m01 stockwatson123635 03 econ part01 1

8/10/2019 M01 StockWatson123635 03 Econ Part01 1

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Introductionto Econometrics

Chapters 1, 2 and 3

The statistical analysis of

economic (and relateddata


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Brief Overview of the Course

! "conomics s#ggests important relationships$ often %ith policyimplications$ t virt#ally never s#ggests '#antitativemagnit#des of ca#sal effects.

What is the quantitativeeffect of red#cing class si)e onst#dent achievement*

+o% does another year of ed#cation change earnings*

What is the price elasticity of cigarettes*

What is the effect on o#tp#t gro%th of a 1 percentage pointincrease in interest rates &y the ,*

What is the effect on ho#sing prices of environmentalimprovements*


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This course is about using data tomeasure causa effects!

! deally$ %e %o#ld li/e an eperiment

What %o#ld &e an eperiment to estimate the effect of class si)e onstandardi)ed test scores*

! #t almost al%ays %e only have o&servational (non-eperimentaldata.

ret#rns to ed#cation

cigarette prices

monetary policy

! ost of the co#rse deals %ith diffic#lties arising from #sing

o&servational to estimate ca#sal effects confo#nding effects (omitted factors

sim#ltaneo#s ca#sality

correlation does not imply ca#sation3


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! 4earn methods for estimating ca#sal effects #singo&servational data

! 5oc#s on applications theory is #sed only as needed to#nderstand the %hys of the methods6

! 4earn to eval#ate the regression analysis of others thismeans yo# %ill &e a&le to read7#nderstand empiricaleconomics papers in other econ co#rses6

! 8et some hands-on eperience %ith regression analysis inyo#r pro&lem sets.

In this course #ou wi$


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! Empirica probem$

Class si)e and ed#cational o#tp#t

Policy '#estion9 What is the effect on test scores (orsome other o#tcome meas#re of red#cing class si)e &yone st#dent per class* &y : st#dents7class*

We m#st #se data to find o#t (is there any %ay to ans%er

this withoutdata*

&eview of 'robabiit# and (tatistics)(* Chapters 2, 3+


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The Caifornia Test (core ata (et

All ;-< and ;-: California school districts (n= >20

?aria&les9

! @thgrade test scores (tanford-B achievement test$com&ined math and reading$ district average

! t#dent-teacher ratio (T, = no. of st#dents in thedistrict divided &y no. f#ll-time e'#ivalent teachers


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Initia oo/ at the data$(You should already know how to interpret this table)

This ta&le doesnt tell #s anything a&o#t the relationship&et%een test scores and the STR.


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o districts with smaer casses havehigher test scores

(catterpot of test score v. st#dent-teacher ratio

What does this figure show?


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*e need to get some numerica evidence on whetherdistricts with ow (T&s have higher test scores but how

1. Compare average test scores in districts %ith lo% T,s to

those %ith high T,s (estimation3

2. Test the n#ll3 hypothesis that the mean test scores in the

t%o types of districts are the same$ against thealternative3 hypothesis that they differ (hypothesis

testing3

D. "stimate an interval for the difference in the mean test

scores$ high v. lo% T, districts (confidence interval3


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Initia data ana#sis$ Compare districts %ith small3 (T, E20 and large3 (T, F 20 class si)es9

1. Estimationof = difference &et%een gro#p means

2. Test the hypothesisthat = 0

3. Constr#ct a confidence intervalfor

Class i)e Average score

(

tandard deviation

(sY

n

mall 1B.> 2D:

4arge


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1! Estimation

=

=

s this a large difference in a real-%orld sense*

tandard deviation across districts = 1B.1

Hifference &et%een


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2! 5#pothesis testing

t=Ys Y

l

ss2

ns

+sl2

nl

=Ys Y

l

SE(Ys Y

l)

