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Here’s another real world data set, this is theSpace Shuttle Challenger and some of youmight remember on January 28, 1986, itexploded and was destroyed and sevenastronauts on board passed away. So,this example of would be one where thequestion is whether a good setdecision, could have predicted this disasterand saved the lives of the astronauts on board.On the day of this tragic launch, thetemperature was unusually low, at 36degrees Fahrenheit. So, the NASAengineers asked themselves, do wehave data from past flights to see if thistemperature could cause problems.They graphed the temperature Xi versusthe number of O-rings that had failedin the past, because those wereknown to be affected by temperature.In seven flights there were failures; infact, sometimes they failed twice andsometimes only one of them failed.And the temperatures were all overthe map, 70, 57, 63, 70, 53, 75 and 58.Now, these were all the cases wherea problem occurred and the questionwas, would the lower temperatureincrease the chances of a problemoccurring? Now, as a good statistician,what do we do? Linear regression?Do we compute confidence intervals?Or do we find a maximum likelihoodestimator. Check exactly onebox of the most appropriate term.
And I would argue this is a clear linear regression case,you have two variables, so this is what you’re going to do.
For linear regression, we computethe quotation B and A,and I give you the formulasB is calculated as a quotientof these terms that I’ve given you beforedivided by those guys here.A as you might rememberis given by this formula.And to compute them, let’s first computethe various components here.Let’s start with N, what is N?
N was easy at 7you just count the number of data points.
What is the sum of the Xis?
And the answer is 446,you just add these number over here.
The sum of the Yis,is a particularly challenging question.
The answer is 9,you just add the Yis.
Now, here is a harder one,sum of the Xi-squareswhich of course is in the formula of b over here.
And finally the term we need isthe sum of the Xis,Yis.What is that?
So now we can take all these five thingsand plug them into the formula for bplease do so,use the formula as shown over hereand calculate for me what b would be.
And the answerfor the data is 0.00143.
Let’s do the same for a.
So this kind of suggests thatas the temperature gets warmer,the chances of failure go up,because b is a positive slope.So you had all reasons to believe that a coldtemperature is actually good for the space shuttleand we should launch.Let’s just for kicks,compute the actual expected numberof O-ring failuresor the Yis for the temperature of 36 degrees.And here’s the formula,it’s 36 times b plus aWhat is that?
And the answer is 1.25.That’s still a significant number,but NASA had now all reason to believecold temperatures are good for you.I should say there’s another kind of flaw here,one we will discuss in a second,but one that’s kind of often isthe only looked at data with failures,so that would give us a numberthat is by definition larger than 1But most flights had no failures.So the actual expectation should be even lower.So NASA was even more confidentthat this thing could be flown safely.How wrong they were.
So in the full dataset there were 23 launchesand many of which had no failures.The sum of Xi was 1,600,sum of Yi is 9 as before.The Xi square is 112,400and the crossover here is 574.Now this is all extracted from the data,we have done it before,and now please calculate for me b.So this is some workbut please calculate for me b againand here is the formulaas before.
And the answer is -0.0475.
And what I now want is a,here’s the same formula as before.1 over N sum Yiminus b times the average Xi.
And the answer is 3.698.
So let’s now plug this inand see for the temperature of36 degreeswhat our predicted failure is.The formula now is simple,36 times b plus a,what do we get?
And the answer is 1.99which is we expect 2 O-rings to fail because of temperatureand those happen to be enoughto destroy Challenger and cause the disaster.