1 Project I The Challenger Disaster. 2 What to do with the zeros?

11

Project I Project I

The Challenger DisasterThe Challenger Disaster

22

What to do with the zeros?What to do with the zeros?

33

The Number of O-Ring Failures Per Launch

0

0.5

1

1.5

2

2.5

3

3.5

30 40 50 60 70 80 90

Temperature, F

Nu

mb

er

44

Number of Failed O-Rings Per Launch Vs. Temperature; No zeros

y = -0.0254x + 3.0465

R2 = 0.0693

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80

Temperature, F

Nu

mb

er

Never Throw Away Data

55

66

Number of O-Ring Failures Per Launch Vs. Temperature

y = -0.0608x + 4.675

R2 = 0.321

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80 90

Temperature, F

Nu

mb

er

Which is the greater sin, (1) throwing away data or running a biased regression?

77

88

Two Regression ApproachesTwo Regression Approaches Probability Models: Qualitative dependentProbability Models: Qualitative dependent

Linear probability modelLinear probability model Non-linearNon-linear

ProbitProbit logitlogit

Number Models: Quantitative dependentNumber Models: Quantitative dependent OLS, biasedOLS, biased Tobit: extension of probitTobit: extension of probit Count modelsCount models

PoissonPoisson

99

Probability ModelsProbability Models

One or more failures per launch coded as One or more failures per launch coded as 11

Zero failures per launch coded as 0Zero failures per launch coded as 0

1010

Low Temperature: 6 launches with failures out of 12 cases

High Temperature: 1 launch with failure out of12 cases

5.012/6ˆ Lp

083.012/1ˆ HP

53.2165.0/)083.05.0(

]}12/)ˆ1(ˆ[]12/)ˆ1(ˆ{[ˆ

ˆ/)]()ˆˆ[(

0.:

0.,:0

z

pppp

ppppz

pepippH

pepippH

HHLL

HLHL

HLHLA

HLHL

1111

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

2]1/)0[(2/1*]2/1[)( zezf

1.645-34.7

5%

1212

k n prob0 12 0.1121571 12 0.2691762 12 0.2960943 12 0.1973964 12 0.0888285 12 0.028425 0.0284249836 12 0.006632 0.0350574797 12 0.001137 0.0361944788 12 0.000142 0.0363366039 12 1.26E-05 0.036349236

10 12 7.58E-0711 12 2.76E-0812 12 4.59E-10

H0: p(low temp) = p(high temp) Binomial Prob(k≥5) in 12 Trials, Given p = 2/12

Power 10

1313

The Probability that O-Rings Fail In a Low Temperature Launch Given Probability of Failure At High Temperature =1/6

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5 6 7 8 9 10 11 12

Number of Launches with Failures

Pro

bab

ility

Power 10

Probability 5 or more fail = 0.036

1414

O-Ring Failures?: Yes or No Vs. Temerature

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90

Temperature

Pro

ba

bili

ty

Ex Post: the event either happens (code 1) or does not (code 2)

1515

Probabilty of One Or More O-Rings Failing Per Launch Vs. Temperature

y = -0.0367x + 2.8583

R2 = 0.3254

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90

Temperature

Pro

ba

bili

ty

Bernoulli

OLS: LPM

Linear (Bernoulli)

LinearApproximation

Ex Ante: what is the probability the event will happen?

1616

Probability Model of O-Ring Failure Per Launch, Yes-No Vs. Temerature

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 30 40 50 60 70 80 90

Temperature, F

Pro

ba

bili

ty

Bern

Logit

1717

1818

1919

Is there another factor besides temperature?

2020

2121

Who Me Worry?Who Me Worry?The Culture at NASA: getting away with The Culture at NASA: getting away with

problems that should be fixedproblems that should be fixedO-ringsO-ringsFoamFoam

NASA may not have been so worried NASA may not have been so worried about o-ring failures. They had about o-ring failures. They had experienced successful returns of the experienced successful returns of the shuttle with as many as three o-ring shuttle with as many as three o-ring failuresfailures

2222

Quantitative ModelsQuantitative Models

2323

The Number of O-Ring Failures Per Launch Vs. Temperature

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80 90

Temperature, F

Nu

mb

er

o-rings

OLS

Tobit

Linear (OLS)

2 obs.

1 obs.

2424

Number of Failed O-Rings Per Launch Vs. Temperature

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80 90

Temperature

Nu

mb

er

O-Rings

OLS

Poisson

Linear (OLS)

2525

Tobit with Dummy Variable for Three Observations Vs. Temperature

0

0.5

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80 90

Temperature, F

Nu

mb

er

O-Rings

Tobit

Tobit with Dummy

2626

2727

Check ListCheck ListDataDataPlotsPlots

7 launches with failures7 launches with failures24 launches24 launches

Linear probability model, logit or probitLinear probability model, logit or probit# of O-Ring failures per launch, OLS, Tobit, Poisson# of O-Ring failures per launch, OLS, Tobit, Poisson

Estimation Results, LabeledEstimation Results, LabeledGoodness of fitGoodness of fitSignificance: t-stat, F-statSignificance: t-stat, F-stat

2828

2929

Probabilty of One Or More O-Rings Failing Per Launch Vs. Temperature

y = -0.0367x + 2.8583

R2 = 0.3254

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90

Temperature

Pro

ba

bili

ty

Bernoulli

OLS: LPM

Linear (Bernoulli)

LinearApproximation

3030