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Tables statistiques Statistique pour ingénieur

Statistique pour ingénieur

Tables statistiques

F. Delacroix

M. Lecom te

, 23 février 2016

Introduction

Dans les pages qui suivent nous proposons quelques tables statistiques classiques. Selon les cas, il s"agira de valeurs de la fonction de répartition d"une loi de probabilité ou de la réciproque de cette fonction de répartition (qu"on appelle fractiles ou quantiles). Dans ce recueil de tables, on a généralement choisi de noterPles valeurs de la fonction de répartition pour les lois continues; on sera donc, dans l"optique de la construction d"intervalles de confiance ou des tests statistiques, à poser fréquemmentP= 1-αou

P= 1-α2

ou encoreP=α2 . Les fractiles correspondants sont généralement notés avec une lettre figurant la loi de probabilité et la valeurPen indice. Pour les lois discrètes, les fractiles sont notés généralementcdans ce recueil. Pour chaque loi, une explication sommaire de la lecture des tables est donnée, suivi des tables ou abaques elles-mêmes.

Enfin, la

section 9 donne p ourles lois év oquéesl"esp érance,la v arianceainsi q uedes formules permettant d"obtenir les valeurs (fonction de répartition ou fractiles) associées

à ces lois. Ces formules ont été testées sur les tableurs Microsoft Excel version 2007 et

LibreOffice Calc 5.0.4 (avec parfois des différences gênantes); et seront à valider dans le cas d"autres tableurs au vu de leurs documentations.Institut Mines-Télécom 1

Statistique pour ingénieur Tables statistiques

Table des matières

Introduction

1

1 Fonction de répartition de la loi normale centrée réduite

4

Table n

o1.1- Fonction de répartition de la loi normale centrée réduite. . . . . 5

Table n

o1.2- Grandes valeurs deΦ. . . . . . . . . . . . . . . . . . . . . . . . .5

2 Fractiles de la loi normale centrée réduite

6

Table n

o2.1- Fractiles de la loi normale centrée réduite. . . . . . . . . . . . . . 7

3 Fractiles de la loi de Student

8

3.1 Définition

8

3.2 Approximation

8

Table n

o3.1- Fractiles de la loi de Student. . . . . . . . . . . . . . . . . . . . . 9

4 Fractiles de la loi duχ210

4.1 Définition

10

4.2 Approximation

10

Table n

o4.1- Fractiles de la loi duχ2. . . . . . . . . . . . . . . . . . . . . . . .11

5 Fractiles de la loi de Fisher-Snedecor

12

Table n

o5.1- Fractiles de la loi de Fisher-Snedecor pourP= 0,95. . . . . . . .13

Table n

o5.2- Fractiles de la loi de Fisher-Snedecor pourP= 0,975. . . . . . .14

Table n

o5.3- Fractiles de la loi de Fisher-Snedecor pourP= 0,99. . . . . . . .15

Table n

o5.4- Fractiles de la loi de Fisher-Snedecor pourP= 0,995. . . . . . .16

6 Probabilités cumulées de la loi binomiale

18

6.1 Définitions

18

6.2 Approximations

18

Table n

o6.1- Probabilités cumulées de la loi binomialeB(n,p). . . . . . . . . .19

7 Intervalle de confiance pour une proportion

20

7.1 Principe

20

7.2 Utilisation

20

Abaque n

o7.1- Intervalle de confiance pour une proportion (1). . . . . . . . . 21

Abaque n

o7.2- Intervalle de confiance pour une proportion (2). . . . . . . . . 22

Abaque n

o7.3- Intervalle de confiance pour une proportion (3). . . . . . . . . 23

8 Probabilités cumulées de la loi de Poisson

24

8.1 Définition

24

8.2 Approximation

24

Table n

o8.1- Probabilités cumulées de la loi de PoissonP(λ)pourλ <10. . .25

Table n

o8.2- Probabilités cumulées de la loi de PoissonP(λ)pour106λ62026

9 Résumé de quelques lois

27 2 Institut Mines-Télécom

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Statistique pour ingénieur Tables statistiques

1 Fonction de répartition de la loi normale centrée

réduite La loi normale centrée réduiteN(0,1)a pour densité de probabilité la fonctionf définie par ?t?R, f(t) =1⎷2πe-t2/2. Une variable aléatoireUsuivant cette loi a pour fonction de répartition la fonctionΦ définie par ?u?R,Φ(u) =P(U6u) =? u -∞f(t)dt.Figure1 - Graphe de la densitéN(0,1) La table 1.1 suiv anteest celle des v aleursde ΦsurR+. Les valeurs deΦsurR-se calculent à l"aide de la propriété de symétrie ?u?R,Φ(-u) = 1-Φ(u). Cette table peut également servir à calculer les valeurs de la fonction de répartition d"une variable aléatoireXsuivant une loi normaleN(μ,σ2)à l"aide de la formule

