Échantillonnage - 2de
L’intervalle de fluctuation avec la loi binomiale
Exercice 1 : Intervalle de fluctuation - Loi binomiale - taille d'échantillon entre 50 et 100
Calculer l’intervalle de fluctuation à \(95\%\) d’une fréquence correspondant à la réalisation, sur un échantillon aléatoire de taille \(n = 75\), d’une variable aléatoire \(X\) suivant une loi binomiale de paramètre \(p = 0,3\).
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).
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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"red"}], [[[4.3, 0.376], [5.7, 0.376]], {"stroke": "red"}], [[[6, 0.391248], [6, 0.394152]], {"stroke": "red"}], [[[5.3, 0.3927], [6.7, 0.3927]], {"stroke": "red"}], [[[7, 0.37324799999999997], [7, 0.376152]], {"stroke": "red"}], [[[6.3, 0.3747], [7.7, 0.3747]], {"stroke": "red"}], [[[8, 0.400548], [8, 0.40345200000000003]], {"stroke": "red"}], [[[7.3, 0.402], [8.7, 0.402]], {"stroke": "red"}], [[[9, 0.386048], [9, 0.388952]], {"stroke": "red"}], [[[8.3, 0.3875], [9.7, 0.3875]], {"stroke": "red"}], [[[10, 0.403648], [10, 0.406552]], {"stroke": "red"}], [[[9.3, 0.4051], [10.7, 0.4051]], {"stroke": "red"}], [[[11, 0.358348], [11, 0.361252]], {"stroke": "red"}], [[[10.3, 0.3598], [11.7, 0.3598]], {"stroke": "red"}], [[[12, 0.399248], [12, 0.402152]], {"stroke": "red"}], [[[11.3, 0.4007], [12.7, 0.4007]], {"stroke": "red"}], [[[13, 0.364948], [13, 0.367852]], {"stroke": "red"}], [[[12.3, 0.3664], [13.7, 0.3664]], {"stroke": "red"}], [[[14, 0.38184799999999997], [14, 0.384752]], {"stroke": "red"}], [[[13.3, 0.3833], [14.7, 0.3833]], {"stroke": "red"}], [[[15, 0.384948], [15, 0.38785200000000003]], {"stroke": "red"}], [[[14.3, 0.3864], [15.7, 0.3864]], {"stroke": "red"}], [[[16, 0.398548], [16, 0.40145200000000003]], {"stroke": "red"}], [[[15.3, 0.4], [16.7, 0.4]], {"stroke": "red"}], [[[17, 0.353148], [17, 0.35605200000000004]], {"stroke": "red"}], [[[16.3, 0.3546], [17.7, 0.3546]], {"stroke": "red"}], [[[18, 0.378748], [18, 0.381652]], {"stroke": "red"}], [[[17.3, 0.3802], [18.7, 0.3802]], {"stroke": "red"}], [[[19, 0.373348], [19, 0.37625200000000003]], {"stroke": "red"}], [[[18.3, 0.3748], [19.7, 0.3748]], {"stroke": "red"}], [[[20, 0.38234799999999997], [20, 0.385252]], {"stroke": "red"}], [[[19.3, 0.3838], [20.7, 0.3838]], {"stroke": "red"}], [[[21, 0.404348], [21, 0.407252]], {"stroke": "red"}], [[[20.3, 0.4058], [21.7, 0.4058]], {"stroke": "red"}], [[[22, 0.398048], [22, 0.40095200000000003]], {"stroke": "red"}], [[[21.3, 0.3995], [22.7, 0.3995]], {"stroke": "red"}], [[[23, 0.354448], [23, 0.357352]], {"stroke": "red"}], [[[22.3, 0.3559], [23.7, 0.3559]], {"stroke": "red"}], [[[24, 0.35104799999999997], [24, 0.353952]], {"stroke": "red"}], [[[23.3, 0.3525], [24.7, 0.3525]], {"stroke": "red"}], [[[25, 0.407148], [25, 0.410052]], {"stroke": "red"}], [[[24.3, 