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Version anglaise de M2/exo5 pour Python/Orgmode
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module2/exo5/exo5_python-en.org
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module2/exo5/exo5_python-en.org
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#+TITLE: Analysis of the risk of failure of the O-rings on the Challenger shuttle
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#+AUTHOR: Arnaud Legrand
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#+LANGUAGE: fr
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/htmlize.css"/>
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="http://www.pirilampo.org/styles/readtheorg/css/readtheorg.css"/>
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#+HTML_HEAD: <script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.3/jquery.min.js"></script>
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#+HTML_HEAD: <script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.4/js/bootstrap.min.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/lib/js/jquery.stickytableheaders.js"></script>
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#+HTML_HEAD: <script type="text/javascript" src="http://www.pirilampo.org/styles/readtheorg/js/readtheorg.js"></script>
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#+LATEX_HEADER: \usepackage{a4}
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#+LATEX_HEADER: \usepackage[french]{babel}
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# #+PROPERTY: header-args :session :exports both
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On January 27, 1986, the day before the takeoff of the shuttle /Challenger/, had
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a three-hour teleconference was held between
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Morton Thiokol (the manufacturer of one of the engines) and NASA. The
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discussion focused on the consequences of the
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temperature at take-off of 31°F (just below
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0°C) for the success of the flight and in particular on the performance of the
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O-rings used in the engines. Indeed, no test
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had been performed at this temperature.
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The following study takes up some of the analyses carried out that
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night with the objective of assessing the potential influence of
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the temperature and pressure to which the O-rings are subjected
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on their probability of malfunction. Our starting point is
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the results of the experiments carried out by NASA engineers
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during the six years preceding the launch of the shuttle
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Challenger.
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* Loading the data
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We start by loading this data:
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#+begin_src python :results value :session *python* :exports both
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import numpy as np
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import pandas as pd
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data = pd.read_csv("shuttle.csv")
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data
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#+end_src
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#+RESULTS:
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#+begin_example
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Date Count Temperature Pressure Malfunction
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0 4/12/81 6 66 50 0
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1 11/12/81 6 70 50 1
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2 3/22/82 6 69 50 0
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3 11/11/82 6 68 50 0
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4 4/04/83 6 67 50 0
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5 6/18/82 6 72 50 0
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6 8/30/83 6 73 100 0
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7 11/28/83 6 70 100 0
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8 2/03/84 6 57 200 1
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9 4/06/84 6 63 200 1
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10 8/30/84 6 70 200 1
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11 10/05/84 6 78 200 0
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12 11/08/84 6 67 200 0
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13 1/24/85 6 53 200 2
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14 4/12/85 6 67 200 0
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15 4/29/85 6 75 200 0
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16 6/17/85 6 70 200 0
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17 7/2903/85 6 81 200 0
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18 8/27/85 6 76 200 0
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19 10/03/85 6 79 200 0
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20 10/30/85 6 75 200 2
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21 11/26/85 6 76 200 0
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22 1/12/86 6 58 200 1
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#+end_example
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The data set shows us the date of each test, the number of O-rings
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(there are 6 on the main launcher), the
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temperature (in Fahrenheit) and pressure (in psi), and finally the
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number of identified malfunctions.
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* Graphical inspection
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Flights without incidents do not provide any information
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on the influence of temperature or pressure on malfunction.
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We thus focus on the experiments in which at least one O-ring was defective.
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#+begin_src python :results value :session *python* :exports both
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data = data[data.Malfunction>0]
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data
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#+end_src
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#+RESULTS:
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: Date Count Temperature Pressure Malfunction
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: 1 11/12/81 6 70 50 1
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: 8 2/03/84 6 57 200 1
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: 9 4/06/84 6 63 200 1
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: 10 8/30/84 6 70 200 1
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: 13 1/24/85 6 53 200 2
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: 20 10/30/85 6 75 200 2
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: 22 1/12/86 6 58 200 1
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We have a high temperature variability but
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the pressure is almost always 200, which should
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simplify the analysis.
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How does the frequency of failure vary with temperature?
