a -es( of 54 car-s -ra+ three at ran-om. >in- the pro$a$i#ity

that there +i## $e e'act#y one ace among them.

Solution:Denote the e!ent 2e are intereste in 5y A he num5er of elements of

iscrete sample space is ,(&C . he num5er of 2ays ra2ing one ace is;

%C . he

num5er of 2ays ra2ing ( others cars < is not ace = is;/

(C . hat means the num5ers

of 2ays ha!ing A to appear shall 5e; ;/

% (.C C . 1o " P


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7./ Some Basic "e#ationships of pro$a$i#ity

A--ition )a+

Union of t+o e@ents* *nion of t2o e!ents A an B is shae in &-th figure a5o!e" thatmeans A appear or A appear

Intersection of t+o e@ents* ntersection of A an B is shae in ;-th figure a5o!e" thatmeans A appear an B also appear

A--ition )a+* < = < = < = < =P A B P A P B P A B = +

N8E: f A an B are e3clusi!e e!ents then< = < = < =P A B P A P B = +


BA

A

B B

A

B

A

B

A

A

A B A B A B A B A B Ac G A

A


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Con-itiona# Pro$a$i#ity

Cor any gi!en e!ent A an B" the pro$a$i#ity of the e@ent A appeare- +ith thecon-ition e@ent B appearis the ratio:

< =

< O =< =

P A BP A B

P B

=

he pro5a5ility of e!ent B appeare 2ith the conition e!ent A appeare is efine 5ysimilar 2aysSome e'amp#es

In-epen-ent E@ents* 2o e!ents A an B are inepenent if< O = < =P A B P A= or < O = < =P B A P B=

Mu#tip#ication )a+

Multiplication 7a2: < = < = < O =P A B P B P A B =

r < = < = < O =P A B P A P B A =


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N8E: f A an B are t2o nepenent e!ents then< = < = < =P A B P A P B =

T%TA) P"%BABI)IT "U)E

A collection of sets E%" E(" " E0such that E% E( E0G 1 is sai to 5ee'c#usi@e

Assume E%" E(" " E0 are 0 mutually e3clusi!e an e3hausti!e e!ents . hen

% % ( (

%

< = < O = < = < O = < = ... < O = < =

< O = < =

k kk

i i

i

P B P B E P E P B E P E P B E P E

P B E P E=

= + + +

=


n the case only t2o e!ents A%an A(2e ha!e


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BAESS THE%"EM

% % % %

%

% % ( (

< = < O = < = < O =

< O = < = < = < O = < = < O =

P A P B A P A P B A

P A B P B P A P B A P A P B A= = +


Using E'ce# to compute Posterior Pro$a$i#itiesE3ample in page %/$

CHAPTE" 5

DISC"EETE P"%BABI)IT DIST"IBUTI%NS

5.1 "an-om ?aria$#es


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Discrete "an-om ?aria$#es* A ranom Faria5le that may either a finite num5er or aninfinite se6uences of !alues such as %"( "

EIAMP7E:E'periment "an-om ?aria$#e Possi$#e ?a#ues

%. 8ontact fi!e Num5er of 8ostumers #.%.(,costumers (. perate a restaurant Num5er of 8ostumers #"%"(".

Continuous "an-om ?aria$#es* A ranom !aria5le that may assume any numerical!alue in an inter!al or collection of inter!als

EIAMP7E:E'periment "an-om ?aria$#e Possi$#e ?a#ues

%. perate a Ban0 ime 5et2een customer 3 # arri!als in minutes(. est a ne2 chemical emperature < min %,##C %,# 3(%(

process an ma3 (%(#C=


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5.4 Discrete Pro$a$i#ity Distri$utions

he Pro$a$i#ity Distri$utionfor a ranom !aria5le escri5es ho2 pro5a5ilities areistri5ute o!er the !alues of the ranom !aria5le .