Hifference-in-means test9 comp#te the t-statistic$

! %here SE( is the standard error3 of $ the

s#&scripts sand lrefer to small3 and large3 T, districts$

and

Ys Yl Ys Yl

ss

2

=

1

ns 1 (Yi Ys )2

i=1

ns


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Comp#te the difference-of-means t-statistic9

= >.0@ItI J 1.B 2D:

large


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3! Confidence interva

A B@L confidence interval for the difference &et%eenthe means is$

( M 1.B M 1.B


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*hat comes ne6t7

! The mechanics of estimation$ hypothesis testing$ andconfidence intervals sho#ld &e familiar

! These concepts etend directly to regression and its variants

! efore t#rning to regression$ ho%ever$ %e %ill revie% someof the #nderlying theory of estimation$ hypothesis testing$and confidence intervals9

Why do these proced#res %or/$ and %hy #se these ratherthan others*

We %ill revie% the intellect#al fo#ndations of statisticsand econometrics


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&eview of (tatistica Theor#

1! The probabiit# framewor/ for statistica inference

2. "stimation

D. Testing

>. Confidence ntervals

1-1


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The probabiit# framewor/ for statisticainference

a Pop#lation$ random varia&le$ and distrition

& oments of a distrition (mean$ variance$ standarddeviation$ covariance$ correlation

c Conditional distritions and conditional means

d Histrition of a sample of data dra%n randomly from apop#lation9 Y1$ O$ Yn


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)a+ 'opuation, random variabe, anddistribution

Population

! The gro#p or collection of all possi&le entities of interest(school districts

! We %ill thin/ of pop#lations as infinitely large ( is an

approimation to very &ig3

Random variable Y

! Q#merical s#mmary of a random o#tcome (district averagetest score$ district T,


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Population distribution of Y

! The pro&a&ilities of different val#es of Ythat occ#r in thepop#lation$ for e. PrRY=


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)b+ 8oments of a popuation distribution$ mean,variance, standard deviation, covariance,correation

mean= epected val#e (epectation of Y= E(Y = Y

= long-r#n average val#e of Yover repeated reali)ations

of Y

Variance= E(Y Y2=

= meas#re of the s'#ared spread of the distrition

tandard deviation= = Y

Y2

variance


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!oments" ctd.

s#e$ness=

= meas#re of asymmetry of a distrition

! skewness= 09 distrition is symmetric

! skewnessJ (E 09 distrition has long right (left tail

#urtosis=

= meas#re of mass in tails= meas#re of pro&a&ility of large val#es

! kurtosis= D9 normal distrition

! skewnessJ D9 heavy tails (lepto#urtotic9+

E Y Y

( )

3

Y

3

E Y Y( )

4

Y

4


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2 random variabes$ :oint distributionsand covariance

! ,andom varia&les"and #have a%oint distribution

! The covariance&et%een"and #is

cov("$# = ER(" "(# #S = "#

! The covariance is a meas#re of the linear association&et%een"and #6 its #nits are #nits of"N#nits of #

! cov("$# J 0 means a positive relation &et%een"and #

! f"and #are independently distrited$ then cov("$# = 0(t not vice versa

! The covariance of a r.v. %ith itself is its variance9

cov("$" = ER(" "(" "S = ER(" "2S = X2


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The covariance &et%een Test S$oreand STRis negative9


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The correlation coefficientis defined in terms ofthe covariance9

corr("$# = = r"#

! 1 corr("$# 1

! corr("$# = 1 mean perfect positive linear association

! corr("$# = 1 means perfect negative linear association

! corr("$# = 0 means no linear association

cov(X,Z)

var(X)var(Z)= XZ

X

Z


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Thecorrelation coefficient measures linearassociation


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)c+ Conditiona distributions andconditiona means

&onditional distributions

! The distrition of Y$ given val#e(s of some other randomvaria&le$"

! "9 the distrition of test scores$ given that T, E 20

&onditional e'pectations and conditional moments

! $onditional ean= mean of conditional distrition

= E(YI"=% (important concept and notation

! $onditional varian$e= variance of conditional distrition

! E%aple9 E(Test s$ores I STRE 20 = the mean of test scoresamong districts %ith small class si)es


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The differen$e in eans is the differen$e between the eans oftwo $onditional distributions!