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#+begin_src python :results output file :var matplot_lib_filename="freq_temp_python.png" :exports both :session *python*
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import matplotlib.pyplot as plt
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plt.clf()
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data["Frequency"]=data.Malfunction/data.Count
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data.plot(x="Temperature",y="Frequency",kind="scatter",ylim=[0,1])
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plt.grid(True)
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plt.savefig(matplot_lib_filename)
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print(matplot_lib_filename)
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#+end_src
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#+RESULTS:
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[[file:freq_temp_python.png]]
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At first glance, the dependence does not look very important, but let's try to
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estimate the impact of temperature $t$ on the probability of O-ring malfunction.
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* Estimation of the temperature influence
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Suppose that each of the six O-rings is damaged with the same
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probability and independently of the others and that this probability
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depends only on the temperature. If $p(t)$ is this probability, the
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number $D$ of malfunctioning O-rings during a flight at
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temperature $t$ follows a binomial law with parameters $n=6$ and
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$p=p(t)$. To link $p(t)$ to $t$, we will therefore perform a
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logistic regression.
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#+begin_src python :results value :session *python* :exports both
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import statsmodels.api as sm
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data["Success"]=data.Count-data.Malfunction
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data["Intercept"]=1
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# logit_model=sm.Logit(data["Frequency"],data[["Intercept","Temperature"]]).fit()
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logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], family=sm.families.Binomial(sm.families.links.logit)).fit()
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logmodel.summary()
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#+end_src
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#+RESULTS:
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#+begin_example
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Generalized Linear Model Regression Results
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==============================================================================
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Dep. Variable: Frequency No. Observations: 7
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Model: GLM Df Residuals: 5
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Model Family: Binomial Df Model: 1
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Link Function: logit Scale: 1.0
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Method: IRLS Log-Likelihood: -3.6370
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Date: Fri, 20 Jul 2018 Deviance: 3.3763
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Time: 16:56:08 Pearson chi2: 0.236
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No. Iterations: 5
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===============================================================================
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coef std err z P>|z| [0.025 0.975]
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-------------------------------------------------------------------------------
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Intercept -1.3895 7.828 -0.178 0.859 -16.732 13.953
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Temperature 0.0014 0.122 0.012 0.991 -0.238 0.240
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===============================================================================
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#+end_example
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The most likely estimator of the temperature parameter is 0.0014
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and the standard error of this estimator is 0.122, in other words we
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cannot distinguish any particular impact and we must take our
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estimates with caution.
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* Estimation of the probability of O-ring malfunction
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The expected temperature on the take-off day is 31°F. Let's try to
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estimate the probability of O-ring malfunction at
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this temperature from the model we just built:
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#+begin_src python :results output file :var matplot_lib_filename="proba_estimate_python.png" :exports both :session *python*
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import matplotlib.pyplot as plt
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data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1})
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data_pred['Frequency'] = logmodel.predict(data_pred)
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data_pred.plot(x="Temperature",y="Frequency",kind="line",ylim=[0,1])
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plt.scatter(x=data["Temperature"],y=data["Frequency"])
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plt.grid(True)
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plt.savefig(matplot_lib_filename)
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print(matplot_lib_filename)
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#+end_src
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#+RESULTS:
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[[file:proba_estimate_python.png]]
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As expected from the initial data, the
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temperature has no significant impact on the probability of failure of the
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O-rings. It will be about 0.2, as in the tests
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where we had a failure of at least one joint. Let's get back to the initial dataset to estimate the probability of failure:
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#+begin_src python :results output :session *python* :exports both
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data = pd.read_csv("shuttle.csv")
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print(np.sum(data.Malfunction)/np.sum(data.Count))
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#+end_src
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#+RESULTS:
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: 0.06521739130434782
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This probability is thus about $p=0.065$. Knowing that there is
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a primary and a secondary O-ring on each of the three parts of the
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launcher, the probability of failure of both joints of a launcher
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is $p^2 \approx 0.00425$. The probability of failure of any one of the
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launchers is $1-(1-p^2)^3 \approximately 1.2%$. That would really be
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bad luck.... Everything is under control, so the takeoff can happen
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tomorrow as planned.
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But the next day, the Challenger shuttle exploded and took away
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with her the seven crew members. The public was shocked and in
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the subsequent investigation, the reliability of the
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O-rings was questioned. Beyond the internal communication problems
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of NASA, which have a lot to do with this fiasco, the previous analysis
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includes (at least) a small problem.... Can you find it?
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You are free to modify this analysis and to look at this dataset
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from all angles in order to to explain what's wrong.
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