Stats Lec2

7/28/2019 Stats Lec2

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ProbabilityandProbability

s r u ons

Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006


2/25

Probabilit

Probabilityunderliesstatisticalinference thedrawingofconclusions

Ifsamplesaredrawnatrandom,theircharacteristics(suchasthe

samplemean)dependuponchance

Henceto

understand

how

to

inter ret

sam le

evidence,

we

need

to

understandchance,orprobability



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Definitionof

Probability

Theprobability

of

an

event

A may

be

defined

in

different

ways:

Thefrequentistview:theproportionoftrialsinwhichtheevent

occurs calculatedasthenumberoftrialsa roachesinfinit

Thesubjectiveview:someonesdegreeofbeliefaboutthe

likelihoodofaneventoccurring



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Probabilities

Witheach

outcome

in

the

sample

space

we

can

associate

a

pro a ty

Example:Tossacoin

Pr(Head)=1/2

Pr(Tail)=

Thisisanexampleofaprobabilitydistribution



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Rulesfor

Probabilities

0 Pr(A) 1

, or100%,summedoveralloutcomes 1p

Pr(notA)

=1

Pr(A)



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ProbabilityDistribution

Weextendtheprobabilityanalysisbyconsideringrandomvariables

These(usually)haveaknownprobabilitydistribution

Oncewe

work

out

the

relevant

distribution,

solving

the

problem

is



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RandomVariables

Moststatistics

(e.g.

the

sample

mean)

are

random

variables

Manyrandomvariableshavewellknownprobabilitydistributions

associatedwiththem

Tounderstand

random

variables,

we

need

to

know

about

probability

distributions



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SomeStandard

Probability Distributions

Binomial distribution

Normaldistributionan

e

s r u on Poissondistribution



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Whendo

They

Arise?

Binomial whentheunderlyingprobabilityexperimenthasonlytwo

ossibleoutcomes

e. .

tossin

acoin

Normal whenmanysmallindependentfactorsinfluenceavariable

Poisson forrareevents,whentheprobabilityofoccurrenceislow



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TheNormal

Distribution

ExamplesofNormallydistributedvariables:

Heights

thesample

mean

sometransformationsofvariables:e.g.naturallogarithmof

incomeisoftennormal



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TheNormal Distribution

(cont.)

TheNormaldistributionis

e s ape

Symmetric

Unimodal

andextendsfrom

x= to+(intheory)



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Parameters ofthe

Distribution

T etwoparameterso t eNorma istri utionaret emean an

thevariance 2

~ 2 ,

e.g.MensheightsareNormallydistributedwithmean174cmand

.

xM ~N(174,92.16)

e.g.WomensheightsareNormallydistributedwithameanof166

cmandvariance40.32

xW , .



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Graphof

Mens

and

Womens

Heights

Women

140 145 150 155 160 165 170 175 180 185 190 195 200

Height in centimetres



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AreasUnder

the

Distribution

Whatistheproportionofwomenthataretallerthan175cm?

Need this area

140 145 150 155 160 165 170 175 180 185 190 195 200

Height in centimetres



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AreasUnder

the Distribution

(cont.)

Howmanystandarddeviationsis175above166?

Onestandard

deviation

is

40.32=6.35,hence

42.1166175

So 175 lies 1.42 standard deviations above the mean

35.6

HowmuchoftheNormaldistributionliesbeyond1.42s.dsabove

themean?

Use

tables...



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TableA2

The

Standard

Normal

Distribution

z 0.00 0.01 0.02 0.03 0.04 0.05

. . . . . . .

0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404

1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885

1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735

1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606



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Answer

7.78%ofwomenaretallerthan175cm.

1. Calculatethezscore,giventhenumberofstandarddeviations

e ween emeanan e es re e g

2. Thenlookthezscoreupintablestogetaprobability

3. Userulesofsymmetrywhereappropriate



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TheDistribution

of

the

Sample

Mean

IfsamplesofsizenarerandomlydrawnfromaNormallydistributed2

,

distributedas

nNx2~

E.g.ifsamplesof50womenarechosen,thesamplemean is 5032.40,166~ Nx

notetheverysmallstandarderror:(40.32/50)=0.897



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TheDistributions

of

xand

of

x

Notethedistinctionbetween

2~ Nx

andnNx

2~

Theformerreferstothedistributionofatypicalmemberofthe

,

We

usually

refer

to

the

square

root

of

the

variance

of

the

sample

meanast estan ar error o t esamp emean,rat ert ant e

standarddeviation



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Example a s epro a yo raw ngasamp eo womenw ose

average heightis>168cm?

23.25032.40

166168

z

z =2.23cutsoff1.29%intheuppertailofthestandardNormal

istri ution,sot ereison yapro a i ityo 1.29%o rawinga

samplewithamean>168cm

Q.whatisprobabilityofdrawingasamplewithamean


21/25

Thesampleproportionalsohasanormaldistribution

1

np ,~

wherepis

the

sample

proportion,

thepopulationproportion,andthevarianceofthesampleproportionis(1 )/n.

since isusuallyunknownweestimateitwithp



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The CentralLimit

Theorem

Ifthesamplesizeislarge(n >25)thepopulationdoesnothavetobe

Normallydistributed,

the

sample

mean

is

(approximately)

Normal

whatevertheshapeofthepopulationdistribution

, .

minimumto

use



23/25

D str ut onsw en

amp es

are

ma :

Usin thetdistribution

When:

Thesamplesizeissmall(


24/25

The tDistribution

Thetdistributionis

bellshaped

symmetric

big n

unimodal extendsfrom

small n

(intheory)

more

spread

out

than

Normal dependsonn1(degreesoffreedom)



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Summary

Moststatisticalproblemsconcernrandomvariables whichhavean

associated

robabilit

distribution

CommondistributionsaretheBinomial,NormalandPoisson(there

manyothers)

Once

the

appropriate

distribution

for

the

problem

is

recognised,

the

solutionisrelativelystraightforward


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