1 midterm review econ 240a. 2 the big picture the classical statistical trail descriptive statistics...

11

Midterm ReviewMidterm Review

Econ 240AEcon 240A

22

The Big PictureThe Big Picture

The Classical Statistical TrailThe Classical Statistical Trail

Descriptive Statistics

Inferential

Statistics

Probability Discrete Random

Variables

Discrete Probability Distributions; Moments

Binomial

Application

Rates &

Proportions

Power 4-#4

44

Descriptive StatisticsDescriptive StatisticsPower One-Lab OnePower One-Lab One

ConceptsConcepts

central tendency: mode, median, meancentral tendency: mode, median, mean

dispersion: range, inter-quartile range, dispersion: range, inter-quartile range, standard deviation (variance)standard deviation (variance)

Are central tendency and dispersion Are central tendency and dispersion enough descriptors?enough descriptors?

55

Concepts Concepts Normal DistributionNormal Distribution– Central tendency: mean or averageCentral tendency: mean or average– Dispersion: standard deviationDispersion: standard deviation

Non-normal distributionsNon-normal distributions

0

20

40

60

80

15 30 45 60 75 90 105 120

Bills

Fre

qu

en

cy

Draw a HistogramDensity Function for the Standardized Normal Variate

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard Deviations

Den

sity


Descriptive Statistics Inferential

Statistics


Variables


Binomial

Application

Rates &

Proportions

Power 4-#4

Classicall Modern

77

Exploratory Data AnalysisExploratory Data Analysis

Stem and Leaf DiagramsStem and Leaf Diagrams

Box and Whiskers PlotsBox and Whiskers Plots

88

Males: 140 145 160 190 155 165 150 190 195 138 160 155 153 145 170 175 175 170 180 135 170 157 130 185 190 155 170 155 215 150 145 155 155 150 155 150 180 160 135 160 130 155 150 148 155 150 140 180 190 145 150 164 140 142 136 123 155

Females: 140 120 130 138 121 125 116 145 150 112 125 130 120 130 131 120 118 125 135 125 118 122 115 102 115 150 110 116 108 95 125 133 110 150 108

Weight Data

1010

Box DiagramBox Diagram

First or lowest quartile;25% of observations below

Upper or highest quartile25% of observations above

median

11113rd Quartile + 1.5* IQR = 156 + 46.5 = 202.5; 1st value below =195



Inferential

Statistics


Variables


Binomial

Application

Rates &

Proportions

Power 4-#4

1313

Power Three - Lab TwoPower Three - Lab Two

ProbabilityProbability

1414

Operations on eventsOperations on events

The event A and the event B both The event A and the event B both occur:occur:

Either the event A or the event B Either the event A or the event B occurs or both do:occurs or both do:

The event A does not occur, i.e.not The event A does not occur, i.e.not A:A:

)( BA

)( BA

A

1515

Probability statementsProbability statements

Probability of either event A or event BProbability of either event A or event B

– if the events are mutually exclusive, thenif the events are mutually exclusive, then

probability of event Bprobability of event B

)()()()( BApBpApBAp

)(1)( BpBp 0)( BAp

1616

Conditional ProbabilityConditional Probability

Example: in rolling two dice, what is Example: in rolling two dice, what is the probability of getting a red one the probability of getting a red one given that you rolled a white one?given that you rolled a white one?– P(R1/W1) ?P(R1/W1) ?

1717

In rolling two dice, what is the probability of getting a red one giventhat you rolled a white one?

1818

Conditional ProbabilityConditional ProbabilityExample: in rolling two dice, what is Example: in rolling two dice, what is the probability of getting a red one the probability of getting a red one given that you rolled a white one?given that you rolled a white one?– P(R1/W1) ?P(R1/W1) ?

)6/1/()36/1()1(/)11()1/1( WpWRpWRp

1919

Independence of two eventsIndependence of two events

p(A/B) = p(A)p(A/B) = p(A)– i.e. if event A is not conditional on event i.e. if event A is not conditional on event

BB– thenthen )(*)( BpApBAp



Inferential

Statistics


Variables


Binomial

Application

Rates &

Proportions

Power 4-#4

2121

Power 4 – Lab TwoPower 4 – Lab Two

2222

H

T

H

T

p

H

T

1-p

p

1 - p

p

H

T

H

T

H

T

H

T

p

1-p

p

1-p

Three flips of a coin; 8 elementary outcomes

3 heads

2 heads

2 heads1 head

2 heads1 head

1 head

0 heads

2323

The Probability of Getting k HeadsThe Probability of Getting k HeadsThe probability of getting k heads The probability of getting k heads (along a given branch) in n trials is: (along a given branch) in n trials is: ppk k *(1-p)*(1-p)n-kn-k