= E(Test s$oresISTRE 20 E(Test s$oresISTRF 20

Uther eamples of conditional means9

! Wages of all female %or/ers (Y= %ages$"= gender

! ortality rate of those given an eperimental treatment(Y= live7die6"= treated7not treated

! f E("I# = const$ then corr("$# = 0

(not necessarily vice versa ho%ever for eample considerthe f#nction

The $onditional ean is a different ter for the failiar idea ofthe group ean

2XY=


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)d+ istribution of a sampe of data drawnrandom# from a popuation$ Y1,7,Yn

(e $ill assume simple random sampling

! Choose an individ#al (district$ entity at random from thepop#lation

Randomness and data

! Prior to sample selection$ the val#e ofYis random &eca#sethe individ#al selected is random

! Unce the individ#al is selected and the val#e of Yiso&served$ then Yis K#st a n#m&er not random

! The data set is (Y1$ Y2$O$ Yn$ %here Yi= val#e of Yfor the ith

individ#al (district$ entity sampled


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)istribution of Y1,7,Ynunder simple random sampling

! eca#se individ#als V1 and V2 are selected at random$ theval#e of Y1has no information content for Y2. Th#s9

Y1and Y2are independently distributed

Y1and Y2come from the same distrition$ that is$ Y1$ Y2

are identically distributed

That is$ #nder simple random sampling$ Y1and Y2are

independently and identically distrited (i.i.d..

ore generally$ #nder simple random sampling$ YiX$ i=

1$O$ n$ are i.i.d.


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1. The pro&a&ility frame%or/ for statistical inference

2! Estimation

D. Testing

>. Confidence ntervals

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This frame$or# allo$s rigorous statistical inferencesabout moments of population distributions using a sampleof data from that population7

Estimation

is the nat#ral estimator of the mean. #t9

a What are the properties of *

& Why sho#ld %e #se rather than some other estimator*! Y1(the first o&servation

! may&e #ne'#al %eights not simple average

! median(Y1$O$ Yn

The starting point is the sampling distrition of O

Y

Y

Y

Y


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)a+ The samping distribution of

is a random varia&le$ and its properties are determined &ythe

sampling distributionof

The individ#als in the sample are dra%n at random.

Th#s the val#es of (Y1$ O$ Yn are random

Th#s f#nctions of (Y1$ O$ Yn$ s#ch as $ are random9 had a

different sample &een dra%n$ they %o#ld have ta/en on adifferent val#e

The distrition of over different possi&le samples of si)e niscalled thesampling distributionof .

The mean and variance of are the mean and variance of itssampling distrition$ E( and var( .

The concept of the samping distribution underpins a ofeconometrics.

Y

YY

YY

Y

Y

Y

Y


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The sampling distribution of " ctd.

E'ample9 #ppose Yta/es on 0 or 1 (a *ernoullirandom varia&le

%ith the pro&a&ility distrition$ PrRY= 0S = .22$ Pr(Y=1 = .G:

Then

E(Y =pN1 Y (1 p N0 =p= .G:

= ERY E(YS2=p(1 p = .G:N (1.G: = 0.1G1:>

Pr( = = 2N.22N.G: = .D>D2

Pr( = 1 = .G:2= .

Y

Y

2

Y

Y

Y

Y

Y


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The sampling distrition of %hen Yis erno#lli (p=.G:9

Y


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Things we want to /now about thesamping distribution$

! What is the mean of *

f E( = tr#e = .G:$ then is an unbiasedestimator of

! What is the variance of *

+o% does var( depend on n?

