L'intervalle de fluctuation

L'échantillonnage - Mathématiques BTS

Exercice 1 : Intervalle de fluctuation pour une précision donnée (Formule Seconde)

Une entreprise X occupe 50% du marché.
Elle procède à un sondage sur 64 clients du marché en leur demandant s'ils sont clients de l'entreprise X.
En utilisant l'intervalle de fluctuation des fréquences à 95%, déterminer l'intervalle de clients qui devraient répondre positivement au sondage.

On donnera les bornes de l'intervalle sous la forme d'un entier.

,
=
+
-
\times
\div
\left(\right)
_{}
^{}
\sqrt{}

\frac{}{}
\pi
\int_{}^{}
\overrightarrow{}
\overline{}
%
\left({}\right)!

\leq
\geq
\lt
\gt
\approx

\cursiveS
\cursiveC
\cursiveA
\cursiveV

\left(\right)
\lcrcbrack{}
\lobrackcparen{}

\N
\Z
\D
\Q
\R
\R^{\star}

\infty
\emptyset
\cup
\cap
\setminus
\in
\notin

\lcrcbrack{}
\lorobrack{}
\lorcbrack{}
\lcrobrack{}
\lcrcbrace{}

€
\$
°
Ω
m^2
m^3

\slash
km \slash h
m \slash s

\cdot
m \cdot s^{-1}

x
y
\alpha
\beta
\gamma
\theta
\lambda
\sigma
\mu

U_n
V_n
u_n
v_n
u_{}
v_{}
\mapsto

\longrightarrow
\rho
\rightleftharpoons
\Delta

Exercice 2 : Intervalle de fluctuation pour une précision donnée (Formule Seconde)

Soit un échantillon de

64

individus pris dans une population, on estime que la probabilité qu'un caractère soit présent chez un individu pris aléatoirement dans la population totale est de

p = 0, 25

.
Calculer l'intervalle de fluctuation des fréquences au seuil

95 %

de de la fréquence de ce caractère.
On arrondira les bornes à $10^{- 2}$ près. Par exemple, $[0, 2386; 0, 6394]$ deviendra $[0, 24; 0, 64]$ .

,
=
+
-
\times
\div
\left(\right)
_{}
^{}
\sqrt{}

\frac{}{}
\pi
\int_{}^{}
\overrightarrow{}
\overline{}
%
\left({}\right)!