The number of branches with k heads in The number of branches with k heads in n trials is given by Cn trials is given by Cnn(k)(k)

So the probability of k heads in n trials So the probability of k heads in n trials is is Prob(k) = CProb(k) = Cnn(k) p(k) pk k *(1-p)*(1-p)n-kn-k

This is the discrete binomial distribution This is the discrete binomial distribution where k can only take on discrete where k can only take on discrete values of 0, 1, …k values of 0, 1, …k

2424

Expected Value of a discrete Expected Value of a discrete random variablerandom variable

E(x) =E(x) =

the expected value of a discrete the expected value of a discrete random variable is the weighted random variable is the weighted average of the observations where average of the observations where the weight is the frequency of that the weight is the frequency of that observationobservation

n

i

ixpix0

)]([*)(

2525

Variance of a discrete random Variance of a discrete random variablevariable

VAR(xVAR(xii) =) =

the variance of a discrete random the variance of a discrete random variable is the weighted sum of each variable is the weighted sum of each observation minus its expected observation minus its expected value, squared,where the weight is value, squared,where the weight is the frequency of that observationthe frequency of that observation

)]([})]([)({[ 2

0

ixpixEixn

i

2626

Lab TwoLab TwoThe Binomial Distribution, Numbers & PlotsThe Binomial Distribution, Numbers & Plots– Coin flips: one, two, …tenCoin flips: one, two, …ten– Die Throws: one, ten ,twentyDie Throws: one, ten ,twenty

The Normal Approximation to the BinomialThe Normal Approximation to the Binomial– As n As n ∞∞, p(k) N[np, np(1-p)], p(k) N[np, np(1-p)]– Sample fraction of successes: Sample fraction of successes:

]/)1(,[~ˆ

/)1()ˆ(,/)ˆ(,/ˆ 2

npppNp

npnppVarpnnppEnkp

2727

Density Function for the Standardized Normal Variate

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard Deviations

Den

sity

1.96-1.96

Z~N(0,1)Prob(-1.96≤z≤1.96)=0.95

npp

pEpz

p

p

/)ˆ1(ˆˆ

ˆ/)]ˆ(ˆ[

ˆ

ˆ

95.0)ˆ*96.1ˆˆ*96.1ˆ(

95.0)ˆ*96.1ˆˆ*96.1(

95.0)ˆ*96.1ˆˆ*96.1(

95.0)96.1ˆ/)]ˆ(ˆ[96.1(

ˆˆ

ˆˆ

ˆˆ

ˆ

pp

pp

pp

p

pppprob

ppprob

ppprob

pEpprob

2.5%2.5%

Lab Three and Power 5,62]1/)0[(2/1*]2/1[)( zezf

2828

Hypothesis Testing: Rates & ProportionsHypothesis Testing: Rates & Proportions

fpH

fpH

a

:

:0

One-tailed test:Step #1:hypotheses

One-tailed test:Step #2: test statistic

npp

fppEpz

p

pp

/)ˆ1(ˆˆ

,ˆ/]ˆ[ˆ/)]ˆ(ˆ[

ˆ

ˆˆ

One-tailed test:Step #3: choose e.g. = 5%


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard Deviations

Den

sity

Z=1.645

5%

Step # 4: this determinesThe rejection region for H0

Reject if645.1ˆ/)ˆ( fp

2929

Remaining TopicsRemaining TopicsInterval estimation and hypothesis Interval estimation and hypothesis testing for population means, using testing for population means, using sample meanssample meansDecision theoryDecision theoryRegressionRegression– EstimatorsEstimators

OLSOLSMaximum lilelihoodMaximum lilelihoodMethod of momentsMethod of moments

– ANOVAANOVA

3030

Midterm Review Cont.Midterm Review Cont.

Econ 240AEcon 240A

3131

Last TimeLast Time



Inferential

Statistics


Variables


Binomial

Application

Rates &

Proportions

Power 4-#4

3333

Remaining TopicsRemaining TopicsInterval estimation and hypothesis Interval estimation and hypothesis testing for population means, using testing for population means, using sample meanssample meansDecision theoryDecision theoryRegressionRegression– EstimatorsEstimators

OLSOLSMaximum lilelihoodMaximum lilelihoodMethod of momentsMethod of moments

– ANOVAANOVA

3434

PopulationRandom variable xDistribution f(f ?

Sample

Sample Statistic:

),(~ 2Nx

Sample Statistic

)1/()( 2

1

2

nxxsn

ii

Pop.