0.4086], [25.7, 0.4086]], {"stroke": "red"}], [[[26, 0.408448], [26, 0.411352]], {"stroke": "red"}], [[[25.3, 0.4099], [26.7, 0.4099]], {"stroke": "red"}], [[[27, 0.40194799999999997], [27, 0.404852]], {"stroke": "red"}], [[[26.3, 0.4034], [27.7, 0.4034]], {"stroke": "red"}], [[[28, 0.411648], [28, 0.41455200000000003]], {"stroke": "red"}], [[[27.3, 0.4131], [28.7, 0.4131]], {"stroke": "red"}], [[[29, 0.374148], [29, 0.377052]], {"stroke": "red"}], [[[28.3, 0.3756], [29.7, 0.3756]], {"stroke": "red"}], [[[30, 0.372448], [30, 0.375352]], {"stroke": "red"}], [[[29.3, 0.3739], [30.7, 0.3739]], {"stroke": "red"}], [[[31, 0.357748], [31, 0.36065200000000003]], {"stroke": "red"}], [[[30.3, 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{"stroke": "red"}], [[[39.3, 0.3758], [40.7, 0.3758]], {"stroke": "red"}], [[[41, 0.333648], [41, 0.336552]], {"stroke": "red"}], [[[40.3, 0.3351], [41.7, 0.3351]], {"stroke": "red"}], [[[42, 0.35564799999999996], [42, 0.358552]], {"stroke": "red"}], [[[41.3, 0.3571], [42.7, 0.3571]], {"stroke": "red"}], [[[43, 0.355448], [43, 0.358352]], {"stroke": "red"}], [[[42.3, 0.3569], [43.7, 0.3569]], {"stroke": "red"}], [[[44, 0.419548], [44, 0.422452]], {"stroke": "red"}], [[[43.3, 0.421], [44.7, 0.421]], {"stroke": "red"}], [[[45, 0.394548], [45, 0.397452]], {"stroke": "red"}], [[[44.3, 0.396], [45.7, 0.396]], {"stroke": "red"}], [[[46, 0.367048], [46, 0.369952]], {"stroke": "red"}], [[[45.3, 0.3685], [46.7, 0.3685]], {"stroke": "red"}], [[[47, 0.36924799999999997], [47, 0.372152]], {"stroke": "red"}], [[[46.3, 0.3707], [47.7, 0.3707]], {"stroke": "red"}], [[[48, 0.356548], [48, 0.359452]], {"stroke": "red"}], [[[47.3, 0.358], [48.7, 0.358]], {"stroke": "red"}], [[[49, 0.356348], [49, 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[[[58, 0.394548], [58, 0.397452]], {"stroke": "red"}], [[[57.3, 0.396], [58.7, 0.396]], {"stroke": "red"}], [[[59, 0.364848], [59, 0.367752]], {"stroke": "red"}], [[[58.3, 0.3663], [59.7, 0.3663]], {"stroke": "red"}], [[[60, 0.378948], [60, 0.381852]], {"stroke": "red"}], [[[59.3, 0.3804], [60.7, 0.3804]], {"stroke": "red"}], [[[61, 0.373848], [61, 0.37675200000000003]], {"stroke": "red"}], [[[60.3, 0.3753], [61.7, 0.3753]], {"stroke": "red"}], [[[62, 0.36874799999999996], [62, 0.371652]], {"stroke": "red"}], [[[61.3, 0.3702], [62.7, 0.3702]], {"stroke": "red"}], [[[63, 0.39894799999999997], [63, 0.401852]], {"stroke": "red"}], [[[62.3, 0.4004], [63.7, 0.4004]], {"stroke": "red"}], [[[64, 0.397248], [64, 0.400152]], {"stroke": "red"}], [[[63.3, 0.3987], [64.7, 0.3987]], {"stroke": "red"}], [[[65, 0.333548], [65, 0.33645200000000003]], {"stroke": "red"}], [[[64.3, 0.335], [65.7, 0.335]], {"stroke": "red"}], [[[66, 0.35964799999999997], [66, 0.362552]], {"stroke": "red"}], [[[65.3, 