P(X6x) = Φ?x-μσ

.4 Institut Mines-Télécom

Tables statistiques Statistique pour ingénieur

Table n

o1.1- Fonction de répartition de la loi normale centrée réduiteu0,000,010,020,030,040,050,060,070,080,090,00,50000,50400,50800,51200,51600,51990,52390,52790,53190,53590,10,53980,54380,54780,55170,55570,55960,56360,56750,57140,57530,20,57930,58320,58710,59100,59480,59870,60260,60640,61030,61410,30,61790,62170,62550,62930,63310,63680,64060,64430,64800,65170,40,65540,65910,66280,66640,67000,67360,67720,68080,68440,68790,50,69150,69500,69850,70190,70540,70880,71230,71570,71900,72240,60,72570,72910,73240,73570,73890,74220,74540,74860,75170,75490,70,75800,76110,76420,76730,77040,77340,77640,77940,78230,78520,80,78810,79100,79390,79670,79950,80230,80510,80780,81060,81330,90,81590,81860,82120,82380,82640,82890,83150,83400,83650,83891,00,84130,84380,84610,84850,85080,85310,85540,85770,85990,86211,10,86430,86650,86860,87080,87290,87490,87700,87900,88100,88301,20,88490,88690,88880,89070,89250,89440,89620,89800,89970,90151,30,90320,90490,90660,90820,90990,91150,91310,91470,91620,91771,40,91920,92070,92220,92360,92510,92650,92790,92920,93060,93191,50,93320,93450,93570,93700,93820,93940,94060,94180,94290,94411,60,94520,94630,94740,94840,94950,95050,95150,95250,95350,95451,70,95540,95640,95730,95820,95910,95990,96080,96160,96250,96331,80,96410,96490,96560,96640,96710,96780,96860,96930,96990,97061,90,97130,97190,97260,97320,97380,97440,97500,97560,97610,97672,00,97720,97780,97830,97880,97930,97980,98030,98080,98120,98172,10,98210,98260,98300,98340,98380,98420,98460,98500,98540,98572,20,98610,98640,98680,98710,98750,98780,98810,98840,98870,98902,30,98930,98960,98980,99010,99040,99060,99090,99110,99130,99162,40,99180,99200,99220,99250,99270,99290,99310,99320,99340,99362,50,99380,99400,99410,99430,99450,99460,99480,99490,99510,99522,60,99530,99550,99560,99570,99590,99600,99610,99620,99630,99642,70,99650,99660,99670,99680,99690,99700,99710,99720,99730,99742,80,99740,99750,99760,99770,99770,99780,99790,99790,99800,99812,90,99810,99820,99820,99830,99840,99840,99850,99850,99860,9986Table n

o1.2- Grandes valeurs deΦuΦ(u)uΦ(u)3,00,9986503,80,9999283,10,9990323,90,9999523,20,9993134,00,9999683,30,9995174,10,9999793,40,9996634,20,9999873,50,9997674,30,9999913,60,9998414,40,9999953,70,9998924,50,999997Institut Mines-Télécom 5

Statistique pour ingénieur Tables statistiques

2 Fractiles de la loi normale centrée réduite

La fonction de répartition deN(0,1)est une bijection croissante deRsur]0,1[et la table 2.1 donne les v aleursde Φ-1. LorsqueUest une variable aléatoire suivant la loiN(0,1)etP?]0,1[, cette table donne la valeur deuP= Φ-1(P), qui est telle que

P(U6uP) =P.

La lecture de la table diffère selon quePest inférieur ou supérieur à0,50: si P60,50, la valeur dePse lit en ajoutant des cellules de la colonne de gauche et la ligne supérieure. Le fractileuPestnégatif(cf.figure 2gauc he). Si P>0,50, la valeur dePse lit en ajoutant des cellules de la colonne de droite

et de la ligne inférieure. Le fractileuPestpositif(cf.figure 2droite). Figure2 - Lecture des fractiles deN(0,1)

Exemple-P ourP= 0,024,uP=-1,9774;-p ourP= 0,976,uP= +1,9774.6 Institut Mines-Télécom

Tables statistiques Statistique pour ingénieur

Table n

o2.1- Fractiles de la loi normale centrée réduiteP0,0000,0010,0020,0030,0040,0050,0060,0070,0080,0090,010+

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3 Fractiles de la loi de Student

3.1 Définition

Une variable aléatoireTsuit la loi de Student àνdegrés de liberté (oùν?N?) si elle

admet pour densité de probabilité la fonctionfdéfinie surRpar f(t) =1⎷νπ

Γ?ν+12

?ν2

1 +t2ν

-ν+12 .Figure3 - Densité de la loi de Student La table 3.1 donne les fractiles de la loi de Studen td"ordre P>60, c"est-à-dire les valeurs detPvérifiant

P(T6tP) =P.