! Hoes it &ecome close to %hen nis large*

4a% of large n#m&ers9 is a consistentestimator of

! appears &ell shaped for nlargeOis this generally tr#e*

n fact$ is approimately normally distrited for nlarge(Central 4imit Theorem

Y

Y

Y

Y

Y

Y


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The mean and variance of the sampingdistribution of

! 8eneral case that is$ for Yii.i.d. from any distrition$ not

K#st erno#lli9

! mean9 E( = E( = = = Y

! ?ariance9 var( = ER E( S2

= ER YS2

= E

= E

Y

Y

1

nYi

i=1

n

1

nE(Y

i)

i=1

n

1

n

Yi=1

n

Y Y Y

1

n Yii=1

n

Y

2

1

n(Y

i

Y)

i=1

n

2

Y


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so var( = E

=

=

=

=

=

1

n(Y

i

Y)

i=1

n

2

Y

E1

n(Y

i

Y)

i=1

n

1

n(Y

j

Y)

j=1

n

1

n

2E (Y

i

Y)(Y

j

Y)

j=1

n

i=1

n

1

n2cov(Y

i,Y

j)

j=1

n

i=1

n

1

n2 Y

2

i=1

n

2

Y

n


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!ean and variance of samplingdistribution of " ctd.

E( = Y

var( =

&pli$ations9

1. is an unbiasedestimator of Y(that is$ E( = Y

2. var( is inversely proportional to n

1. the spread of the sampling distrition is

proportional to 17

2. Th#s the sampling #ncertainty associated %ithis proportional to 17 (larger samples$ less#ncertainty$ t s'#are-root la%

Y

Y

Y

Y

2

n

Y

n

n

Y

Y


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The samping distribution of when nisarge

5or small sample si)es$ the distrition of is complicated$ tif nis large$ the sampling distrition is simpleZ

1. As nincreases$ the distrition of &ecomes moretightly centered aro#nd Y(the 'aw of 'arge ubers

2. oreover$ the distrition of Y&ecomes normal

(the entral 'iit Theore

Y

Y

Y

Y


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The +a$ of +arge ,umbers$

An estimator is consistentif the pro&a&ility that its falls %ithin aninterval of the tr#e pop#lation val#e tends to one as the sample si)eincreases.

f (Y1$O$Yn are i.i.d. and E $ then is a consistent estimator of

Y$ that is$

PrRI YI E S 1 as n

%hich can &e %ritten$ Y

( Y3 means converges in pro&a&ility toY3.

(the ath9 as n$ var( = 0$ %hich implies thatPrRI YI E S 1.

Y2 Y

Y

p

Y

Y Y

p

Y

Y

2

nY


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The &entral +imit Theorem)C;T+9

f (Y1$O$Yn are i.i.d. and 0 E E $ then %hen nis large

the distrition of is %ell approimated &y a normaldistrition.

[ (Y$

normally distrited %ith mean Yand variance 7n3

( Y7Yis \ approimately distrited (0$1

(standard normal

That is$ standardi)ed3 = = is

approimately distrited as (0$1

The larger the n$ the &etter is the approimation.

n Y

Y2

Y

Y

Y

2

n

Y2

YYE(Y)

var(Y)

Y Y

Y / n


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ampling distrition of %hen Yis erno#lli$p= 0.G:9

Y


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(ummar#$ The (amping istribution of

5or Y1$O$Yni.i.d. %ith 0 E E $

! The eact (finite sample sampling distrition of has mean Y(

is an #n&iased estimator of Y3 and variance 7n

! Uther than its mean and variance$ the eact distrition of is

complicated and depends on the distrition of Y(the pop#lationdistrition

! When nis large$ the sampling distrition simplifies9

Y

Y2

Y

Y

2

Y

*Y (4a% of large n#m&ers

p

Y

YE(Y)

var(Y) is approimately (0$1 (C4T

Y


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)b+ *h#


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Estimator of the variance of Y$

= = sample variance of Y3

5act9

f (Y1$O$Yn are i.i.d. and E(Y> E $ then

Why does the la% of large n#m&ers apply*

! eca#se is a sample average6

! Technical note9 %e ass#me E(Y> E &eca#se herethe average is not of Yi$ t of its s'#are6

sY2

1n 1

(Yi Y)

