topic statistics

8/6/2019 Topic Statistics

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Engineering Mathematics 2

STATISTICS


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Mathematics for Technology II (CT034-3-2) Statistics Slide 3 (of 59)

Learning Outcomes

A the end of this topic, You should be able to:

Understand the concepts of sampling

theory and Central limit theorem.

Perform statistical inference using

estimation and hypothesis testing.


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K ey Terms you must be able to use

If you have mastered this topic, you should beable to use the following terms correctly in your assignments and final exam :


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Mathematics for Technology II (CT034-3-2) Statistics

Central Limit Theorem

If

1 2, 3

2

1 2 3

2

1

, , ... , ...is a sequence of independent random variables

with ( ) var( ) , 1, 2,3, 4,5,6........ and if ... ,...then under certain general conditions

and variance

n

i i i i

n n

n

i

i

X X X X

E X X i nS X X X X

Q W

Q Q W !

! ! !!

! 21

as n tends to infinity.n

i

i

W !

!


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Sampling Theory- Population or universe

An aggregate of objects (a nim ate/ inanim ate)und er stu dy is calle d populat ion or u niv erse. Itis thus a collect ion of individ uals or of the irattr ibutes or of results of operat ions wh ichcan be num er ically spec if ied .

A univ erse co n ta inin g a f ini te num ber of individ uals or m em bers is calle d f ini teuniv erse . For exa m ple , the u niv erse of th ewe ight of the stu den ts in a part icular class.


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Sampling Theory

A univ erse of co ncrete objects is a n ex iste n tuniv erse .

The collect ion of all poss ible wa ys in wh ich aspec if ied even t ca n happe n is calle d ahypothet ical u niv erse.

The u niv erse of hea ds a nd ta ils obta ined by toss ing a co in in an in f ini te num ber of t im es.


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Sampling Theory

The stat ist ician is ofte n con fro n te d with theproble m of discuss ing univ erse of wh ich he

cann ot exa min e e ver y m em ber i.e. of wh ichcom plete e num erat ion is im pract icable.

For exa m ple if we wa n t to ha ve a n idea of theaverage per cap ital incom e of the people of Indi a ,e num erat ion of e ver y ear nin g individ ualin the cou n tr y is a ver y di ff icult task.


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Sampling Theory

N aturall y the quest ion ar ises:What ca n be sa id about a u niv erse of wh ich

we ca n exa min e o n ly a limi te d num ber of m em bers?The quest ion is the or igin of the THEORY OF

SAMPLING

.


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Sampling Theory

Sam ple: A f ini te subset is calle d a sa m ple. A sa m ple is thus a s m all port ion of the u niv erse.

Sam ple s ize: The num ber of individ uals in asa m ple is calle d the sa m ple s ize.

Sam pling : The process of select ing a sa m plefro m a u niv erse is calle d a sa m pling.


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Sampling Theory

The theor y of sa m pling is a stu dy of relat ionsh ip ex ist ing betwee n a populat ion

and sa m ples d raw n fro m the populat ion .The fu nd am en tal object of sa m pling is to getthe in for m at ion as poss ible of the wholeuniv erse b y exa minin g on ly a part of it.An atte m pt of is thus m ade through sa m plingto g ive the m axim um infor m at ion about thepare n t u niv erse w ith the minim um effort .


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Sampling Theory

Sam pling is qu ite ofte n use d in our da ily life.

For exa m ple , in a shop we asses the qual ity of sugar , r ice or a ny other pro duct o n ly by tak ing it fro m the bag a nd the n dec idewhether to purchase it or not .


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Sampling Theory

A house w ife nor m ally tests the cooke d pro ducts to f ind if the y are properl y cooke d

and con ta in the proper qual ity of salt or sugar,by tak ing a spoo n ful of it.