Les fractiles d"ordreP60,4s"obtiennent par la relation de symétrie t

P=-t1-P.

3.2 Approximation

Pourν >100, la loi de Student peut être approchée par la loi normale centrée réduite

N(0,1).8 Institut Mines-Télécom

Tables statistiques Statistique pour ingénieur

Table n

o3.1- Fractiles de la loi de Student1-P→0,400,300,200,100,050,0250,010,0010,0005ν

↓P→0,600,700,800,900,950,9750,990,9990,999510,3250,7271,3763,0786,31412,70631,821318,309636,61920,2890,6171,0611,8862,9204,3036,96522,32731,59930,2770,5840,9781,6382,3533,1824,54110,21512,92440,2710,5690,9411,5332,1322,7763,7477,1738,61050,2670,5590,9201,4762,0152,5713,3655,8936,86960,2650,5530,9061,4401,9432,4473,1435,2085,95970,2630,5490,8961,4151,8952,3652,9984,7855,40880,2620,5460,8891,3971,8602,3062,8964,5015,04190,2610,5430,8831,3831,8332,2622,8214,2974,781100,2600,5420,8791,3721,8122,2282,7644,1444,587110,2600,5400,8761,3631,7962,2012,7184,0254,437120,2590,5390,8731,3561,7822,1792,6813,9304,318130,2590,5380,8701,3501,7712,1602,6503,8524,221140,2580,5370,8681,3451,7612,1452,6243,7874,140150,2580,5360,8661,3411,7532,1312,6023,7334,073160,2580,5350,8651,3371,7462,1202,5833,6864,015170,2570,5340,8631,3331,7402,1102,5673,6463,965180,2570,5340,8621,3301,7342,1012,5523,6103,922190,2570,5330,8611,3281,7292,0932,5393,5793,883200,2570,5330,8601,3251,7252,0862,5283,5523,850210,2570,5320,8591,3231,7212,0802,5183,5273,819220,2560,5320,8581,3211,7172,0742,5083,5053,792230,2560,5320,8581,3191,7142,0692,5003,4853,768240,2560,5310,8571,3181,7112,0642,4923,4673,745250,2560,5310,8561,3161,7082,0602,4853,4503,725260,2560,5310,8561,3151,7062,0562,4793,4353,707270,2560,5310,8551,3141,7032,0522,4733,4213,690280,2560,5300,8551,3131,7012,0482,4673,4083,674290,2560,5300,8541,3111,6992,0452,4623,3963,659300,2560,5300,8541,3101,6972,0422,4573,3853,646320,2550,5300,8531,3091,6942,0372,4493,3653,622340,2550,5290,8521,3071,6912,0322,4413,3483,601360,2550,5290,8521,3061,6882,0282,4343,3333,582380,2550,5290,8511,3041,6862,0242,4293,3193,566400,2550,5290,8511,3031,6842,0212,4233,3073,551500,2550,5280,8491,2991,6762,0092,4033,2613,496600,2540,5270,8481,2961,6712,0002,3903,2323,460700,2540,5270,8471,2941,6671,9942,3813,2113,435800,2540,5260,8461,2921,6641,9902,3743,1953,416900,2540,5260,8461,2911,6621,9872,3683,1833,4021000,2540,5260,8451,2901,6601,9842,3643,1743,3902000,2540,5250,8431,2861,6531,9722,3453,1313,3405000,2530,5250,8421,2831,6481,9652,3343,1073,310Institut Mines-Télécom 9

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4 Fractiles de la loi duχ2

4.1 Définition

Une variable aléatoireZsuit la loi duχ2(ou Loi de Pearson) àνdegrés de libertés (oùν?N?) si elle admet pour densité de probabilité la fonctionfdéfinie surRpar f(t) =? ??12 ν2

Γ?ν2

e-t2 tν2 -1sit>0

0sinon.

C"est un cas particulier de loiΓ, celle de paramètres?12 ,ν2 ?.Figure4 - Densité de probabilité de la loi duχ2 SiU1,...,Unsontnvariables aléatoires indépendantes suivant toutes la loiN(0,1), alors la variable aléatoire Z=n i=1U2i suit la loi duχ2àndegrés de liberté. La table 4.1 donne, p our16ν630et certaines valeurs deP, les fractiles de la loi duχ2, c"est-à-dire les valeurs deχ2Ptelles que P ?Z6χ2P?=P.

