11

O-ring Failure & O-ring Failure & Temperature DependenceTemperature Dependence

January 28, 1986January 28, 1986

2222

33

Divide Sample Into 12 Low Temp & 12 Divide Sample Into 12 Low Temp & 12 High Temp LaunchesHigh Temp Launches

70700 0 F is boundary temperature dividing low temp & F is boundary temperature dividing low temp & high temp launches. There are 4 launches at 70high temp launches. There are 4 launches at 700 0 , , two with o-ring failurestwo with o-ring failures

There are three possibilities: zero, one & two high There are three possibilities: zero, one & two high temp launches with o-ring failure at 70temp launches with o-ring failure at 7000

What are probabilities of the corresponding number What are probabilities of the corresponding number of low temp launches with o-ring failures for these of low temp launches with o-ring failures for these three scenarios? ( I,e prob. Of o-ring failure of 1/12 three scenarios? ( I,e prob. Of o-ring failure of 1/12 (at high temp), or 2/12 or 3/12 if these high temp (at high temp), or 2/12 or 3/12 if these high temp probabilities apply to low temp launches probabilities apply to low temp launches

44

Assume only one high temp launch Assume only one high temp launch with o-ring failure ( @ 75with o-ring failure ( @ 7500 F) F)

What is probability that half of the low What is probability that half of the low temp launches will have o-ring failure if we temp launches will have o-ring failure if we assume the probability of o-ring failure is assume the probability of o-ring failure is the probability at high temp, i.e 1/12?the probability at high temp, i.e 1/12?

prob(k=6) = [n!/k!(n-k)!] pprob(k=6) = [n!/k!(n-k)!] pk k (1-p)(1-p)n-kn-k

Prob(k=6) = (12!/6!6!) (1/12)Prob(k=6) = (12!/6!6!) (1/12)66 (11/12) (11/12)66

Prob(k=6) ~ 2 in 10,000, or a rare event, Prob(k=6) ~ 2 in 10,000, or a rare event, so reject hypothesis that p=1/12so reject hypothesis that p=1/12

55

66

probabilty of o-ring failure in 12 launches if the probability of failure is 1/12

k binomial

n probability cumulative

12 0 0.351996 0.351996

12 1 0.383995 0.735991

12 2 0.191998 0.927988

12 3 0.058181 0.98617

12 4 0.011901 0.99807

12 5 0.001731 0.999801

12 6 0.000184 0.999985

12 7 1.43E-05 0.999999

12 8 8.13E-07 1

12 9 3.28E-08 1

12 10 8.96E-10 1

12 11 1.48E-11 1

12 12 1.12E-13 1

77

Binomial Distribution: Probability of 6 Binomial Distribution: Probability of 6 Low Temp Launches with O-ring Low Temp Launches with O-ring

Failure out of 12, p=1/12Failure out of 12, p=1/12Binomial Distribution: Probability of 6 launches with o-ring Failure

0

0.2

0.4

0.6

0.8

1

1.2

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

Low temp launches with O-Ring Failure

Pro

bab

ility

probability

cumulative

88

Assume two high temp launches Assume two high temp launches with o-ring failure ( @ 75with o-ring failure ( @ 7500 F & 70 F & 7000))

What is probability that 5 of the low temp What is probability that 5 of the low temp launches will have o-ring failure if we launches will have o-ring failure if we assume the probability of o-ring failure is assume the probability of o-ring failure is the probability at high temp?the probability at high temp?

prob(k=5) = [n!/k!(n-k)!] pprob(k=5) = [n!/k!(n-k)!] pk k (1-p)(1-p)n-kn-k

Prob(k=5) = (12!/5!7!) (2/12)Prob(k=5) = (12!/5!7!) (2/12)55 (10/12) (10/12)77

Prob(k=5) ~ 3 in 100, still an unlikely Prob(k=5) ~ 3 in 100, still an unlikely event, so reject hypothesis that p=2/12event, so reject hypothesis that p=2/12

99

1010

Probabiliy of 5 launches with o-ring failure out of 12 launches

binomial

n probability cumulative

12 0 0.112157 0.112157

12 1 0.269176 0.381333

12 2 0.296094 0.677426

12 3 0.197396 0.874822

12 4 0.088828 0.96365

12 5 0.028425 0.992075

12 6 0.006632 0.998707

12 7 0.001137 0.999844

12 8 0.000142 0.999987

12 9 1.26E-05 0.999999

12 10 7.58E-07 1

12 11 2.76E-08 1

1111

Binomial Distribution: Probability of 5 Binomial Distribution: Probability of 5 Low Temp launches with O-Ring Low Temp launches with O-Ring

Failure out of 12, p=2/12Failure out of 12, p=2/12Probability of up to 5 launches with o-ring failure out of 12