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Random Sampling

The select ion of a n individ ual fro m theuniv erse in such a wa y that each individ ual of

the u niv erse has the sa m e cha nce of be ingselecte d is calle d R andom SamplingA sa m ple obta ined by the ra nd om sa m pling iscalle d a ra nd om sa m pleThe s im plest m etho d wh ich is nor m ally use d for ra nd om sa m pling is the lotter y syste m


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Random Sampling

Sam pling of attr ibutes: The sa m pling of attr ibutes m ay be regar ded as the d raw ing of

sa m ples fro m a u niv erse whose num berspossess the attr ibutes A or B

The u niv erse is thus divid ed in to two m utuall y exclus ive a nd collect ively exhaust ive classes-one class possess ing the attr ibute A and theother class not possess ing the attr ibute A


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Random Sampling

The prese nce of a part icular attr ibute in sa m ple d uni t m ay be ter m ed as success a nd

its abse nce is fa ilure.


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Random Sampling

By sim ple sa m pling we m ea n ra nd om sa m pling in wh ich each e ven t has the sa m e

probab ility of success a nd the probab ility of even t is ind epe nd en t of the success or fa ilureof e ven ts in the prece din g tr ials.Thus the s im ple sa m pling is a spec ial case of ra nd om sa m pling in wh ich the tr ials areind epe nd en t a nd probab ility of success isconsta n t.


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Random Sampling

For exa m ple , cou n t ing the num ber of successes in the throw ing of a dice or toss ing

of a co in is a case of s im ple sa m pling since theprobab ility of gett ing hea ds w ith the a co in isunaffecte d by the pre vious tr ials a nd re m a insconsta n t irrespect ive of the num ber of tr ialsm ade pro vid ed the co in re m a ins u nb iase d .


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Random Sampling

Sam pling , through ra nd om , nee d not besim ple.

Rand om sa m pling fro m an in f ini te u niv ersealwa ys s im ple but ra nd om sa m pling fro m af ini te u niv erse m ay or m ay not be s im pleaccor din g as the m em bers d raw n are replace d or not.


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Random Sampling

The co ndi t ions of s im ple sa m pling viz,consta n t probab ility p, a nd ind epe nd en t

even ts sat isf y the bas ic assu m pt ions of theBinomi al Distr ibut ion .

The b inomi al probab ility distr ibut ion thusdeter min ed is calle d the sa m pling distr ibut ion of the num ber of successes in the sa m ple.


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Students t test

G osset , who wrote u nd er the pe n nam e of stu den t , der ived a theoret ical distr ibut ion wh ich has co m e to be k now n asStu den t s t distr ibut ion .

The qua n t ity t is def ined as

n = number of the obser vat ions in the sa m ple

Where is the sa mple m ea n i s the populat ion m ea n.S is the sa m ple sta nd ar d deviat ion .

xt

sn

Q!

x

Q


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Application of Students t test

To test the s igni f icance of a sa m ple m ea n

To test the s igni f icance of the di ffere ncebetwee n two sa m ple m ea ns

To test the s igni f icance of the coeff icien t of correlat ion .


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SNEDECORS F-test-For equality of Population Variances

Let

Let be the sa m ple m ea ns.

Let

Then

( 1, 2,... ) ( 1, 2, ... )

var .i j x i n and y j n be the values of two independent random samples

drawn from the normal populations withthe same iance

! !

xand y

1 222

2 2

1 11 2

1 1&

1 1

n n

x i y ji i

S x x S y yn n

! !

! !

22 2

2

22 2

2

x x y

y

y y x

x

S F if S S

S

or

S F if S S

S

! "

! "


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SNEDECORS F-test-For equalityof Population Variances

Tak ing the h ypothes is that the two sa m ples ha ved raw n fro m nor m al populat ions w ith the sa m evar iance , we co m pare the calculate d value F w ith itstable value.If the calculate d value of F > the table value of F w ith5% level of s ignif icance , the rat io is s igni f ican t a nd the h ypothes is m ay be rejecte d If the calculate d value of F < the table value of F w ith5% level of s ignif icance ,the h ypothes is m ay berejecte d


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Chi Square Test

Whe n a co in is tosse d 200 t im es , the theoret icalconsiderat ions lea d us to expect 100 hea ds a nd 100 ta ils.But in pract ice the results are rarel y ach ieved .

The qua n t ity a G reek letter , pro nou nce d as CHI SQUAREdescr ibes the m agni tu de of discrepa ncy betwee n the theor y and obser vat ion .Then

2 G

2

2

1

( 1 , 2 ,... )

( 1 , 2 ,... )

ni i

i i

i

i

O

O i n Ob seved frequency

E i n Expected frequency

G !