4.2 Approximation

Pourν >30, on peut admettre que la variable aléatoire⎷2Z-⎷2ν-1suit approxi- mativement la loi normale centrée réduiteN(0,1).10 Institut Mines-Télécom

Tables statistiques Statistique pour ingénieur

Table n

o4.1- Fractiles de la loi duχ21-P→0,9990,9950,9750,950,900,500,100,050,0250,010,0050,001ν

↓P→0,0010,0050,0250,050,100,500,900,950,9750,990,9950,99910,000,000,000,000,020,452,713,845,026,637,8810,8320,000,010,050,100,211,394,615,997,389,2110,6013,8230,020,070,220,350,582,376,257,819,3511,3412,8416,2740,090,210,480,711,063,367,789,4911,1413,2814,8618,4750,210,410,831,151,614,359,2411,0712,8315,0916,7520,5260,380,681,241,642,205,3510,6412,5914,4516,8118,5522,4670,600,991,692,172,836,3512,0214,0716,0118,4820,2824,3280,861,342,182,733,497,3413,3615,5117,5320,0921,9526,1291,151,732,703,334,178,3414,6816,9219,0221,6723,5927,88101,482,163,253,944,879,3415,9918,3120,4823,2125,1929,59111,832,603,824,575,5810,3417,2819,6821,9224,7226,7631,26122,213,074,405,236,3011,3418,5521,0323,3426,2228,3032,91132,623,575,015,897,0412,3419,8122,3624,7427,6929,8234,53143,044,075,636,577,7913,3421,0623,6826,1229,1431,3236,12153,484,606,267,268,5514,3422,3125,0027,4930,5832,8037,70163,945,146,917,969,3115,3423,5426,3028,8532,0034,2739,25174,425,707,568,6710,0916,3424,7727,5930,1933,4135,7240,79184,906,268,239,3910,8617,3425,9928,8731,5334,8137,1642,31195,416,848,9110,1211,6518,3427,2030,1432,8536,1938,5843,82205,927,439,5910,8512,4419,3428,4131,4134,1737,5740,0045,31216,458,0310,2811,5913,2420,3429,6232,6735,4838,9341,4046,80226,988,6410,9812,3414,0421,3430,8133,9236,7840,2942,8048,27237,539,2611,6913,0914,8522,3432,0135,1738,0841,6444,1849,73248,089,8912,4013,8515,6623,3433,2036,4239,3642,9845,5651,18258,6510,5213,1214,6116,4724,3434,3837,6540,6544,3146,9352,62269,2211,1613,8415,3817,2925,3435,5638,8941,9245,6448,2954,05279,8011,8114,5716,1518,1126,3436,7440,1143,1946,9649,6455,482810,3912,4615,3116,9318,9427,3437,9241,3444,4648,2850,9956,892910,9913,1216,0517,7119,7728,3439,0942,5645,7249,5952,3458,303011,5913,7916,7918,4920,6029,3440,2643,7746,9850,8953,6759,70Institut Mines-Télécom 11

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5 Fractiles de la loi de Fisher-Snedecor

Une variable aléatoireFsuit la loi de Fisher-Snedecor(ν1,ν2)degrés de liberté si elle admet pour densité de probabilité la fonctionfdéfinie surRpar f(t) =? ?ν1+ν22 ?ν12 ?Γ?ν22 12

1νν

22

2tν

12 -1(ν1t+ν2)ν

1+ν22

sit>0

0sinon.Figure5 - Densité de probabilité de la loi de Fisher-Snedecor

SiZ1etZ2sont deux variables aléatoires indépendantes suivant les lois duχ2respec- tivement àν1etν2degrés de liberté, alors la variable aléatoire

F=Z1/ν1Z

2/ν2

suit la loi de Fisher-Snedecor à(ν1,ν2)degrés de liberté.

Les tables

5.1 5.2 5.3 et 5.4 donnen t,p ourquelques v aleursde Pet en fonction de

1etν2les fractiles de la loi de Fisher-Snedecor à(ν1,ν2)degrés de liberté, c"est-à-dire

les valeurs defP(ν1,ν2)telles que

P(F6fP(ν1,ν2)) =P.

N"y figurent que les fractiles supérieurs à1; pour ceux inférieurs à1on pourra utiliser la