Lab ThreePower 7

3535

x0 1

f(x)

f(x) in this example is UniformX~U(0.5, 1/12)E(x) = 0.5Var(x) = 1/12

Nonetheless, from the central Limit theorem, the sample meanHas a normal density

nxz

xExz

nNx

x

/)12/1(/]5.0[

/)]([

]/)12/1(,5.0[~


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard Deviations

Den

sity

3636

Histogram of 50 sample means

05

1015

20

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0.85

0.95

Mor

e

Sample Mean

Fre

qu

en

cy

Histogram of 50 Sample Means, Uniform, U(0.5, 1/12)

Average of the 50 sample means: 0.4963

3737

InferenceInference

95.0))/(96.1)()/(96.1(Pr

95.0))/(96.1)()/(96.1(Pr

95.0))/(96.1)()/(96.1(Pr

95.0)96.1)//()(96.1(Pr

95.0)96.196.1(Pr

)//(][,

]/,[~

lim

),(~,

2

2

nxnxob

nxnob

nxnob

nxob

zob

nxzso

nNx

eknowittheoremwtrafromthecen

fxingeneral


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard Deviations

Den

sity

Z=1.96

2.5%

Z=-1.96

2.5%

3838

Confidence IntervalsConfidence IntervalsIf the population variance is known, use the If the population variance is known, use the normal distribution and znormal distribution and z

If the population variance is unknown, use If the population variance is unknown, use Student’s t-distribution and tStudent’s t-distribution and t

)//()( nxz

ni

ii nxxswhere

nsxt

1

2 )1/()(,

)//()(

3939

Text p.253Normal compared to t

t-distribution

t distributionas smple size grows

4040

Appendix BTable 4p. B-9

4141

Hypothesis testsHypothesis tests

vH

vH

a

:

:0

)//(][

/)]([

nvxz

xExz x

Step One: state the hypotheses

Step two: choose the test statistic

Step Three: choose the sizeOf the Type I error, =0.05


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-5 -4 -3 -2 -1 0 1 2 3 4 5

Standard Deviations

Den

sity

Z=1.96

2.5%

Z=-1.96

2.5%

Step four: reject the null hypothesis if the test statistic is in the Rejection region

2-tailed test

You choose v

4242

Decision

Accept null

Reject null

True State of Nature

p = 0.5 P > 0.5

No Error 1 -

Type I error C(I)

No Error 1 -

Type II error C(II)

E[C] = C(I)* + C(II)*

4343

Regression EstimatorsRegression Estimators

Minimize the sum of squared Minimize the sum of squared residualsresiduals

Maximum likelhood of the sampleMaximum likelhood of the sample

Method of momentsMethod of moments

4444

Minimize the sum of squared residuals

xbayae

xbay

yyeMin

n

ii

i

n

ii

i

n

ii

n

ii

ˆˆ,0ˆ/ˆ

)ˆˆ(

)ˆ(ˆ.

1

2

2

1

2

11

2

4545

n

ii neLik

1

222 /]ˆ[ˆ,0/ln

Maximum likelihood

Method of moments

n

i

n

iiii xxxxyyb

1 1

2)(/))((ˆ

n

i

n

iiii xxxxyyb

1 1

2)(/))((ˆ

4646

Inference in RegressionInference in RegressionInterval estimationInterval estimation

95.0)ˆˆˆˆ(

95.0)ˆˆˆ(

95.0)ˆˆˆ(

95.0)ˆ/]ˆ[(

ˆ/]ˆ[ˆ/]ˆˆ[

975.ˆ025.ˆ

975.ˆ025.ˆ

975.ˆ025.ˆ

975.ˆ025.

ˆˆ

tbbtbprob

tbbtprob

tbbtprob

tbbtprob

bbbEbt

bb

bb

bb

b

bb

4747

Estimated Coefficients, Power 8Estimated Coefficients, Power 8

CoefficientsStandard

Error t Stat P-value Lower 95%Upper

95%

Intercept -1.02377776 0.727626534 -1.40701 0.167999648 -2.499472762 0.451917

X Variable 1 0.06565026 0.001086328 60.43316 8.58311E-38 0.063447085 0.067853

a

b

41.1727.0/)0204.1(ˆ/)]ˆ(ˆ[ ˆˆ aa aEat

4848

Appendix BTable 4p. B-9

4949

Inference in RegressionInference in RegressionHypothesis testingHypothesis testing

Step OneState the hypothesis

0:

0:0

bHa

bH

Step TwoChoose the test statistic

bbbt ˆˆ/]ˆ[

Step ThreeChoose the size of theType I error,

Step FourReject the null hypothesis if the Test statistic is in the rejection region

1 midterm review econ 240a. 2 the big picture the classical statistical trail descriptive statistics...

Documents

probability slide

probability of event

median slide

histogram slide

classicallmodern slide

weight data slide

big picture slide

descriptive statistics