! ! !! !


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Chi Square Test

Degrees of free dom = n -1Chi square test is o ne of the s im plest a nd

m ost ge neral test k now n .It is appl icable to a ver y large num ber of proble m s in pract ice wh ich ca n be su mm ed up

und er the follow ing hea ds:Chi square test of goo dn ess of f itChi square test of ind epe nd ence of attr ibutes.


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Chi Square Test

Condi t ions for appl yin g ch i square test:Follow ing are the co ndi t ions wh ich shoul d be

sat isf ied before Ch i square test is appl ied .N , the total num ber of freque ncies shoul d belarge.

It is di ff icult to sa y what co nst itutes large ness,but as a n arb itrar y f igure , we m ay sa y that N shoul d be at least 50, howe ver , few the cells.


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Chi Square Test

N o theoret ical cell freque ncy shoul d be s m all.

5 shoul d be regar ded as the ver y minim um and 10 is better.

It is im porta n t to re m em ber that the num berof degrees of free dom is deter min ed with thenum ber of classes after regroup ing


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R egression

Regress ion is the est im at ion or pre dict ion of unknow n values of o ne var iable fro m know n

values of a nother var iable.After establ ish ing the fact of correlat ion betwee n two var iables , it is natural cur ios ity to k now the exte n t to wh ich o ne var iablevar ies in respo nse to a g iven var iat ion in another var iable


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R egression

To know about the nature of relat ionsh ipbetwee n the two var iables.

R egression measures the nature and extentof correlation.


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Linear R egression

If two var iables x a nd y are correlate d i.e.there ex ists a n assoc iat ion or relat ionsh ip

betwee n the m , the n the scatter diagra m willbe m ore or less co nce n trate d rou nd a cur ve.

This cur ve is calle d the cur ve of regress ion .The relat ion sh ip is sa id to be expresse d by m ea ns of cur vilinear regress ion .


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Linear R egression

Whe n the cur ve is a stra ight l ine , it is calle d aline of regress ion and the regress ion is sa id to

be l inearA line of regression is the straight line whichgives the best fit in the least square sense tothe given frequency.


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Regression Lines

If the l ine of regress ion is so chose n that thesu m of squares of deviat ion parallel to the ax is

of y is minimiz ed , it is calle d the l ine of regress ion of y on x

It gives the best est im ate of y for a ny given values of x


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Regression Lines

If the l ine of regress ion is so chose n that thesu m of squares of deviat ions parallel to the

axis of x is minimiz ed , it is calle d the l ine of regress ion of x o n y

It gives the best est im ate of x for a ny given values of y


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Lines of Regression

Y on X

X on Y

( ) y yx x

y y b r x xW

W

! !

( ) x xy y

x x r y yW

W

! !


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Regression Lines

If r = 0 , the two l ines of regress ion beco m es

The two stra ight l ines are parallel to X a nd Y axes respect ively and pass ing through the m ea ns

They are m utuall y perpe ndi cular

If , the two l ines of regress ion will coincide.

, y y x x

! !

1r ! s


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Properties of Regression coefficients

Correlat ion coeff icien ts is the geo m etr ic m ea n betwee n the regress ion coeff icien ts.

If one of the regress ion coeff icien ts is greatertha n uni ty, the other m ust be less tha n uni ty.

Arith m et ic m ea n of regress ion coeff icien ts isgreater tha n the correlat ion coeff icien t.


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Properties of Regression coefficients

Regress ion coeff icien ts are ind epe nd en t of cha nge of or igin but not of scale.

The correlat ion coeff icien t a nd the tworegress ion coeff icien ts ha ve the sa m e s ign


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A ngle between two R egression lines

Angle betwee n two regress ion lines

2

2 2

1an x y

x y

r

r

W W U

W W !


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Q uick R eview Q uestion

?


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T opic OutlineCen tral Limit theore mTest ing of h ypothes isCorrelat ionRegress ion

S ummary of Main Teaching Points


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Q & A

Q uestion and Answer S ession


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Matrices

and Determin

ants

N ext S ession

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