relation f

P(ν1,ν2) =1f

1-P(ν2,ν1).12 Institut Mines-Télécom

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Table n

o5.1- Fractiles de la loi de Fisher-Snedecor pourP= 0,95ν 2ν

1123456789101214161820304060100500

1161,4199,5215,7224,6230,2234,0236,8238,9240,5241,9243,9245,4246,5247,3248,0250,1251,1252,2253,0254,1218,5119,0019,1619,2519,3019,3319,3519,3719,3819,4019,4119,4219,4319,4419,4519,4619,4719,4819,4919,49310,139,559,289,129,018,948,898,858,818,798,748,718,698,678,668,628,598,578,558,5347,716,946,596,396,266,166,096,046,005,965,915,875,845,825,805,755,725,695,665,6456,615,795,415,195,054,954,884,824,774,744,684,644,604,584,564,504,464,434,414,3765,995,144,764,534,394,284,214,154,104,064,003,963,923,903,873,813,773,743,713,6875,594,744,354,123,973,873,793,733,683,643,573,533,493,473,443,383,343,303,273,2485,324,464,073,843,693,583,503,443,393,353,283,243,203,173,153,083,043,012,972,9495,124,263,863,633,483,373,293,233,183,143,073,032,992,962,942,862,832,792,762,72104,964,103,713,483,333,223,143,073,022,982,912,862,832,802,772,702,662,622,592,55114,843,983,593,363,203,093,012,952,902,852,792,742,702,672,652,572,532,492,462,42124,753,893,493,263,113,002,912,852,802,752,692,642,602,572,542,472,432,382,352,31134,673,813,413,183,032,922,832,772,712,672,602,552,512,482,462,382,342,302,262,22144,603,743,343,112,962,852,762,702,652,602,532,482,442,412,392,312,272,222,192,14154,543,683,293,062,902,792,712,642,592,542,482,422,382,352,332,252,202,162,122,08164,493,633,243,012,852,742,662,592,542,492,422,372,332,302,282,192,152,112,072,02174,453,593,202,962,812,702,612,552,492,452,382,332,292,262,232,152,102,062,021,97184,413,553,162,932,772,662,582,512,462,412,342,292,252,222,192,112,062,021,981,93194,383,523,132,902,742,632,542,482,422,382,312,262,212,182,162,072,031,981,941,89204,353,493,102,872,712,602,512,452,392,352,282,222,182,152,122,041,991,951,911,86214,323,473,072,842,682,572,492,422,372,322,252,202,162,122,102,011,961,921,881,83224,303,443,052,822,662,552,462,402,342,302,232,172,132,102,071,981,941,891,851,80234,283,423,032,802,642,532,442,372,322,272,202,152,112,082,051,961,911,861,821,77244,263,403,012,782,622,512,422,362,302,252,182,132,092,052,031,941,891,841,801,75254,243,392,992,762,602,492,402,342,282,242,162,112,072,042,011,921,871,821,781,73264,233,372,982,742,592,472,392,322,272,222,152,092,052,021,991,901,851,801,761,71284,203,342,952,712,562,452,362,292,242,192,122,062,021,991,961,871,821,771,731,67304,173,322,922,692,532,422,332,272,212,162,092,041,991,961,931,841,791,741,701,64504,033,182,792,562,402,292,202,132,072,031,951,891,851,811,781,691,631,581,521,461003,943,092,702,462,312,192,102,031,971,931,851,791,751,711,681,571,521,451,391,31Institut Mines-Télécom 13

Statistique pour ingénieur Tables statistiques

Table n

o5.2- Fractiles de la loi de Fisher-Snedecor pourP= 0,975ν 2ν

1123456789101214161820304060100500

1647,8799,5864,2899,6921,8937,1948,2956,7963,3968,6976,7982,5986,9990,3993,11001,41005,61009,81013,21017,2238,5139,0039,1739,2539,3039,3339,3639,3739,3939,4039,4139,4339,4439,4439,4539,4639,4739,4839,4939,50317,4416,0415,4415,1014,8814,7314,6214,5414,4714,4214,3414,2814,2314,2014,1714,0814,0413,9913,9613,91412,2210,659,989,609,369,209,078,988,908,848,758,688,638,598,568,468,418,368,328,27510,018,437,767,397,156,986,856,766,686,626,526,466,406,366,336,236,186,126,086,0368,817,266,606,235,995,825,705,605,525,465,375,305,245,205,175,075,014,964,924,8678,076,545,895,525,295,124,994,904,824,764,674,604,544,504,474,364,314,254,214,1687,576,065,425,054,824,654,534,434,364,304,204,134,084,034,003,893,843,783,743,6897,215,715,084,724,484,324,204,104,033,963,873,803,743,703,673,563,513,453,403,35106,945,464,834,474,244,073,953,853,783,723,623,553,503,453,423,313,263,203,153,09116,725,264,634,284,043,883,763,663,593,533,433,363,303,263,233,123,063,002,962,90126,555,104,474,123,893,733,613,513,443,373,283,213,153,113,072,962,912,852,802,74136,414,974,354,003,773,603,483,393,313,253,153,083,032,982,952,842,782,722,672,61146,304,864,243,893,663,503,383,293,213,153,052,982,922,882,842,732,672,612,562,50156,204,774,153,803,583,413,293,203,123,062,962,892,842,792,762,642,592,522,472,41166,124,694,083,733,503,343,223,123,052,992,892,822,762,722,682,572,512,452,402,33176,044,624,013,663,443,283,163,062,982,922,822,752,702,652,622,502,442,382,332,26185,984,563,953,613,383,223,103,012,932,872,772,702,642,602,562,442,382,322,272,20195,924,513,903,563,333,173,052,962,882,822,722,652,592,552,512,392,332,272,222,15205,874,463,863,513,293,133,012,912,842,772,682,602,552,502,462,352,292,222,172,10215,834,423,823,483,253,092,972,872,802,732,642,562,512,462,422,312,252,182,132,06225,794,383,783,443,223,052,932,842,762,702,602,532,472,432,392,272,212,142,092,02235,754,353,753,413,183,022,902,812,732,672,572,502,442,392,362,242,182,112,061,99245,724,323,723,383,152,992,872,782,702,642,542,472,412,362,332,212,152,082,021,95255,694,293,693,353,132,972,852,752,682,612,512,442,382,342,302,182,122,052,001,92265,664,273,673,333,102,942,822,732,652,592,492,422,362,312,282,162,092,031,971,90285,614,223,633,293,062,902,782,692,612,552,452,372,322,272,232,112,051,981,921,85305,574,183,593,253,032,872,752,652,572,512,412,342,282,232,202,072,011,941,881,81505,343,973,393,052,832,672,552,462,382,322,222,142,082,031,991,871,801,721,661,571005,183,833,252,922,702,542,422,322,242,182,082,001,941,891,851,711,641,561,481,3814 Institut Mines-Télécom

