stats lec2
TRANSCRIPT
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ProbabilityandProbability
s r u ons
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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Probabilit
Probabilityunderliesstatisticalinference thedrawingofconclusions
Ifsamplesaredrawnatrandom,theircharacteristics(suchasthe
samplemean)dependuponchance
Henceto
understand
how
to
inter ret
sam le
evidence,
we
need
to
understandchance,orprobability
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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Rulesfor
Probabilities
0 Pr(A) 1
, or100%,summedoveralloutcomes 1p
Pr(notA)
=1
Pr(A)
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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ProbabilityDistribution
Weextendtheprobabilityanalysisbyconsideringrandomvariables
These(usually)haveaknownprobabilitydistribution
Oncewe
work
out
the
relevant
distribution,
solving
the
problem
is
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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SomeStandard
Probability Distributions
Binomial distribution
Normaldistributionan
e
s r u on Poissondistribution
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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Whendo
They
Arise?
Binomial whentheunderlyingprobabilityexperimenthasonlytwo
ossibleoutcomes
e. .
tossin
acoin
Normal whenmanysmallindependentfactorsinfluenceavariable
Poisson forrareevents,whentheprobabilityofoccurrenceislow
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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TheNormal
Distribution
ExamplesofNormallydistributedvariables:
Heights
thesample
mean
sometransformationsofvariables:e.g.naturallogarithmof
incomeisoftennormal
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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TheNormal Distribution
(cont.)
TheNormaldistributionis
e s ape
Symmetric
Unimodal
andextendsfrom
x= to+(intheory)
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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 , .
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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...
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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Answer
7.78%ofwomenaretallerthan175cm.
1. Calculatethezscore,giventhenumberofstandarddeviations
e ween emeanan e es re e g
2. Thenlookthezscoreupintablestogetaprobability
3. Userulesofsymmetrywhereappropriate
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
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Thesampleproportionalsohasanormaldistribution
1
np ,~
wherepis
the
sample
proportion,
thepopulationproportion,andthevarianceofthesampleproportionis(1 )/n.
since isusuallyunknownweestimateitwithp
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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The CentralLimit
Theorem
Ifthesamplesizeislarge(n >25)thepopulationdoesnothavetobe
Normallydistributed,
the
sample
mean
is
(approximately)
Normal
whatevertheshapeofthepopulationdistribution
, .
minimumto
use
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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D str ut onsw en
amp es
are
ma :
Usin thetdistribution
When:
Thesamplesizeissmall(
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The tDistribution
Thetdistributionis
bellshaped
symmetric
big n
unimodal extendsfrom
small n
(intheory)
more
spread
out
than
Normal dependsonn1(degreesoffreedom)
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006
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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
Barrow, Statistics for Economics, Accounting and Business Studies, 4th edition Pearson Education Limited 2006