Cor a iscrete ranom !aria5le 3" the Pro5a5ility Distri5ution is efine 5y apro$a$i#ity functionenote 5y f


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E'pecte- ?a#ue or mean of a ranom !aria5le is a measure of the central locationfor the ranom !aria5le

?ariance

We use the Faria5le ?ariance in chapter & to summari?e the !aria5ility in ata. No22e use ?ariance to summari?e the !aria5ility in the !alues of a ranom !aria5le

E'pecte- ?a#ue or mean of a Discrete ranom !aria5le:

< = < =E x xf x= =

?ariance of a Discrete ranom !aria5le:

( (< = < = < =Var x x f x = =


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Some e'amp#es

Using E'ce# to compute the E'pecte- ?a#ue; ?ariance an- Stan-ar- De@iation

Eunction


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< =

< =


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4. he occurrence or non-occurrence in any inter!al is inepenent of theoccurrence or non-occurrence in any other inter!al

Poisson Pro$a$i#ity >unction

< =J

xef x

x

=

2here

is e3pecte !alue or mean num5er of occurrence in inter!al

Some e'amp#es an- using E'ce# to compute Poisson Pro$a$i#ities

< =J

xef x

x

= :P11Nrom a set of n e#ements +hich contains r e#ements +ith the property A ta(e a

samp#e N e#ements at ran-om. )et F $e the e'act#y ' e#ements +ith property A


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among the samp#e. F is ca##e- to $e a Hyper ,eometric Distri$ution ran-om

@aria$#e

Hypergeometric Pro$a$i#ity >unction*

< =x n x

r N r

n

N

r N r

x n xC Cf x

NC

n

= =

for # 3 r=

Some e'amp#es an- using E'ce# to compute Hypergeometric Pro$a$i#ities

< =x n x

r N r

n

N

r N r

x n xC Cf x

NC

n

= =

:H4PEMD1


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8.1 The Pro$a$i#ity Density >unction

Cor e!ery )anom !aria5le I " a funcion f


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N8E: Cor The Uniform Pro$a$i#ity Density >unction on inter@a# Ga;$

< =(

a bE x

+= an-

(< =

< =%(

b aVar x

=

8./ Norma# Pro$a$i#ity Distri$ution

Norma# Pro$a$i#ity Density >unction N; ( 3*

(

(

< =

(%

< =

(

x

f x e

=

Stan-ar- Norma# Pro$a$i#ity Distri$ution N2;13


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Stan-ar- Norma# Density >unction*

(

(%

< =(

x

f x e

=

N8E: Area as a Measure of Pro$a$i#ity

Because 2e ha!e < = < =P X f x dx

= so" < =P X is the Area uner the cur!eof the f


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1uppose I has the Normal Pro5a5ility Distri5ution N< &'",## " ,##(= . Cin P< I R;#"### = " P< I ;#"###= G K

Solution: We ha!e P< I R ;#"### = G P


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he Point Estimation of is(

%

< =

%

n

i

i

x x

sn

=

=

he Point Estimation of Pro$a$i#ity p is fre&uency xn

6./ SAMP)IN, DIST"IBUTI%NS

he ranom !aria5les I%" I(" " In are aran-om samp#eof si?e n ifa he In>s are inepenent ranom !aria5les

5 E!ery IL has the same pro5a5ility istri5ution

A statisticis any function of the o5ser!ations in a ranom sample

he pro5a5ility istri5ution of a statistic is calle a samp#ing -istri$ution


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f I%" I(" " In are a ranom sample of si?e n ta0en from a population 2ith

mean an finite !ariance ( then

X

n

= as n " is the stanar Normal

istri5ution

6./ Samp#ing Distri$ution of p

he sample proportion p is the point estimator of the population p.

E


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f np S , an n


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inter!al. Ho2e!er" the conUence inter!al is constructe so that 2e ha!e highconUence that it oes contain the un0no2n population parameter. 8onUenceinter!als are 2iely use in engineering an the sciences.