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Table n

o5.3- Fractiles de la loi de Fisher-Snedecor pourP= 0,99ν 2ν

1123456789101214161820304060100500

298,599,099,299,299,399,399,499,499,499,499,499,499,499,499,499,599,599,599,599,5334,1230,8229,4628,7128,2427,9127,6727,4927,3527,2327,0526,9226,8326,7526,6926,5026,4126,3226,2426,15421,2018,0016,6915,9815,5215,2114,9814,8014,6614,5514,3714,2514,1514,0814,0213,8413,7513,6513,5813,49516,2613,2712,0611,3910,9710,6710,4610,2910,1610,059,899,779,689,619,559,389,299,209,139,04613,7510,929,789,158,758,478,268,107,987,877,727,607,527,457,407,237,147,066,996,90712,259,558,457,857,467,196,996,846,726,626,476,366,286,216,165,995,915,825,755,67811,268,657,597,016,636,376,186,035,915,815,675,565,485,415,365,205,125,034,964,88910,568,026,996,426,065,805,615,475,355,265,115,014,924,864,814,654,574,484,414,331010,047,566,555,995,645,395,205,064,944,854,714,604,524,464,414,254,174,084,013,93119,657,216,225,675,325,074,894,744,634,544,404,294,214,154,103,943,863,783,713,62129,336,935,955,415,064,824,644,504,394,304,164,053,973,913,863,703,623,543,473,38139,076,705,745,214,864,624,444,304,194,103,963,863,783,723,663,513,433,343,273,19148,866,515,565,044,694,464,284,144,033,943,803,703,623,563,513,353,273,183,113,03158,686,365,424,894,564,324,144,003,893,803,673,563,493,423,373,213,133,052,982,89168,536,235,294,774,444,204,033,893,783,693,553,453,373,313,263,103,022,932,862,78178,406,115,184,674,344,103,933,793,683,593,463,353,273,213,163,002,922,832,762,68188,296,015,094,584,254,013,843,713,603,513,373,273,193,133,082,922,842,752,682,59198,185,935,014,504,173,943,773,633,523,433,303,193,123,053,002,842,762,672,602,51208,105,854,944,434,103,873,703,563,463,373,233,133,052,992,942,782,692,612,542,44218,025,784,874,374,043,813,643,513,403,313,173,072,992,932,882,722,642,552,482,38227,955,724,824,313,993,763,593,453,353,263,123,022,942,882,832,672,582,502,422,33237,885,664,764,263,943,713,543,413,303,213,072,972,892,832,782,622,542,452,372,28247,825,614,724,223,903,673,503,363,263,173,032,932,852,792,742,582,492,402,332,24257,775,574,684,183,853,633,463,323,223,132,992,892,812,752,702,542,452,362,292,19267,725,534,644,143,823,593,423,293,183,092,962,862,782,722,662,502,422,332,252,16287,645,454,574,073,753,533,363,233,123,032,902,792,722,652,602,442,352,262,192,09307,565,394,514,023,703,473,303,173,072,982,842,742,662,602,552,392,302,212,132,03507,175,064,203,723,413,193,022,892,782,702,562,462,382,322,272,102,011,911,821,711006,904,823,983,513,212,992,822,692,592,502,372,272,192,122,071,891,801,691,601,47Institut Mines-Télécom 15

