1. 1 O-ring Failure & Temperature Dependence January 28, 1986

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3. 3 Divide Sample Into 12 Low Temp & 12 High Temp Launches 70 0 F is boundary temperature dividing low temp & high temp launches. There are 4 launches at 70 0, two with o-ring failures 70 0 F is boundary temperature dividing low temp & high temp launches. There are 4 launches at 70 0, two with o-ring failures There are three possibilities: zero, one & two high temp launches with o-ring failure at 70 0 There are three possibilities: zero, one & two high temp launches with o-ring failure at 70 0 What are probabilities of the corresponding number of low temp launches with o-ring failures for these 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 probabilities apply to low temp launches What are probabilities of the corresponding number of low temp launches with o-ring failures for these 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 probabilities apply to low temp launches

4. 4 Assume only one high temp launch with o-ring failure ( @ 75 0 F) What is probability that half of the low temp launches will have o-ring failure if we assume the probability of o-ring failure is the probability at high temp, i.e 1/12? What is probability that half of the low temp launches will have o-ring failure if we assume the probability of o-ring failure is the probability at high temp, i.e 1/12? prob(k=6) = [n!/k!(n-k)!] p k (1-p) n-k prob(k=6) = [n!/k!(n-k)!] p k (1-p) n-k Prob(k=6) = (12!/6!6!) (1/12) 6 (11/12) 6 Prob(k=6) = (12!/6!6!) (1/12) 6 (11/12) 6 Prob(k=6) ~ 2 in 10,000, or a rare event, so reject hypothesis that p=1/12 Prob(k=6) ~ 2 in 10,000, or a rare event, so reject hypothesis that p=1/12

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6. 6 probabilty of o-ring failure in 12 launches if the probability of failure is 1/12 kbinomial nprobabilitycumulative 1200.351996 1210.3839950.735991 1220.1919980.927988 1230.0581810.98617 1240.0119010.99807 1250.0017310.999801 1260.0001840.999985 1271.43E-050.999999 1288.13E-071 1293.28E-081 12108.96E-101 12111.48E-111 12 1.12E-131

7. 7 Binomial Distribution: Probability of 6 Low Temp Launches with O-ring Failure out of 12, p=1/12

8. 8 Assume two high temp launches with o-ring failure ( @ 75 0 F & 70 0 ) What is probability that 5 of the low temp launches will have o-ring failure if we assume the probability of o-ring failure is the probability at high temp? What is probability that 5 of the low temp launches will have o-ring failure if we assume the probability of o-ring failure is the probability at high temp? prob(k=5) = [n!/k!(n-k)!] p k (1-p) n-k prob(k=5) = [n!/k!(n-k)!] p k (1-p) n-k Prob(k=5) = (12!/5!7!) (2/12) 5 (10/12) 7 Prob(k=5) = (12!/5!7!) (2/12) 5 (10/12) 7 Prob(k=5) ~ 3 in 100, still an unlikely event, so reject hypothesis that p=2/12 Prob(k=5) ~ 3 in 100, still an unlikely event, so reject hypothesis that p=2/12

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10. 10 Probabiliy of 5 launches with o-ring failure out of 12 launches binomial nprobabilitycumulative 1200.112157 1210.2691760.381333 1220.2960940.677426 1230.1973960.874822 1240.0888280.96365 1250.0284250.992075 1260.0066320.998707 1270.0011370.999844 1280.0001420.999987 1291.26E-050.999999 12107.58E-071 12112.76E-081

11. 11 Binomial Distribution: Probability of 5 Low Temp launches with O-Ring Failure out of 12, p=2/12

12. 12 Assume three high temp launches with o-ring failure ( @ 75 0 F & 2@ 70 0 ) What is probability that 4 of the low temp launches will have o-ring failure if we assume the probability of o-ring failure is the probability at high temp, i.e 3/12? What is probability that 4 of the low temp launches will have o-ring failure if we assume the probability of o-ring failure is the probability at high temp, i.e 3/12? prob(k=4) = [n!/k!(n-k)!] p k (1-p) n-k prob(k=4) = [n!/k!(n-k)!] p k (1-p) n-k Prob(k=4) = (12!/4!8!) (3/12) 4 (9/12) 8 Prob(k=4) = (12!/4!8!) (3/12) 4 (9/12) 8 Prob(k=4) ~ 19 in 100, not so unlikely an event, so should we accept p = 3/12 for both high & low temperature launches? Prob(k=4) ~ 19 in 100, not so unlikely an event, so should we accept p = 3/12 for both high & low temperature launches?

