L'échantillonage et les estimations

Pour aller plus loin (Ancien programme) - Mathématiques Spécialité

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

En France, le 1er janvier 2012, 24,2% des foyers d'une ville possédaient au moins un écran plat de télévision. Une étude s'intéresse à un échantillon de 170 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 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 \(p=0.18\). 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 \([0,24 ; 0,64]\).

Exercice 3 : 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 \(26,3\)% 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 \(175\) individus de la population étudiée. Sur cet échantillon, \(47\) 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 4 : 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

\( N = \) 800

\(N =\) 1600

\( N = \) 4500

Exercice 5 : Détermination de proportion par intervalle de confiance

On procède à un contrôle technique de 135 scooters constituant un échantillon représentatif des scooters circulant dans une ville.
74 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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