Statistique pour ingénieur Tables statistiques

Table n

o5.4- Fractiles de la loi de Fisher-Snedecor pourP= 0,995ν 2ν

1123456789101214161820304060100500

2198,5199,0199,2199,2199,3199,3199,4199,4199,4199,4199,4199,4199,4199,4199,4199,5199,5199,5199,5199,5355,5549,8047,4746,1945,3944,8444,4344,1343,8843,6943,3943,1743,0142,8842,7842,4742,3142,1542,0241,87431,3326,2824,2623,1522,4621,9721,6221,3521,1420,9720,7020,5120,3720,2620,1719,8919,7519,6119,5019,36522,7818,3116,5315,5614,9414,5114,2013,9613,7713,6213,3813,2113,0912,9812,9012,6612,5312,4012,3012,17618,6314,5412,9212,0311,4611,0710,7910,5710,3910,2510,039,889,769,669,599,369,249,129,038,91716,2412,4010,8810,059,529,168,898,688,518,388,188,037,917,837,757,537,427,317,227,10814,6911,049,608,818,307,957,697,507,347,217,016,876,766,686,616,406,296,186,095,98913,6110,118,727,967,477,136,886,696,546,426,236,095,985,905,835,625,525,415,325,211012,839,438,087,346,876,546,306,125,975,855,665,535,425,345,275,074,974,864,774,671112,238,917,606,886,426,105,865,685,545,425,245,105,004,924,864,654,554,454,364,251211,758,517,236,526,075,765,525,355,205,094,914,774,674,594,534,334,234,124,043,931311,378,196,936,235,795,485,255,084,944,824,644,514,414,334,274,073,973,873,783,671411,067,926,686,005,565,265,034,864,724,604,434,304,204,124,063,863,763,663,573,461510,807,706,485,805,375,074,854,674,544,424,254,124,023,953,883,693,583,483,393,291610,587,516,305,645,214,914,694,524,384,274,103,973,873,803,733,543,443,333,253,141710,387,356,165,505,074,784,564,394,254,143,973,843,753,673,613,413,313,213,123,011810,227,216,035,374,964,664,444,284,144,033,863,733,643,563,503,303,203,103,012,901910,077,095,925,274,854,564,344,184,043,933,763,643,543,463,403,213,113,002,912,80209,946,995,825,174,764,474,264,093,963,853,683,553,463,383,323,123,022,922,832,72219,836,895,735,094,684,394,184,013,883,773,603,483,383,313,243,052,952,842,752,64229,736,815,655,024,614,324,113,943,813,703,543,413,313,243,182,982,882,772,692,57239,636,735,584,954,544,264,053,883,753,643,473,353,253,183,122,922,822,712,622,51249,556,665,524,894,494,203,993,833,693,593,423,303,203,123,062,872,772,662,572,46259,486,605,464,844,434,153,943,783,643,543,373,253,153,083,012,822,722,612,522,41269,416,545,414,794,384,103,893,733,603,493,333,203,113,032,972,772,672,562,472,36289,286,445,324,704,304,023,813,653,523,413,253,123,032,952,892,692,592,482,392,28309,186,355,244,624,233,953,743,583,453,343,183,062,962,892,822,632,522,422,322,21508,635,904,834,233,853,583,383,223,092,992,822,702,612,532,472,272,162,051,951,821008,245,594,543,963,593,333,132,972,852,742,582,462,372,292,232,021,911,791,681,5316 Institut Mines-Télécom

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Institut Mines-Télécom 17

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6 Probabilités cumulées de la loi binomiale

6.1 Définitions

Une variable aléatoireKsuit la loi binomialeB(n,p)(oùn?N?etp?]0,1[) si ses valeurs possibles sont les entiers entre0etnet si ?k? {0,...,n}P(K=k) =Cknpk(1-p)n-k. La table 6.1 donne les probabilités cum uléesde la loi B(n,p), c"est-à-dire, pour tout c? {0,...,n}, la probabilité

P(K6c) =c

k=0Cknpk(1-p)n-k.

6.2 Approximations

Théorème (Moivre-Laplace)Sinp(1-p)>18on peut approcher la loi binomialeB(n,p)par la loi normale demêmes espéranceμ=npet varianceσ2=np(1-p).Sip <0,10, on peu approcher la loiB(n,p)par la loi de Poisson de paramètreλ=np

avec une erreur pratiquement négligeable sinest grand (n >30).18 Institut Mines-Télécom

Tables statistiques Statistique pour ingénieur

Table n

o6.1- Probabilités cumulées de la loi binomialeB(n,p)n ↓c ↓p→1%2%3%4%5%6%7%8%9%10%