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"red"}], [[[83.3, 0.3599], [84.7, 0.3599]], {"stroke": "red"}], [[[85, 0.397648], [85, 0.400552]], {"stroke": "red"}], [[[84.3, 0.3991], [85.7, 0.3991]], {"stroke": "red"}], [[[86, 0.37274799999999997], [86, 0.375652]], {"stroke": "red"}], [[[85.3, 0.3742], [86.7, 0.3742]], {"stroke": "red"}], [[[87, 0.348948], [87, 0.351852]], {"stroke": "red"}], [[[86.3, 0.3504], [87.7, 0.3504]], {"stroke": "red"}], [[[88, 0.383748], [88, 0.386652]], {"stroke": "red"}], [[[87.3, 0.3852], [88.7, 0.3852]], {"stroke": "red"}], [[[89, 0.372448], [89, 0.375352]], {"stroke": "red"}], [[[88.3, 0.3739], [89.7, 0.3739]], {"stroke": "red"}], [[[90, 0.370848], [90, 0.37375200000000003]], {"stroke": "red"}], [[[89.3, 0.3723], [90.7, 0.3723]], {"stroke": "red"}], [[[91, 0.394148], [91, 0.397052]], {"stroke": "red"}], [[[90.3, 0.3956], [91.7, 0.3956]], {"stroke": "red"}], [[[92, 0.38934799999999997], [92, 0.392252]], {"stroke": "red"}], [[[91.3, 0.3908], [92.7, 0.3908]], {"stroke": "red"}], [[[93, 0.377448], [93, 0.380352]], {"stroke": "red"}], [[[92.3, 0.3789], [93.7, 0.3789]], {"stroke": "red"}], [[[94, 0.378748], [94, 0.381652]], {"stroke": "red"}], [[[93.3, 0.3802], [94.7, 0.3802]], {"stroke": "red"}], [[[95, 0.352148], [95, 0.35505200000000003]], {"stroke": "red"}], [[[94.3, 0.3536], [95.7, 0.3536]], {"stroke": "red"}], [[[96, 0.423548], [96, 0.426452]], {"stroke": "red"}], [[[95.3, 0.425], [96.7, 0.425]], {"stroke": "red"}], [[[97, 0.374948], [97, 0.377852]], {"stroke": "red"}], [[[96.3, 0.3764], [97.7, 0.3764]], {"stroke": "red"}], [[[98, 0.393148], [98, 0.396052]], {"stroke": "red"}], [[[97.3, 0.3946], [98.7, 0.3946]], {"stroke": "red"}], [[[99, 0.351948], [99, 0.354852]], {"stroke": "red"}], [[[98.3, 0.3534], [99.7, 0.3534]], {"stroke": "red"}], [[[100, 0.361048], [100, 0.363952]], {"stroke": "red"}], [[[99.3, 0.3625], [100.7, 0.3625]], {"stroke": "red"}]]}
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.
Exercice 3 : Intervalle de fluctuation - Loi binomiale
Calculer l’intervalle de fluctuation à \(95\%\) d’une fréquence correspondant à la réalisation, sur un échantillon aléatoire de taille \(n = 6\), d’une variable aléatoire \(X\) suivant une loi binomiale de paramètre \(p = 0,9\).
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).
Exercice 4 : Intervalle de fluctuation - Loi binomiale - taille d'échantillon entre 50 et 100
Calculer l’intervalle de fluctuation à \(95\%\) d’une fréquence correspondant à la réalisation, sur un échantillon aléatoire de taille \(n = 87\), d’une variable aléatoire \(X\) suivant une loi binomiale de paramètre \(p = 0,3\).
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).
On arrondira les bornes à \(10^{-3}\) près. Par exemple, \([0,2627 ; 0,6648]\) deviendra \([0,263 ; 0,665]\).
Exercice 5 : 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.