\leq
\geq
\lt
\gt
\approx

\cursiveS
\cursiveC
\cursiveA
\cursiveV

\left(\right)
\lcrcbrack{}
\lobrackcparen{}

\N
\Z
\D
\Q
\R
\R^{\star}

\infty
\emptyset
\cup
\cap
\setminus
\in
\notin

\lcrcbrack{}
\lorobrack{}
\lorcbrack{}
\lcrobrack{}
\lcrcbrace{}

€
\$
°
Ω
m^2
m^3

\slash
km \slash h
m \slash s

\cdot
m \cdot s^{-1}

x
y
\alpha
\beta
\gamma
\theta
\lambda
\sigma
\mu

U_n
V_n
u_n
v_n
u_{}
v_{}
\mapsto

\longrightarrow
\rho
\rightleftharpoons
\Delta

Exercice 3 : Échantillonnage et intervalle de fluctuation

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

{"graphInit": {"range": [[-3, 101], [0.29400000000000004, 0.486]], "scale": [6.0, 2343.7500000000005], "axisOpacity": 0.5, "axisArrows": "->", "gridOpacity": 0.15, "gridStep": [100, 0.005], "tickStep": [1, 1], "labelStep": [1, 10], "xLabel": "N\u00b0 d'\u00e9chantillon", "yLabel": "Fr\u00e9quences"}, "label": [[[0.20000000000000018, 0.29], "0,29", "left", {"font-size": 13}]], "path": [[[[0, 0.39], [100, 0.39]]], [[[1, 0.396988], [1, 0.401212]], {"stroke": "red"}], [[[0.29999999999999993, 0.3991], [1.7000000000000002, 0.3991]], {"stroke": "red"}], [[[2, 0.365988], [2, 0.370212]], {"stroke": "red"}], [[[1.2999999999999998, 0.3681], [2.7, 0.3681]], {"stroke": "red"}], [[[3, 0.385288], [3, 0.389512]], {"stroke": "red"}], [[[2.3, 0.3874], [3.7, 0.3874]], {"stroke": "red"}], [[[4, 0.402088], [4, 0.406312]], {"stroke": "red"}], [[[3.3, 0.4042], [4.7, 0.4042]], {"stroke": "red"}], [[[5, 0.410088], [5, 0.414312]], {"stroke": "red"}], [[[4.3, 0.4122], [5.7, 0.4122]], {"stroke": "red"}], [[[6, 0.348088], [6, 0.352312]], {"stroke": "red"}], [[[5.3, 0.3502], [6.7, 0.3502]], {"stroke": "red"}], [[[7, 0.398788], [7, 0.403012]], {"stroke": "red"}], [[[6.3, 0.4009], [7.7, 0.4009]], {"stroke": "red"}], [[[8, 0.432888], [8, 0.437112]], {"stroke": "red"}], [[[7.3, 0.435], [8.7, 0.435]], {"stroke": "red"}], [[[9, 0.378188], [9, 0.38241200000000003]], {"stroke": "red"}], [[[8.3, 0.3803], [9.7, 0.3803]], {"stroke": "red"}], [[[10, 0.401288], [10, 0.405512]], {"stroke": "red"}], [[[9.3, 0.4034], [10.7, 0.4034]], {"stroke": "red"}], [[[11, 0.367788], [11, 0.372012]], {"stroke": "red"}], [[[10.3, 0.3699], [11.7, 0.3699]], {"stroke": "red"}], [[[12, 0.400188], [12, 0.404412]], {"stroke": "red"}], [[[11.3, 0.4023], [12.7, 0.4023]], {"stroke": "red"}], [[[13, 0.392188], [13, 0.396412]], {"stroke": "red"}], [[[12.3, 0.3943], [13.7, 0.3943]], {"stroke": "red"}], [[[14, 0.365288], [14, 0.369512]], {"stroke": "red"}], [[[13.3, 0.3674], [14.7, 0.3674]], {"stroke": "red"}], [[[15, 0.417688], [15, 0.421912]], {"stroke": "red"}], [[[14.3, 0.4198], [15.7, 0.4198]], {"stroke": "red"}], [[[16, 0.413988], [16, 0.41821200000000003]], {"stroke": "red"}], [[[15.3, 0.4161], [16.7, 0.4161]], {"stroke": "red"}], [[[17, 0.376088], [17, 0.380312]], {"stroke": "red"}], [[[16.3, 0.3782], [17.7, 0.3782]], {"stroke": "red"}], [[[18, 0.41798799999999997], [18, 0.422212]], {"stroke": "red"}], [[[17.3, 0.4201], [18.7, 0.4201]], {"stroke": "red"}], [[[19, 0.390088], [19, 0.394312]], {"stroke": "red"}], [[[18.3, 0.3922], [19.7, 0.3922]], {"stroke": "red"}], [[[20, 0.357988], [20, 0.362212]], {"stroke": "red"}], [[[19.3, 0.3601], [20.7, 0.3601]], {"stroke": "red"}], [[[21, 0.357788], [21, 0.362012]], {"stroke": "red"}], [[[20.3, 