500,95100,90390,85870,81540,77380,73390,69570,65910,62400,590510,99900,99620,99150,98520,97740,96810,95750,94560,93260,918521,00000,99990,99970,99940,99880,99800,99690,99550,99370,991431,00001,00001,00001,00000,99990,99990,99980,99970,999541,00001,00001,00001,00001,00001000,90440,81710,73740,66480,59870,53860,48400,43440,38940,348710,99570,98380,96550,94180,91390,88240,84830,81210,77460,736120,99990,99910,99720,99380,98850,98120,97170,95990,94600,929831,00001,00000,99990,99960,99900,99800,99640,99420,99120,987241,00001,00000,99990,99980,99970,99940,99900,998451,00001,00001,00001,00000,99990,999961,00001,00001500,86010,73860,63330,54210,46330,39530,33670,28630,24300,205910,99040,96470,92700,88090,82900,77380,71680,65970,60350,549020,99960,99700,99060,97970,96380,94290,91710,88700,85310,815931,00000,99980,99920,99760,99450,98960,98250,97270,96010,944441,00000,99990,99980,99940,99860,99720,99500,99180,987351,00001,00000,99990,99990,99970,99930,99870,997861,00001,00001,00000,99990,99980,999771,00001,00001,00002000,81790,66760,54380,44200,35850,29010,23420,18870,15160,121610,98310,94010,88020,81030,73580,66050,58690,51690,45160,391720,99900,99290,97900,95610,92450,88500,83900,78790,73340,676931,00000,99940,99730,99260,98410,97100,95290,92940,90070,867041,00000,99970,99900,99740,99440,98930,98170,97100,956851,00000,99990,99970,99910,99810,99620,99320,988761,00001,00000,99990,99970,99940,99870,997671,00001,00000,99990,99980,999681,00001,00000,999991,00005000,60500,36420,21810,12990,07690,04530,02660,01550,00900,005210,91060,73580,55530,40050,27940,19000,12650,08270,05320,033820,98620,92160,81080,67670,54050,41620,31080,22600,16050,111730,99840,98220,93720,86090,76040,64730,53270,42530,33030,250340,99990,99680,98320,95100,89640,82060,72900,62900,52770,431251,00000,99950,99630,98560,96220,92240,86500,79190,70720,616160,99990,99930,99640,98820,97110,94170,89810,84040,770271,00000,99990,99920,99680,99060,97800,95620,92320,877981,00000,99990,99920,99730,99270,98330,96720,942191,00000,99980,99930,99780,99440,98750,9755101,00000,99980,99940,99830,99570,9906111,00000,99990,99950,99870,9968121,00000,99990,99960,9990131,00000,99990,9997141,00000,9999151,0000Institut Mines-Télécom 19

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7 Intervalle de confiance pour une proportion

Les abaques des pages suivantes permettent de déterminer un intervalle de confiance pour une proportion à partir d"une lecture graphique, à partir de la fréquencef=kn

observée de la caractéristique étudiée dans un échantillon de taillen, et ce pour différents

niveaux de confiance.

7.1 Principe

Ces abaques sont établies à partir de la loi binomialeB(n,p)suivie par la variable aléatoireKdont la valeur estk. Pour un niveau de confiance1-α, les bornes inférieurep1et supérieurep2de l"intervalle de confiance pour la proportion inconnuepsont établies à partir des équations suivantes :

P(K>k) =n?

j=kCjnpj

1(1-p1)n-j=?

?α/2pour un intervalle bilatéral

αpour un intervalle unilatéral à droite

avec par conventionp1= 0sik= 0, et

P(K6k) =k

j=0Cjnpj

2(1-p2)n-j=?

?α/2pour un intervalle bilatéral

αpour un intervalle unilatéral à gauche

avec également la conventionp2= 1lorsquek=n.

7.2 Utilisation

Pour déterminer un intervalle de confiance pour une proportionpà partir d"une lecture graphique des abaques des pages suivantes : c hoisirl"abaque corresp ondantau niv eaude co nfiance1-αvoulu et au type d"intervalle souhaité (bilatéral ou unilatéral); iden tifierla courb e(p ourun in tervalleunilatéral) ou les courb es(p ourun in tervalle bilatéral) correspondant à la taillende l"échantillon; re porteren abscisse la fréquence f=kn observée sur l"échantillon; lire la b orneinférieure p1sur la courbe du bas, la borne supérieurep2sur la courbe du haut; for merl"in tervallede confiance v oulu: Ic

1-α(p) =?

??[p1,p2]pour un intervalle bilatéral, [p1,1]pour un intervalle unilatéral à droite, [0,p2]pour un intervalle unilatéral à gauche.20 Institut Mines-Télécom

Tables statistiques Statistique pour ingénieur

Abaque n

o7.1- Intervalle de confiance pour une proportion (1) Intervalle bilatéral : niveau de confiance1-α= 0,90 Intervalles unilatéraux : niveau de confiance1-α= 0,9555 66
88

101012

1215
1520
2030
3050
50100

100250

250Proportion p dans la populationFréquence f observée dans l'échantillonInstitut Mines-Télécom 21

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