2

i=1

n

sY2

p

Y2

sY

2


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1.The pro&a&ility frame%or/ for statistical inference

2."stimation

3!Testing

>.Confidence intervals

1-"0



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5#pothesis Testing

The hypothesis testing pro&lem(for the mean9 ma/e aprovisional decision &ased on the evidence at hand %hether an#ll hypothesis is tr#e$ or instead that some alternativehypothesis is tr#e. That is$ test

+09 E(Y = Y$0vs. +19 E(Y J Y$0(1-sided$ J

+09 E(Y = Y$0vs. +19 E(Y E Y$0(1-sided$ E

+09 E(Y = Y$0vs. +19 E(Y ] Y$0(2-sided


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T#pes of errors$

! Type "rror - n a hypothesis test$ a type error occ#rs %henthe n#ll hypothesis is reKected %hen it is in fact tr#e6 that is$ +0

is %rongly reKected.

! Type "rror - n a hypothesis test$ a type error occ#rs %hen

the n#ll hypothesis +0$ is not reKected %hen it is in fact false.

! The follo%ing ta&le gives a s#mmary of possi&le res#lts of anyhypothesis test9

1-%4

ecision

&e:ect 54 on=t re:ect 54

Truth54 Type "rror ,ight Hecision

51 ,ight Hecision Type "rror


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ome terminology for testing statisticalhypotheses-

p-value= pro&a&ility of dra%ing a statistic (e.g. at leastas adverse to the n#ll as the val#e act#ally comp#ted %ith yo#rdata$ ass#ming that the n#ll hypothesis is tr#e.

Thesignificance level(or the alpha level of a test is a pre-

specified pro&a&ility of incorrectly reKecting the n#ll$ %hen then#ll is tr#e.

&alculating the pvalue&ased on 9

p-val#e =

Where is the val#e of act#ally o&served (nonrandom

PrH0 [Y Y,0 >Yact

Y,0 !

Y

Y

Yact Y


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&alculating the pvalue" ctd.

! To comp#te thep-val#e$ yo# need the to /no% the sampling

distrition of $ %hich is complicated if nis small.

! f nis large$ yo# can #se the normal approimation (C4T9

p-val#e = $

=

=

pro&a&ility #nder leftYright (0$1 tails

%here = std. dev.of the distrition of = Y7 .

Y

PrH

0

[Y Y,0

>Yact Y,0

!

PrH

0

[Y

Y,0

Y

/ n>

Yact Y,0

Y

/ n!

PrH

0

[Y

Y,0

Y

>Yact

Y,0

Y

!

Y Y n

[=


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&alculating the pvalue $ith Y#no$n-

! 5or large n$p-val#e = the pro&a&ility that a (0$1 randomvaria&le falls o#tside I( Y$07 I

! n practice$ is #n/no%n it m#st &e estimated

Yact

Y

Y


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&omputing the pvalue $ith estimated9

p-val#e = $

=

(large n

so

pro&a&ility #nder normal tails o#tside Ita$tI

%here t= (the #s#al t-statistic

Y2

PrH0 [YY,0 >Y

act Y,0 !

PrH

0

[Y

Y,0

Y

/ n>

Yact Y,0

Y

/ n!

PrH

0

[ Y Y,0s

Y/ n

> Yact

Y,0s

Y/ n

!

Y Y,0

sY/ n

[=

Pr

H0

[ t> tact ! Y2p-val#e = ( estimated

[=


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*hat is the in/ between thep-vaue andthe significance eve

! The significance level is prespecified.

! 5or eample$ if the prespecified significance level is @L$ yo#reKect the n#ll hypothesis if ItI F 1.B


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The (tudent t distribution

f Yi$ i= 1$O$ nis i.i.d. (Y$ $ then the t-statistic has the

t#dent t-distrition %ith n 1 degrees of freedom.

t=

The critical val#es of the t#dent t-distrition is talated inthe &ac/ of all statistics &oo/s.