0.3599], [21.7, 0.3599]], {"stroke": "red"}], [[[22, 0.353388], [22, 0.357612]], {"stroke": "red"}], [[[21.3, 0.3555], [22.7, 0.3555]], {"stroke": "red"}], [[[23, 0.372388], [23, 0.376612]], {"stroke": "red"}], [[[22.3, 0.3745], [23.7, 0.3745]], {"stroke": "red"}], [[[24, 0.407288], [24, 0.411512]], {"stroke": "red"}], [[[23.3, 0.4094], [24.7, 0.4094]], {"stroke": "red"}], [[[25, 0.40438799999999997], [25, 0.408612]], {"stroke": "red"}], [[[24.3, 0.4065], [25.7, 0.4065]], {"stroke": "red"}], [[[26, 0.405788], [26, 0.410012]], {"stroke": "red"}], [[[25.3, 0.4079], [26.7, 0.4079]], {"stroke": "red"}], [[[27, 0.426588], [27, 0.43081200000000003]], {"stroke": "red"}], [[[26.3, 0.4287], [27.7, 0.4287]], {"stroke": "red"}], [[[28, 0.403188], [28, 0.407412]], {"stroke": "red"}], [[[27.3, 0.4053], [28.7, 0.4053]], {"stroke": "red"}], [[[29, 0.408588], [29, 0.412812]], {"stroke": "red"}], [[[28.3, 0.4107], [29.7, 0.4107]], {"stroke": "red"}], [[[30, 0.371588], [30, 0.375812]], {"stroke": "red"}], [[[29.3, 0.3737], [30.7, 0.3737]], {"stroke": "red"}], [[[31, 0.402488], [31, 0.406712]], {"stroke": "red"}], [[[30.3, 0.4046], [31.7, 0.4046]], {"stroke": "red"}], [[[32, 0.373488], [32, 0.377712]], {"stroke": "red"}], [[[31.3, 0.3756], [32.7, 0.3756]], {"stroke": "red"}], [[[33, 0.425388], [33, 0.429612]], {"stroke": "red"}], [[[32.3, 0.4275], [33.7, 0.4275]], {"stroke": "red"}], [[[34, 0.428688], [34, 0.432912]], {"stroke": "red"}], [[[33.3, 0.4308], [34.7, 0.4308]], {"stroke": "red"}], [[[35, 0.397688], [35, 0.401912]], {"stroke": "red"}], [[[34.3, 0.3998], [35.7, 0.3998]], {"stroke": "red"}], [[[36, 0.395688], [36, 0.399912]], {"stroke": "red"}], [[[35.3, 0.3978], [36.7, 0.3978]], {"stroke": "red"}], [[[37, 0.34288799999999997], [37, 0.347112]], {"stroke": "red"}], [[[36.3, 0.345], [37.7, 0.345]], {"stroke": "red"}], [[[38, 0.354788], [38, 0.359012]], {"stroke": "red"}], [[[37.3, 0.3569], [38.7, 0.3569]], {"stroke": "red"}], [[[39, 0.416188], [39, 0.420412]], {"stroke": "red"}], [[[38.3, 0.4183], [39.7, 0.4183]], {"stroke": "red"}], [[[40, 0.395088], [40, 0.399312]], {"stroke": "red"}], [[[39.3, 0.3972], [40.7, 0.3972]], {"stroke": "red"}], [[[41, 0.414088], [41, 0.418312]], {"stroke": "red"}], [[[40.3, 0.4162], [41.7, 0.4162]], {"stroke": "red"}], [[[42, 0.409388], [42, 0.413612]], {"stroke": "red"}], [[[41.3, 0.4115], [42.7, 0.4115]], {"stroke": "red"}], [[[43, 0.387388], [43, 0.391612]], {"stroke": "red"}], [[[42.3, 0.3895], [43.7, 0.3895]], {"stroke": "red"}], [[[44, 0.421388], [44, 0.425612]], {"stroke": "red"}], [[[43.3, 0.4235], [44.7, 0.4235]], {"stroke": "red"}], [[[45, 0.357388], [45, 0.361612]], {"stroke": "red"}], [[[44.3, 0.3595], [45.7, 0.3595]], {"stroke": "red"}], [[[46, 0.360188], [46, 0.364412]], {"stroke": "red"}], [[[45.3, 0.3623], [46.7, 0.3623]], {"stroke": "red"}], [[[47, 0.399488], [47, 0.403712]], {"stroke": "red"}], [[[46.3, 0.4016], [47.7, 0.4016]], {"stroke": "red"}], [[[48, 0.400688], [48, 0.404912]], {"stroke": "red"}], [[[47.3, 0.4028], [48.7, 0.4028]], {"stroke": "red"}], [[[49, 0.412988], [49, 0.417212]], {"stroke": "red"}], [[[48.3, 0.4151], [49.7, 0.4151]], {"stroke": "red"}], [[[50, 0.399788], [50, 0.404012]], {"stroke": "red"}], [[[49.3, 0.4019], [50.7, 0.4019]], {"stroke": "red"}], [[[51, 