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14. 14 Probabilty of 4 launches with o-ring failure out of 12 launches binomial nprobabilitycumulative 1200.031676 1210.1267050.158382 1220.2322930.390675 1230.2581040.648779 1240.1935780.842356 1250.1032410.945598 1260.0401490.985747 1270.0114710.997218 1280.002390.999608 1290.0003540.999962 12103.54E-050.999998 12112.15E-061 12 5.96E-081

15. 15 Binomial Distribution: Probability 4 Low temp Launches with O-Ring Failure, p=3/12

16. 16 Hypothesis Testing Formulate null hypothesis: probability of a launch with o-ring failure is same at low temperature as at high temperature, p low T Formulate null hypothesis: probability of a launch with o-ring failure is same at low temperature as at high temperature, p low T = p high T p low T Formulate an alternative hypothesis: probability of a launch with o-ring failure is higher at low temperature than at high temperature, p low T > p high T

17. 17 Hypothesis Testing, Continued Choose a test statistic, e.g. the probability of k launches with o-ring failure out of n launches at low temperature, conditional on the probability of failure at high temperature,i.e. under the null that the probability of failure is the same at high & low temperatures, p low T Choose a test statistic, e.g. the probability of k launches with o-ring failure out of n launches at low temperature, conditional on the probability of failure at high temperature,i.e. under the null that the probability of failure is the same at high & low temperatures, p low T = p high 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.

18. 18 Decision Theory State of Nature: Null, H 0 is true State of Nature: Null, H 0 is False OK Prob = 1-α Type II error Prob = β Type I error Prob = α OK Prob = 1 - β Choice Accept H 0 Reject H 0

19. 19 The economics of decision theory Minimize the expected cost of making errors: Min. Prob(Type I error)* Cost(Type I error) + Prob(Type II error) * Cost(Type II error) Minimize the expected cost of making errors: Min. Prob(Type I error)* Cost(Type I error) + Prob(Type II error) * Cost(Type II error) Min α * C(Type I error) + β * (type II error) Min α * C(Type I error) + β * (type II error) For the engineers at NASA on the eve of the launch, they did not want to reject the null of p low T For the engineers at NASA on the eve of the launch, they did not want to reject the null of p low T = p high T, i.e. no temperature dependence since rejection might mean no launch

20. 20 Economics of Decision Theory Cont. For the 7 Astronauts, they do not want the null accepted if it is false since the cost of a type II error for them could be death! For the 7 Astronauts, they do not want the null accepted if it is false since the cost of a type II error for them could be death! So, the question of who is in charge, making the decision to accept or reject the hypothesis of no temperature dependence and whose perception of the relative costs of a type I error versus the costs of a type II error is critical So, the question of who is in charge, making the decision to accept or reject the hypothesis of no temperature dependence and whose perception of the relative costs of a type I error versus the costs 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 heard? Are the Astronauts aware of the situation and being heard?

21. 21 Economics of Decision Theory, Cont. Why woudn’t the NASA engineers want to make α, the probability of rejecting temperature dependence very small? Why woudn’t the NASA engineers want to make α, the probability of rejecting temperature dependence very small? They would, but the smaller you make α, the larger you make β They would, but the smaller you make α, the larger you make β

22. 22 What happened the evening of January 27, 1986? The NASA engineers argued with mid- level engineers at Thiokol, the makers of the o-rings that sealed the sections of the booster rockets about the physical chemistry properties of the o-rings. The mid-level technical experts at Thiokol warned that the o-rings became harder at low temperatures and would not seal as well! The NASA engineers argued with mid- level engineers at Thiokol, the makers of the o-rings that sealed the sections of the booster rockets about the physical chemistry properties of the o-rings. The mid-level technical experts at Thiokol warned that the o-rings became harder at low temperatures and would not seal as well!

23. 23 What happened? Superiors at NASA persuaded the upper level bosses at Thiokol to sign off on the launch Superiors at NASA persuaded the upper level bosses at Thiokol to sign off on the launch As for temperature dependence from previous launch experience, the engineers at NASA threw out all the launches with no o-ring failure and saw no pattern with temperature in the remaining lauches that experienced o-ring failure As for temperature dependence from previous launch experience, the engineers at NASA threw out all the launches with no o-ring failure and saw no pattern with temperature in the remaining lauches that experienced o-ring failure

24. 2424 Never Throw Away Data

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26. 26 So what happened? They launched Challenger at 34 0 F They launched Challenger at 34 0 F STS-51-L crew: (front row) Michael J. Smith, Dick Scobee, Ronald McNair;Michael J. SmithDick ScobeeRonald McNair (back row) Ellison Onizuka, Christa McAuliffe, Gregory Jarvis, Judith Resnik.Ellison OnizukaChrista McAuliffeGregory JarvisJudith Resnik

27. 27 These sequential photos show a fiery plume escaping from the right solid rocket booster as the space shuttle Challenger ascends to the sky on Jan. 28, 1986.

28. 28 Myth #2: Challenger exploded The 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.

29. 29 Myth #3: The crew died instantly The 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.

30. 30 http://www.google.com/search?hl=ensource=hp&q=video+of+Challenger+spacecraft+launch&btnG=Google+Search&aq=f&oq=&aqi=

31. 31 http://www.google.com/search?hl=en&source=hp&q=video+of+Challenger+spacecraft+launch&btnG=Google+Search&aq=f&oq=&aqi=