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variable aléatoire suivant la loi normale $\mathcal{N}(\mu, \sigma)$ alors +\begin{itemize} + \item $ P(\mu - \sigma < X < \mu + \sigma) = 0,683$ + \item $ P(\mu - 2\sigma < X < \mu + 2\sigma) = 0.954$ + \item $ P(\mu - 3\sigma < X < \mu + 3 \sigma) = 0,997$ + \item $ P(\mu - 1,96\sigma < X < \mu + 1,96 \sigma) = 0,950$ +\end{itemize} + +\subsubsection*{Remarque} +Les 3 premières valeurs sont à connaître et la dernière nous servira pour définir les outils statistiques dans la suite du chapitre. + +\section{Approximation de la loi binomiale par la loi normale} + +\subsection*{Propriété} + +Soit $X \sim \mathcal{B}(n, p)$ et $Y\sim \mathcal{N} (np, \sqrt{np(1-p)})$. + +On considèrera que $Y$ est une bonne approximation de $X$ quand est $n$ est assez grand ce qui ce traduira par +\[ + n \geq 30 \qquad np \geq 5 \qquad n(1-p) \geq 5 +\] + +\end{document} diff --git a/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/1E_binomiale_normal.pdf 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et de fluctuation (suite)} +\tribe{Terminale TESL} +\date{Mai 2020} + +\pagestyle{empty} + +\begin{document} + +\maketitle + +\setcounter{section}{2} +\section{Intervalle de fluctuation} + +\subsection*{Vocabulaire} + +\begin{itemize} + \item On étudie un \textbf{caractère} d'une \textbf{population} et on appelle en générale $p$ la proportion d'individus possédant ce caractère dans la population. + \item On prélève un échantille de $n$ individus. On supposera que ce tirage sera équivalent à un tirage avec remise et donc que la taille de la population est très grand par rapport à la taille de l'échantillon. + \item On notera $X_n$ la variable aléatoire correspondant au nombre d'individus possédant ce caractère et $F_n$ la proportion correspondante. +\end{itemize} + +\subsection*{Définition - Intervalle de fluctuation} + +En reprenant les notations précédente, on appelle \textbf{intervalle de fluctuation asymptotique au seuil 95\%} l'intervalle +\[ + I_n = \intFF{p - 1,96\sqrt{\frac{p(1-p)}{n}}}{p + 1,96\sqrt{\frac{p(1-p)}{n}}} +\] + +\subsection*{Propriété} + +Quand $n$ et $p$ correspondent au cas où la loi binomiale peut être approchée par une loi normale, c'est à dire +\[ + n \geq 30 \qquad np \geq 5 \qquad n(1-p) \geq 5 +\] +Alors la probabilité que $F_n$ appartienne à $I_n$ est égale à 0,95. +\[ + P(F_n \in I_n) = 0,95 +\] + +\subsubsection*{Exemple} +\afaire{} +On suppose que un quart de la population française est brun. On prélève un échantillon de 100 personnes au hasard. Calculer l'intervalle de fluctuation. + + + + +\end{document} diff --git a/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/2E_parite.pdf b/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/2E_parite.pdf new file mode 100644 index 0000000000000000000000000000000000000000..692fdd9dcb87047850d398943c81728ebba2de7a GIT binary patch literal 45872 zcmce-V~{A#wk_JWZQHhOueNR5w(aiKSZ&+3ZQHi)`p!A~#=G%e#Jhj?u86A0%AA=| zBgU*5b7qYqRS*%QWu#+;BAs0vT7_aEU?8wFw1nc}fudLOus0!~SC==mG%<39qE~h? zbpEd%F*`>aD0(3wJ9h#thQIcV4F7fNK+*p*@?WzUq3C4|oEYy3YWe_Q`NI{V*+|1%Ic{#_6_|6>ps{~_n! zLIdMJl>cXVU}XBYf%wlk6XUu0(?6vDTil6>`9BBZpCI_}0m1g41M!dN z{#!sW|MRx}%borwd^0gKG5vS^{{Ii(Q&`e=2g7gIdVSJ$NC>jWyaV}y{?}X~gV`jMjB86khY^c)DM-hKG8fwLcw$PV&NUks5%A`#=NW zWi#}L5ak}oe?E_n?cCURekLUaA*e#~hqg)rl|i73r3dS%uFv-e&Yv$b>wh_EBR8!# zZ!~GMUcmk`7G3t+{sIdRNz?|`9)n1_TQHo)2Rz^C$f9Q#fceDygi`!Xs|jh|@{tAa zw_gxCu(OLu?Q)0j<)X(hl|XX5SN*y&b+clq6_2DS`zN|bPr0r<_Ueonxn}Q4%N)iM zR7gXyKuO8sbkklko~(Qo0Tv>J_)%vGHXkWgtz+0HxBNp#Ip%tzd24Zs;jyr1(T%)? 