0

0.2

0.4

0.6

0.8

1

1.2

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

Launches with o-ring faiure

Pro

bab

ility

probability

cumulative

1212

Assume three high temp launches with Assume three high temp launches with o-ring failure ( @ 75o-ring failure ( @ 7500 F & 2@ 70 F & 2@ 7000))

What is probability that 4 of the low temp What is probability that 4 of the low temp launches will have o-ring failure if we launches will have o-ring failure if we assume the probability of o-ring failure is assume the probability of o-ring failure is the probability at high temp, i.e 3/12?the probability at high temp, i.e 3/12?

prob(k=4) = [n!/k!(n-k)!] pprob(k=4) = [n!/k!(n-k)!] pk k (1-p)(1-p)n-kn-k

Prob(k=4) = (12!/4!8!) (3/12)Prob(k=4) = (12!/4!8!) (3/12)44 (9/12) (9/12)88

Prob(k=4) ~ 19 in 100, not so unlikely an Prob(k=4) ~ 19 in 100, not so unlikely an event, so should we accept p = 3/12 for event, so should we accept p = 3/12 for both high & low temperature launches?both high & low temperature launches?

1313

1414

Probabilty of 4 launches with o-ring failure out of 12 launches

binomial

n probability cumulative

12 0 0.031676 0.031676

12 1 0.126705 0.158382

12 2 0.232293 0.390675

12 3 0.258104 0.648779

12 4 0.193578 0.842356

12 5 0.103241 0.945598

12 6 0.040149 0.985747

12 7 0.011471 0.997218

12 8 0.00239 0.999608

12 9 0.000354 0.999962

12 10 3.54E-05 0.999998

12 11 2.15E-06 1

12 12 5.96E-08 1

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Binomial Distribution: Probability 4 Low Binomial Distribution: Probability 4 Low temp Launches with O-Ring Failure, temp Launches with O-Ring Failure,

p=3/12p=3/12Binomial Distribution: Probability of up to 4 Launches with O-Ring Failure Out of 12 Launches

0

0.2

0.4

0.6

0.8

1

1.2

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

Launches with O-ring Failure

Pro

bab

ility

probability

cumulative

1616

Hypothesis TestingHypothesis Testing

Formulate null hypothesis: probability of a Formulate null hypothesis: probability of a launch with o-ring failure is same at low launch with o-ring failure is same at low temperature as at high temperature, temperature as at high temperature, pplow T low T = phigh T

Formulate an alternative hypothesis: probability of a launch with o-ring failure is higher at low temperature than at high temperature, pplow T low T > phigh T

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Hypothesis Testing, ContinuedHypothesis Testing, Continued Choose a test statistic, e.g. the probability of k Choose a test statistic, e.g. the probability of k

launches with o-ring failure out of n launches at launches with o-ring failure out of n launches at low temperature, conditional on the probability of low temperature, conditional on the probability of failure at high temperature,i.e. under the null that failure at high temperature,i.e. under the null that the probability of failure is the same at high & the probability of failure is the same at high & low temperatures, plow temperatures, p low T low T = phigh T , i.e. there is no temperature dependence

Reject the null hypothesis if you observe a rare event, i.e. one not likely to happen by chance.

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Decision TheoryDecision Theory

State of Nature:State of Nature:

Null, HNull, H0 0 is trueis true

State of Nature:State of Nature:

Null, HNull, H0 0 is False is False

OKOK

Prob = 1-Prob = 1-αα

Type II errorType II error

Prob = Prob = ββ

Type I errorType I error

Prob = Prob = αα

OKOK

Prob = 1 - Prob = 1 - ββ

Choice

Accept H0

Reject H0

1919

The economics of decision theoryThe economics of decision theoryMinimize the expected cost of making Minimize the expected cost of making

errors: Min. Prob(Type I error)* Cost(Type I errors: Min. Prob(Type I error)* Cost(Type I error) + Prob(Type II error) * Cost(Type II error) + Prob(Type II error) * Cost(Type II error)error)

Min Min αα * C(Type I error) + * C(Type I error) + ββ * (type II error) * (type II error)For the engineers at NASA on the eve of For the engineers at NASA on the eve of

the launch, they did not want to reject the the launch, they did not want to reject the null of null of pplow T low T = phigh T , i.e. no temperature dependence since rejection might mean no launch

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Economics of Decision Theory Cont.Economics of Decision Theory Cont. For the 7 Astronauts , they do not want the null For the 7 Astronauts , they do not want the null

accepted if it is false since the cost of a type II accepted if it is false since the cost of a type II error for them could be death!error for them could be death!