=.1 C%N>IDENCE INTE"?A) %N THE MEAN %> A N%"MA)DIST"IBUTI%N; ?A"IANCE KN%!N

If x is the samp#e mean of a ran-om samp#e of sie nfrom a norma# popu#ation

+ith (no+n @ariance 4

; a 12213L CI on is gi@en $y

A( A(A Ax ! n x ! n +

!here A(! is the upper 1224 percentage point of the stan-ar- norma#

-istri$ution.

E'amp#e

=.4 C%N>IDENCE INTE"?A) %N THE MEAN %> A N%"MA)

DIST"IBUTI%N; ?A"IANCE UNKN%!N


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If x is the samp#e mean of a ran-om samp#e of sie nfrom a norma# popu#ation

+ith un(no+n @ariance; a 12213L CI on is gi@en $y

A( A(A Ax t s n x t s n +!here A(t is the upper 122 4 percentage point of the t -istri$ution +ith n1

-egrees of free-om .

E'amp#es

=./ Determining the samp#e Sie

If x is use- as an estimate of ; +e can $e 12213L conJ-ent that the errorx +i## not e'cee- a speciJe- amountE +hen the samp#e sie is

(A(< =!

n

E

;


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E'amp#es

=/ C%N>IDENCE INTE"?A) %N THE ?A"IANCE AND STANDA"D

DE?IATI%N %> A N%"MA) DIST"IBUTI%N

7et I%" I("" In5e a ranom sample from a normal istri5ution 2ith un0no2n!ariance ( . f s(is the sample then a 12213L confi-ence inter@a# on ( is

( (

(

( (

A(" % % A(" %

< %= < %=

n n

n s n s

2here(

A(" %n an(

% A(" %n are the upper an the lo2er %## A ( percentage points

of the chi-s6uare istri5ution 2ith


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=.7 A)A",ESAMP)EC%N>IDENCEINTE"?A)>%"A

P%PU)ATI%N P"%P%"TI%N

f p is the proportion of o5ser!ation in a ranom sample of si?e n that 5elong to aclass of interest " an appro'imate 12213L confi-ence inter@a# on theproportion p of the popu#ationthat 5e long to this class is

A( A(


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(A (< = #.(,

!n

E

=

Some e'amp#es


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CHAPTE" ai#ing to reOect the nu## hypothesisH2 +hen it is fai#se is -efine-as a type II error

= Ptype I error3 : PreOectH2 +henH2 is true3


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Sometimes the type I error pro$a$i#ity is ca##e- the significance

#e@e#; or the error; or thesie of the test.

The po+er of a statistica# test is the pro$a$i#ity of reOecting the nu## hypothesis H2+hen the a#ternati@e hypothesis is true.

T+o Tai#e- Tests*

H2* # = H1* #


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his est is calle a t2o-sie test" 5ecause it is important to etect ifferences from thehypothesi?e !alue of the mean # that lie on either sie of #. n such a test" the criticalregion is split into t2o parts" 2ith


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the pro5lem conte3t.

AN%"MA)DIST"IBUTI%N; KN%!N

1uppose that 2e 2ish to test the hypotheses H#: # H%: # =

Test Statistics#

#

X

n

=

f the null hypothesis"#: # = is true" EIG # " an it follo2s that the istri5ution of#

is the stanar normal istri5ution enoteN


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istri5ution of#2oul 5e unusual if"#: # = is true therefore" it is an inication that

"# is false. hus" 2e shoul reLect"# if the o5ser!e !alue of the test statistic Y# is

either #R

-A(

or#S A(


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We may also e!elop proceures for testing hypotheses on the mean 2herethe

a#ternati@e hypothesis is onesi-e-.1uppose that 2e specify the hypotheses as

H#: # = H%: #

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Bai Giangbusiness Statistics Hai Phong Dang Soan (1)