Quiz — Lycée Pilote Sfax - Statistiques 2.pdf

Question 1 sur 10 20:00

[{"id":39080,"question":"Soit X une variable aléatoire suivant une loi normale N(0,1). Quelle est la probabilité P(X ≤ 1) ?","option_a":"0,6826","option_b":"0,8413","option_c":"0,9772","option_d":"0,3413","option_e":"","option_f":"","bonne_reponse":"b","explication":"La probabilité P(X ≤ 1) pour une loi normale centrée réduite est donnée par la table de la loi normale et vaut environ 0,8413.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0,6826\", \"b\": \"0,8413\", \"c\": \"0,9772\", \"d\": \"0,3413\"}}","_debug_options_count":4},{"id":39081,"question":"Si deux événements A et B sont indépendants, alors P(A ∩ B) = P(A) + P(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour des événements indépendants, P(A ∩ B) = P(A) × P(B). La formule P(A) + P(B) est incorrecte.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39082,"question":"Quelle est l'espérance d'une variable aléatoire X suivant une loi binomiale B(n=10, p=0,3) ?","option_a":"3","option_b":"7","option_c":"10","option_d":"0,3","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'espérance d'une loi binomiale B(n,p) est donnée par E(X) = n × p = 10 × 0,3 = 3.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3\", \"b\": \"7\", \"c\": \"10\", \"d\": \"0,3\"}}","_debug_options_count":4},{"id":39083,"question":"Un intervalle de confiance à 95% pour une proportion p est donné par [p̂ - 1,96√(p̂(1-p̂)\/n) ; p̂ + 1,96√(p̂(1-p̂)\/n)].","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette formule est correcte pour un intervalle de confiance d'une proportion avec un niveau de confiance de 95%.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39084,"question":"Soit X une variable aléatoire d'espérance μ = 5 et d'écart-type σ = 2. Quelle est la probabilité P(3 ≤ X ≤ 7) si X suit une loi normale ?","option_a":"0,6826","option_b":"0,9544","option_c":"0,9974","option_d":"0,3413","option_e":"","option_f":"","bonne_reponse":"b","explication":"Pour une loi normale, environ 95% des valeurs sont dans l'intervalle [μ - 2σ ; μ + 2σ], soit [1 ; 9]. Ici, 3 et 7 sont dans cet intervalle, donc P(3 ≤ X ≤ 7) ≈ 0,9544.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"0,6826\", \"b\": \"0,9544\", \"c\": \"0,9974\", \"d\": \"0,3413\"}}","_debug_options_count":4},{"id":39085,"question":"La variance d'une variable aléatoire X est toujours positive ou nulle.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance mesure la dispersion des valeurs autour de l'espérance. Elle est toujours positive ou nulle (si X est constante).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39086,"question":"Dans un échantillon de taille n = 100, on observe une proportion p̂ = 0,4. Quel est l'intervalle de confiance à 95% pour la proportion p ?","option_a":"[0,302 ; 0,498]","option_b":"[0,352 ; 0,448]","option_c":"[0,320 ; 0,480]","option_d":"[0,380 ; 0,420]","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'intervalle de confiance est calculé par p̂ ± 1,96√(p̂(1-p̂)\/n) = 0,4 ± 1,96√(0,4×0,6\/100) ≈ 0,4 ± 0,096, soit [0,304 ; 0,496].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"[0,302 ; 0,498]\", \"b\": \"[0,352 ; 0,448]\", \"c\": \"[0,320 ; 0,480]\",","_debug_options_count":4},{"id":39087,"question":"Si X suit une loi binomiale B(n,p), alors la variance de X est égale à n × p × (1-p).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La variance d'une loi binomiale B(n,p) est bien V(X) = n × p × (1-p).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":39088,"question":"Quelle est la probabilité qu'une variable aléatoire X suivant une loi normale N(10,4) soit supérieure à 12 ?","option_a":"0,1587","option_b":"0,3085","option_c":"0,6915","option_d":"0,8413","option_e":"","option_f":"","bonne_reponse":"a","explication":"On calcule P(X \u003E 12) = 1 - P(X ≤ 12). En standardisant, Z = (12-10)\/2 = 1, donc P(X ≤ 12) = P(Z ≤ 1) ≈ 0,8413. Ainsi, P(X \u003E 12) ≈ 1 - 0,8413 = 0,1587.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0,1587\", \"b\": \"0,3085\", \"c\": \"0,6915\", \"d\": \"0,8413\"}}","_debug_options_count":4},{"id":39089,"question":"Un test d'hypothèse permet de conclure avec certitude qu'une hypothèse est vraie ou fausse.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un test d'hypothèse ne permet pas de conclure avec certitude, mais d'accepter ou de rejeter une hypothèse avec un certain niveau de confiance.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]