So, the question of who is in charge, making the So, the question of who is in charge, making the decision to accept or reject the hypothesis of no decision to accept or reject the hypothesis of no temperature dependence and whose perception temperature dependence and whose perception of the relative costs of a type I error versus the of the relative costs of a type I error versus the costs of a type II error is criticalcosts of a type II error is critical Are the NASA engineers in charge?Are the NASA engineers in charge? Are the Astronauts aware of the situation and being Are the Astronauts aware of the situation and being

heard?heard?

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Economics of Decision Theory, Cont.Economics of Decision Theory, Cont.Why woudn’t the NASA engineers want to Why woudn’t the NASA engineers want to

make make αα, the probability of rejecting , the probability of rejecting temperature dependence very small?temperature dependence very small?They would, but the smaller you make They would, but the smaller you make αα, the , the

larger you make larger you make ββ

2222

What happened the evening of What happened the evening of January 27, 1986?January 27, 1986?

The NASA engineers argued with mid-The NASA engineers argued with mid-level engineers at Thiokol, the makers of level engineers at Thiokol, the makers of the o-rings that sealed the sections of the the o-rings that sealed the sections of the booster rockets about the physical booster rockets about the physical chemistry properties of the o-rings. The chemistry properties of the o-rings. The mid-level technical experts at Thiokol mid-level technical experts at Thiokol warned that the o-rings became harder at warned that the o-rings became harder at low temperatures and would not seal as low temperatures and would not seal as well!well!

2323

What happened?What happened?Superiors at NASA persuaded the upper Superiors at NASA persuaded the upper

level bosses at Thiokol to sign off on the level bosses at Thiokol to sign off on the launchlaunch

As for temperature dependence from As for temperature dependence from previous launch experience, the engineers previous launch experience, the engineers at NASA threw out all the launches with no at NASA threw out all the launches with no o-ring failure and saw no pattern with o-ring failure and saw no pattern with temperature in the remaining lauches that temperature in the remaining lauches that experienced o-ring failureexperienced o-ring failure

24242424

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

25252525

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

2626

So what happened?So what happened?They launched Challenger at 34They launched Challenger at 340 0 FF

STS-51-L crew: (front row) Michael J. Smith, Dick Scobee, Ronald McNair; (back row) Ellison Onizuka, Christa McAuliffe, Gregory Jarvis, Judith Resnik.

2727

These sequential photos show a fiery plume These sequential photos show a fiery plume escaping from the right solid rocket booster escaping from the right solid rocket booster as the space shuttle Challenger ascends to as the space shuttle Challenger ascends to

the sky on Jan. 28, 1986.the sky on Jan. 28, 1986.

2828

Myth #2: Challenger explodedThe shuttle did not explode in the common definition of that word. There was no shock wave, no detonation, no "bang" — viewers on the ground just heard the roar of the engines stop as the shuttle’s fuel tank tore apart, spilling liquid oxygen and hydrogen which formed a huge fireball at an altitude of 46,000 ft. (Some television documentaries later added the sound of an explosion to these images.) But both solid-fuel strap-on boosters climbed up out of the cloud, still firing and unharmed by any explosion. Challenger itself was torn apart as it was flung free of the other rocket components and turned broadside into the Mach 2 airstream. Individual propellant tanks were seen exploding — but by then, the spacecraft was already in pieces.

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Myth #3: The crew died instantlyThe flight, and the astronauts’ lives, did not end at that point, 73

seconds after launch. After Challenger was torn apart, the pieces continued upward from their own momentum, reaching a peak altitude of 65,000 ft before arching back down into the water. The cabin hit the surface 2 minutes and 45 seconds after breakup, and all investigations indicate the crew was still alive until then.

What's less clear is whether they were conscious. If the cabin depressurized (as seems likely), the crew would have had difficulty breathing. In the words of the final report by fellow astronauts, the crew “possibly but not certainly lost consciousness”, even though a few of the emergency air bottles (designed for escape from a smoking vehicle on the ground) had been activated.

The cabin hit the water at a speed greater than 200 mph, resulting in a force of about 200 G’s — crushing the structure and destroying everything inside. If the crew did lose consciousness (and the cabin may have been sufficiently intact to hold enough air long enough to prevent this), it’s unknown if they would have regained it as the air thickened during the last seconds of the fall. Official NASA commemorations of “Challenger’s 73-second flight” subtly deflect attention from what was happened in the almost three minutes of flight (and life) remaining AFTER the breakup.

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http://www.google.com/search?hl=ensource=hp&q=video+of+Challenger+spacecraft+launch&btnG=Google+Search&aq=f&oq=&aqi=

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http://www.google.com/search?hl=en&source=hp&q=video+of+Challenger+spacecraft+launch&btnG=Google+Search&aq=f&oq=&aqi=