0.385688], [51, 0.389912]], {"stroke": "red"}], [[[50.3, 0.3878], [51.7, 0.3878]], {"stroke": "red"}], [[[52, 0.357888], [52, 0.362112]], {"stroke": "red"}], [[[51.3, 0.36], [52.7, 0.36]], {"stroke": "red"}], [[[53, 0.376988], [53, 0.381212]], {"stroke": "red"}], [[[52.3, 0.3791], [53.7, 0.3791]], {"stroke": "red"}], [[[54, 0.396688], [54, 0.400912]], {"stroke": "red"}], [[[53.3, 0.3988], [54.7, 0.3988]], {"stroke": "red"}], [[[55, 0.384788], [55, 0.389012]], {"stroke": "red"}], [[[54.3, 0.3869], [55.7, 0.3869]], {"stroke": "red"}], [[[56, 0.417388], [56, 0.421612]], {"stroke": "red"}], [[[55.3, 0.4195], [56.7, 0.4195]], {"stroke": "red"}], [[[57, 0.349988], [57, 0.354212]], {"stroke": "red"}], [[[56.3, 0.3521], [57.7, 0.3521]], {"stroke": "red"}], [[[58, 0.399388], [58, 0.403612]], {"stroke": "red"}], [[[57.3, 0.4015], [58.7, 0.4015]], {"stroke": "red"}], [[[59, 0.38018799999999997], [59, 0.384412]], {"stroke": "red"}], [[[58.3, 0.3823], [59.7, 0.3823]], {"stroke": "red"}], [[[60, 0.40688799999999997], [60, 0.411112]], {"stroke": "red"}], [[[59.3, 0.409], [60.7, 0.409]], {"stroke": "red"}], [[[61, 0.380288], [61, 0.384512]], {"stroke": "red"}], [[[60.3, 0.3824], [61.7, 0.3824]], {"stroke": "red"}], [[[62, 0.384088], [62, 0.388312]], {"stroke": "red"}], [[[61.3, 0.3862], [62.7, 0.3862]], {"stroke": "red"}], [[[63, 0.385288], [63, 0.389512]], {"stroke": "red"}], [[[62.3, 0.3874], [63.7, 0.3874]], {"stroke": "red"}], [[[64, 0.424088], [64, 0.428312]], {"stroke": "red"}], [[[63.3, 0.4262], [64.7, 0.4262]], {"stroke": "red"}], [[[65, 0.367188], [65, 0.371412]], {"stroke": "red"}], [[[64.3, 0.3693], [65.7, 0.3693]], {"stroke": "red"}], [[[66, 0.378788], [66, 0.383012]], {"stroke": "red"}], [[[65.3, 0.3809], [66.7, 0.3809]], {"stroke": "red"}], [[[67, 0.34288799999999997], [67, 0.347112]], {"stroke": "red"}], [[[66.3, 0.345], [67.7, 0.345]], {"stroke": "red"}], [[[68, 0.406088], [68, 0.410312]], {"stroke": "red"}], [[[67.3, 0.4082], [68.7, 0.4082]], {"stroke": "red"}], [[[69, 0.386488], [69, 0.390712]], {"stroke": "red"}], [[[68.3, 0.3886], [69.7, 0.3886]], {"stroke": "red"}], [[[70, 0.380488], [70, 0.384712]], {"stroke": "red"}], [[[69.3, 0.3826], [70.7, 0.3826]], {"stroke": "red"}], [[[71, 0.352088], [71, 0.356312]], {"stroke": "red"}], [[[70.3, 0.3542], [71.7, 0.3542]], {"stroke": "red"}], [[[72, 0.360188], [72, 0.364412]], {"stroke": "red"}], [[[71.3, 0.3623], [72.7, 0.3623]], {"stroke": "red"}], [[[73, 0.411888], [73, 0.416112]], {"stroke": "red"}], [[[72.3, 0.414], [73.7, 0.414]], {"stroke": "red"}], [[[74, 0.393388], [74, 0.397612]], {"stroke": "red"}], [[[73.3, 0.3955], [74.7, 0.3955]], {"stroke": "red"}], [[[75, 0.375488], [75, 0.379712]], {"stroke": "red"}], [[[74.3, 0.3776], [75.7, 0.3776]], {"stroke": "red"}], [[[76, 0.39228799999999997], [76, 0.396512]], {"stroke": "red"}], [[[75.3, 0.3944], [76.7, 0.3944]], {"stroke": "red"}], [[[77, 0.427088], [77, 0.43131200000000003]], {"stroke": "red"}], [[[76.3, 0.4292], [77.7, 0.4292]], {"stroke": "red"}], [[[78, 0.388288], [78, 0.392512]], {"stroke": "red"}], [[[77.3, 0.3904], [78.7, 0.3904]], {"stroke": "red"}], [[[79, 0.380788], [79, 0.385012]], {"stroke": "red"}], [[[78.3, 0.3829], [79.7, 0.3829]], {"stroke": "red"}], [[[80, 0.410588], [80, 