Pour aller plus loin (Ancien programme) - Spécialité

L'échantillonage et les estimations

Exercice 1 : Déterminer un intervalle de fluctuation (Formule Terminale)

En France, le 1er janvier 2014, 20,9% des foyers d'une ville possédaient au moins un écran plat de télévision. Une étude s'intéresse à un échantillon de 810 foyers de cette ville.
Donner un intervalle de fluctuation à 95% de la fréquence de ces foyers possédant un écran plat.
On arrondira les bornes à \(10^{-2}\) près. Par exemple, \([0,2386 ; 0,6394]\) deviendra \([0,24 ; 0,64]\).

Exercice 2 : Intervalle de confiance : Trouver un échantillon minimum (formule de Seconde)

On veut estimer la proportion de foyers disposant en France d'un abonnement internet.

On veut que l'intervalle de confiance obtenu soit de largeur 0.2% au seuil de 0.95.
En utilisant la formule de l'intervalle de confiance, déterminer la taille minimale de l'échantillon \(n\) pour obtenir un intervalle d'une telle précision.

Exercice 3 : Échantillonnage et intervalle de fluctuation

On étudie la fréquence d’un événement grâce au graphique ci-dessous représentant \( 100 \) échantillons.

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Déduire de ce graphique une valeur approchée de la taille \( N \) des échantillons puis choisir la valeur exacte la plus proche parmis les choix suivant.

\( N = \) 330
\( N = \) 500
\(N =\) 1600
\( N = \) 4500
\( N = \) 800

Exercice 4 : Prise de décision sur une hypothèse à l'aide de l'intervalle de fluctuation (Formule Terminale)

Un chercheur observe la répartition d'un caractère aléatoire dans une population. Il fait l'hypothèse que \(77\)% des individus de cette population possèdent ce caractère.
Il souhaite corroborer son hypothèse par l'observation, et que la probabilité que sa conclusion soit fausse soit de \(5\)%.
Il possède les données d'un échantillon de \(100\) individus de la population étudiée. Sur cet échantillon, \(91\) possèdent le caractère en question.
Déterminer l'intervalle de fluctuation asymptotique adapté au problème du chercheur.
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).

Au vu de cette observation, le chercheur doit-il rejeter son hypothèse ?

Exercice 5 : Intervalle de fluctuation pour une précision donnée (Formule Terminale)

On estime que la probabilité qu'un caractère soit présent chez un individu pris aléatoirement dans une population est de \(p=0.26\). Soit un échantillon de \(100\) individus,
Calculer l'intervalle de fluctuation asymptotique au seuil 95% de la fréquence de ce caractère dans cet échantillon.
On arrondira les bornes à \(10^{-2}\) près. Par exemple, \([0,2386 ; 0,6394]\) deviendra \([0,24 ; 0,64]\).

Revenir au choix du niveau