L'échantillonage et les estimations

Pour aller plus loin (Ancien programme) - Mathématiques Spécialité

Exercice 1 : 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 \(0\mbox{,}1\:\%\) au seuil de \(0\mbox{,}95\).

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 2 : 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

\(N =\) 1600

\( N = \) 800

\( N = \) 500

\( N = \) 4500

Exercice 3 : Déterminer un intervalle de fluctuation (Formule Terminale)

En France, le 1er janvier 2013, 50,7% des foyers d'une ville possédaient au moins un écran plat de télévision. Une étude s'intéresse à un échantillon de 750 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 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 \(34,2\)% 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 \(225\) individus de la population étudiée. Sur cet échantillon, \(85\) 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 : Détermination de proportion par intervalle de confiance

On procède à un contrôle technique de 80 scooters constituant un échantillon représentatif des scooters circulant dans une ville.
9 de ces scooters sont déclarés en mauvais état.
À partir de ce résultat, on souhaite estimer la proportion de scooters en mauvais état circulant dans la ville.
Déterminer un intervalle de confiance, au niveau de confiance de 95%, pour la proportion de scooters en mauvais état dans la ville. On donnera les bornes arrondies au centième.

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