0.414812]], {"stroke": "red"}], [[[79.3, 0.4127], [80.7, 0.4127]], {"stroke": "red"}], [[[81, 0.364488], [81, 0.368712]], {"stroke": "red"}], [[[80.3, 0.3666], [81.7, 0.3666]], {"stroke": "red"}], [[[82, 0.399788], [82, 0.404012]], {"stroke": "red"}], [[[81.3, 0.4019], [82.7, 0.4019]], {"stroke": "red"}], [[[83, 0.358488], [83, 0.362712]], {"stroke": "red"}], [[[82.3, 0.3606], [83.7, 0.3606]], {"stroke": "red"}], [[[84, 0.40588799999999997], [84, 0.410112]], {"stroke": "red"}], [[[83.3, 0.408], [84.7, 0.408]], {"stroke": "red"}], [[[85, 0.384188], [85, 0.388412]], {"stroke": "red"}], [[[84.3, 0.3863], [85.7, 0.3863]], {"stroke": "red"}], [[[86, 0.408888], [86, 0.413112]], {"stroke": "red"}], [[[85.3, 0.411], [86.7, 0.411]], {"stroke": "red"}], [[[87, 0.374788], [87, 0.379012]], {"stroke": "red"}], [[[86.3, 0.3769], [87.7, 0.3769]], {"stroke": "red"}], [[[88, 0.429288], [88, 0.433512]], {"stroke": "red"}], [[[87.3, 0.4314], [88.7, 0.4314]], {"stroke": "red"}], [[[89, 0.398188], [89, 0.402412]], {"stroke": "red"}], [[[88.3, 0.4003], [89.7, 0.4003]], {"stroke": "red"}], [[[90, 0.411188], [90, 0.415412]], {"stroke": "red"}], [[[89.3, 0.4133], [90.7, 0.4133]], {"stroke": "red"}], [[[91, 0.374688], [91, 0.378912]], {"stroke": "red"}], [[[90.3, 0.3768], [91.7, 0.3768]], {"stroke": "red"}], [[[92, 0.405788], [92, 0.410012]], {"stroke": "red"}], [[[91.3, 0.4079], [92.7, 0.4079]], {"stroke": "red"}], [[[93, 0.374388], [93, 0.378612]], {"stroke": "red"}], [[[92.3, 0.3765], [93.7, 0.3765]], {"stroke": "red"}], [[[94, 0.400788], [94, 0.405012]], {"stroke": "red"}], [[[93.3, 0.4029], [94.7, 0.4029]], {"stroke": "red"}], [[[95, 0.348688], [95, 0.352912]], {"stroke": "red"}], [[[94.3, 0.3508], [95.7, 0.3508]], {"stroke": "red"}], [[[96, 0.387088], [96, 0.391312]], {"stroke": "red"}], [[[95.3, 0.3892], [96.7, 0.3892]], {"stroke": "red"}], [[[97, 0.380988], [97, 0.385212]], {"stroke": "red"}], [[[96.3, 0.3831], [97.7, 0.3831]], {"stroke": "red"}], [[[98, 0.377788], [98, 0.382012]], {"stroke": "red"}], [[[97.3, 0.3799], [98.7, 0.3799]], {"stroke": "red"}], [[[99, 0.396088], [99, 0.400312]], {"stroke": "red"}], [[[98.3, 0.3982], [99.7, 0.3982]], {"stroke": "red"}], [[[100, 0.414388], [100, 0.418612]], {"stroke": "red"}], [[[99.3, 0.4165], [100.7, 0.4165]], {"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 4 : Intervalle de fluctuation pour une précision donnée (Formule Seconde)

Une entreprise X occupe 20% du marché.
Elle procède à un sondage sur 64 clients du marché en leur demandant s'ils sont clients de l'entreprise X.
En utilisant l'intervalle de fluctuation des fréquences à 95%, déterminer l'intervalle de clients qui devraient répondre positivement au sondage.

On donnera les bornes de l'intervalle sous la forme d'un entier.

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

Soit un échantillon de

324

individus pris dans une population, on estime que la probabilité qu'un caractère soit présent chez un individu pris aléatoirement dans la population totale est de

p = 0, 35

.
Calculer l'intervalle de fluctuation des fréquences au seuil

95 %

de de la fréquence de ce caractère.
On arrondira les bornes à $10^{- 2}$ près. Par exemple, $[0, 2386; 0, 6394]$ deviendra $[0, 24; 0, 64]$ .

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