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--git a/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/index.rst b/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/index.rst index 79cb6db..41e7493 100644 --- a/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/index.rst +++ b/Tsti2d/Probabilite/Intervalle_fluctuation_confiance/index.rst @@ -1,10 +1,46 @@ -Intervalles de fluctuation et de confiance pour l'année 2019-2020 en terminale STI2D +Intervalles de fluctuation et de confiance pour l'année 2019-2020 en terminale sti2d #################################################################################### :date: 2020-05-28 -:modified: 2020-05-28 +:modified: 2020-06-01 :authors: Bertrand Benjamin :category: Tsti2d :tags: Probabilité, Statistiques -:summary: Intervalles de fluctuation et de confiance pour l'année 2019-2020 en terminale STI2D +:summary: Passage de la loi binomiale à la loi normale et intevalle en statistiques pour l'année 2019-2020 en terminale Tsti2d + +Étape 1: De la loi binomiale à la loi normale +============================================= + +Remobilisation des notions autour de la loi binomiale et la loi normale. + +.. image:: ./1E_binomiale_normal.pdf + :height: 200px + :alt: Écart-type pour la loi normale et approximation + +Bilan + +.. image:: ./1B_binomiale_normale.pdf + :height: 200px + :alt: Bilan écart-type pour la loi normale et approximation + + +Étape 2: Intervalles de fluctuation +=================================== + +Prolème ouvert sur le respect de la parité. Les élèves sont amenés à se poser la question de comment déterminer si oui ou non une entreprise respecte la parité. Pour cela ils vont devoir construire un critère en se basant sur les effectifs. + +.. image:: ./2E_parite.pdf + :height: 200px + :alt: Problème ouvert sur le respect de la parité. + +On pourra espéré que les élèves aient évoqué la notion de proportion et d'intervalle. Nous formaliserons ensuite ces notions. + +Bilan + +.. image:: ./2B_intervalle_fluctuation.pdf + :height: 200px + :alt: Bilan sur l'intervalle de fluctuation + +Étape 3: Intervalles de confiance +================================= diff --git a/Tsti2d/semaines.rst b/Tsti2d/semaines.rst index 60c14a0..370aa01 100644 --- a/Tsti2d/semaines.rst +++ b/Tsti2d/semaines.rst @@ -2,7 +2,7 @@ Programmes semaine par semaine pendant le confinement avec les Tsti2d ##################################################################### :date: 2020-04-14 -:modified: 2020-05-28 +:modified: 2020-06-01 :authors: Bertrand Benjamin :category: Tsti2d :tags: Progression @@ -30,14 +30,14 @@ Cette organisation est amenée à évoluer en fonction de mon ressenti et surtou Semaine 23: 01/06 -> 05/06 ========================== +Vous aurez besoin du tableur pour jeudi. Le travail à rendre sera un travail sur tableur. + S23 - Programme Visio (2h) -------------------------- -On commencera exactement à l'heure précisée donc essayez de vous connecter au moins 5min avant pour résoudre les éventuels problèmes techniques (je donnerai le lien 15 min avant). On commencera directement par la correction des questions flashs alors faites-les avant! - -- Mardi 13h45-14h15: présentation de la semaine, -- Mercredi 11h15-11h45: -- Jeudi 15h45-16h45: +- Mardi 13h45-14h15: présentation de la semaine, travail sur des probabilités remarquables de la loi normale +- **Mercredi 11h15-11h45: réunion de préparation de la reprise, pas de visio** +- Jeudi 15h45-16h45: approximation de la loi binomiale avec la loi normale sur tableur. S23 - Questions flashs (3x5min) ------------------------------- @@ -50,10 +50,6 @@ S23 - Questions flashs (3x5min) - Mercredi -.. image:: ./Questions_Flash/P5/QF_20_06_01-2.pdf - :height: 200px - :alt: Questions flash - - Jeudi .. image:: ./Questions_Flash/P5/QF_20_06_01-3.pdf @@ -63,8 +59,17 @@ S23 - Questions flashs (3x5min) S23 - Intervalle de fluctuation ------------------------------- -S23 - Équations différentielles d'ordre 2 ------------------------------------------ +Exercices sur des probabilités à connaître sur la loi normale et approximation de la loi binomiale avec la loi normale. + +.. image:: ./Probabilite/Intervalle_fluctuation_confiance/1E_binomiale_normal.pdf + :height: 200px + :alt: Exercices sur des probabilités à connaître sur la loi normale et approximation de la loi binomiale avec la loi normale. + +Cours: Bilan sur les valeurs de probabilités à connaître sur la loi normale et l'approximation + +.. image:: ./Probabilite/Intervalle_fluctuation_confiance/1B_binomiale_normale.pdf + :height: 200px + :alt: Exercices sur des probabilités à connaître sur la loi normale et approximation de la loi binomiale avec la loi normale. Semaine 22: 25/05 -> 28/05