From d39b2c2ed1569d099d09db79a75ac4c2d888da6b Mon Sep 17 00:00:00 2001 From: Bertrand Benjamin Date: Fri, 29 May 2020 08:31:32 +0200 Subject: [PATCH] Feat: Cours et exercices sur l'intervalle de fluctuation --- .../2B_intervalle_fluctuation.pdf | Bin 0 -> 64432 bytes .../2B_intervalle_fluctuation.tex | 50 ++++++++++++++++++ .../2E_parite.pdf | Bin 24109 -> 45872 bytes .../banque.tex | 22 ++++++++ .../index.rst | 16 +++++- 5 files changed, 87 insertions(+), 1 deletion(-) create mode 100644 TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.pdf create mode 100644 TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.tex diff --git a/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.pdf b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.pdf new file mode 100644 index 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literal 0 HcmV?d00001 diff --git a/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.tex b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.tex new file mode 100644 index 0000000..8562220 --- /dev/null +++ b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2B_intervalle_fluctuation.tex @@ -0,0 +1,50 @@ +\documentclass[a4paper,10pt]{article} +\usepackage{myXsim} + +\title{Intervalles de confiance 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/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2E_parite.pdf b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/2E_parite.pdf index 44dba99c765b983cdb84786b59609284c70f0722..692fdd9dcb87047850d398943c81728ebba2de7a 100644 GIT binary patch delta 36540 zcmY(JQ*fXSu%=_%wr$&)7!!M9+sPN(b~3@lwrx$EiEZ22|7_LSs_m<{tNOOPy5D}P zyFlxX!0M!+AlTU1*piJQXn<)vIfwm`*DJ$*xq37dg(Jbi0?~jgJ}c+bKaZq+_DD@* zmuLwx)ifRJz_vpkp6ahPgLCDVf;oYnXWfzEUUSy33e(1PL3(jYYm_!7CHdb+E#y}`4m^Q?xSE`T0-^IFS#vmWO; z;!jh_Wnk+QDk3yV4_t2?Ci!;3WQG{@Y`rs^nOg+m1NQ??_zI>n8$%+4g^4+C3YZDyP6wRg??e-msAR*vgA8(G%T6#w zMHba4C8sC^MF8|JK<3Fg2UimeN3UllnMO3`Ow$`JK~%&|oh-EZ*QZ4U77EpqPXtbU z=Nx7oH2050ml$kKjx+{Up4gPt_H#}%?^E}aU^;I0IleEJ2gPkEB%sK-^7cQJ~< z_lLDwbpQj4Hd@A!jHC9mi1^mTP3OP0SgZyTZrNbkE2KF7^@ZUtZ=7JJN|fA zAh6|J$!=RA`ioMPMVYkErW?m9FN)hHQgf;Q4nd?2dqn)g17x_zFYe>aXD3qV(7)FIUf_ZQGXNAqm1XA7OS+eR$i 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zoNWI^ZdBlyr5!9BNjTa6Pa#I4%MbX)$}Pgi&dDOi#?JjqoQ+352WYPMSFhcgOLrA24b^Yqf6D=7hlLIK5!+V(6 zATJkP_ANri)CdOGe%G+IVVCnEi>RttqXN`!-JtFo)NNGK>gCMn5{44TNq_W!obU68`cAsK}^!Twyl3*GM5Lnqbd062nDa92f;Qkl) CDtGPx diff --git a/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/banque.tex b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/banque.tex index 60c2699..80104e6 100644 --- a/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/banque.tex +++ b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/banque.tex @@ -81,4 +81,26 @@ \item Comment pourrait-on construire un critère pour déterminer si oui ou non une entreprise respecte la parité en se basant uniquement sur ses effectifs? \textit{Cette questions est une question ouverte. C'est à vous de déterminer le ou les critères qui vous semblent pertinents ou à minima un mécanisme de décision}. \end{enumerate} \end{exercise} + +\begin{exercise}[subtitle={Construction de l'intervalle de fluctuation}, step={2}, topics={Statistiques}] + On reprend le cadre de l'exercice sur la parité et l'on définit la parité de la façon suivante: + \begin{quote} + Chaque employé a une chance sur deux d'être une femme. + \end{quote} + On concidère une entreprise avec 100 salariés et on suppose qu'elle respecte la parité. Et on note $X$ la variable aléatoire qui compte le nombre de femme dans cette entreprise. + \begin{enumerate} + \item Quelle loi suit la variable aléatoire $X$? Préciser ses paramètres. + \item On approche $X$ par une loi normale. Expliquer pourquoi cette opération est possible et préciser les paramètres de cette loi. Dans la suite on concidèrera que $X$ suit cette loi normale. + \item Déterminer $a$ et $b$ pour avoir + \[ + P(a < X < b) = 0.95 + \] + \item On définit $I = \dfrac{X}{100}$ la variable aléatoire qui modélise la proportion de femme dans cette entreprise. En vous aidant de la question précédente determiner $c$ et $d$ tel que + \[ + P(c < I < d) = 0.95 + \] + \item Interpréter la question précédente. + \item On condidère une deuxième entreprise qui a elle aussi 100 employés dont 39 femmes. D'après ce qui a été fait précedement, peut-on concidéré qu'elle respecte la parité avec un niveau de confiance de 95\%? + \end{enumerate} +\end{exercise} \collectexercisesstop{banque} diff --git a/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/index.rst b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/index.rst index c483cb7..91a7a7c 100644 --- a/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/index.rst +++ b/TES/Probabilte_statistiques/Intervalle_fluctuation_confiance/index.rst @@ -2,7 +2,7 @@ Intervalles de fluctuation et de confiance pour l'année 2019-2020 en terminale ################################################################################# :date: 2020-05-28 -:modified: 2020-05-28 +:modified: 2020-05-29 :authors: Bertrand Benjamin :category: TESL :tags: Probabilité, Statistiques @@ -28,5 +28,19 @@ Bilan 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 =================================