1. Comp#te the t-statistic

2. Comp#te the degrees of freedom$ %hich is n 1

D. 4oo/ #p the @L critical val#e

>. f the t-statistic eceeds (in a&sol#te val#e this criticalval#e$ reKect the n#ll hypothesis.

Y2

Y Y,0

sY/ n


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&omments on tudent t distribution

! f the sample si)e is moderate (several do)en or large(h#ndreds or more$ the difference &et%een the t-distrition and Q(0$1 critical val#es is negligi&le. +ere aresome @L critical val#es for 2-sided tests9

degrees of freedom

(n 1

@L t-distrition

critical val#e

10 2.2D

20 2.0BD0 2.0>


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The tudentt distribution / ummary

! The ass#mption that Yis distrited (Y$ is rarely pla#si&lein practice (ncome* Q#m&er of children*

! 5or nJ D0$ the t-distrition and (0$1 are very close (as ngro%s large$ the tn(1distrition converges to (0$1

! Thet-distrition is an artifact from days %hen sample si)es %eresmall and comp#ters3 %ere people

! 5or historical reasons$ statistical soft%are typically #ses the t-distrition to comp#tep-val#es t this is irrelevant %hen thesample si)e is moderate or large.

! 5or these reasons$ in this class %e %ill foc#s on the large-napproimation given &y the C4T

Y2


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(ome other distributions

! Ta&les of +ypothesis Tests

1-4
http://mathnstats.com/index.php/hypothesis-testing/130-table-of-hypothesis-tests.htmlhttp://mathnstats.com/index.php/hypothesis-testing/130-table-of-hypothesis-tests.html


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&eguar e6ampes of h#pothesis testing

! Airport sec#rity systems are designed to detect %eapons and&om&s. When yo# %al/ thro#gh an airport metal detector$ thesystem is trying to discriminate &et%een the hypothesis ^thisperson is not carrying a %eapon^ and the hypothesis ^this person iscarrying a %eapon$^ on the &asis of electromagnetic fieldmeas#rements.

! A dr#g company %ants to determine %hether a ne% headacheremedy %or/s. The concl#sion %ill &e &ased on %hat happens %hena gro#p of s#&Kects #se the remedy or a place&o to treatheadaches.

1-1


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1.The pro&a&ility frame%or/ for statistical inference

2."stimation

D.Testing

"!Confidence intervas

1-2



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Confidence Intervas

! A B@L confidence intervalfor Yis an interval thatcontains the tr#e val#e of Yin B@L of repeated samples.

! ,igression9 What is random here* The val#es of Y1$...$Ynand

th#s any f#nctions of them incl#ding the confidence

interval. The confidence interval %ill differ from one sampleto the net. The pop#lation parameter$ Y$ is not random6

%e K#st dont /no% it.


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&onfidence intervals" ctd.

A B@L confidence interval can al%ays &e constr#cted as theset of val#es of Ynot reKected &y a hypothesis test %ith a

@L significance level.

Y9 1.B


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(ummar#9

5rom the t%o ass#mptions of9

1. simple random sampling of a pop#lation$ that is$

Yi$ i=1$O$nX are i.i.d.

2. 0 E E(Y> E

%e developed$ for large samples (large n9 Theory of estimation (sampling distrition of

Theory of hypothesis testing (large-n distrition of t-statistic and comp#tation of thep-val#e

Theory of confidence intervals (constr#cted &y invertingthe test statistic

Are ass#mptions (1 _ (2 pla#si&le in practice* >es

Y

;et?s go bac/ to the origina poic#


66/66

;et?s go bac/ to the origina poic#@uestion$

What is the effect on test scores of red#cing T, &y onest#dent7class*

+ave we answered this question?

m01 stockwatson123635 03 econ part01 1

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