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USACH Facultad de Ciencia Dpto. Matemática y CC. Área de Estadísticas

ALGUNOS MODELOS DE VARIABLES ALEATORIAS DISCRETAS MODELO (Parámetros) (*2) BERNOULLI (𝑝)

FUNCIÓN DE CUANTÍA

ESPERANZA 𝐸[𝑥]

VARIANZA 𝑉[𝑥]

𝑝

𝑝𝑞

𝑛𝑝

𝑛𝑝𝑞

𝑥 = 1,2, …

1 𝑝

𝑞 𝑝2

𝑁𝑝 𝑁𝑞 )( ) 𝑥 𝑛 −𝑥 𝑝(𝑥) = 𝑁 ( ) 𝑛

𝑛𝑝

𝑝(𝑥) = 𝑝𝑥 𝑞1−𝑥 𝑥 = 0,1

BINOMIAL (𝑛, 𝑝) 𝑛 ∈ 𝑍+

𝑛 𝑝(𝑥) = ( ) 𝑝𝑥 𝑞𝑛−𝑥 𝑥 𝑥 = 0, … , 𝑛

GEOMÉTRICA (𝑝)

𝑝(𝑥) = 𝑝 𝑞 𝑥−1

ASIMETRÍA 𝛼3 [𝑥]

CURTOSIS 𝛼4 [𝑥]

1 − 2𝑝 √𝑝𝑞 1 − 2𝑝 √𝑛𝑝𝑞

F.GEN.M. 𝑀𝑋 [𝑡]

3+

[1 − 6𝑝𝑞] 𝑝𝑞

(𝑝𝑒 𝑡 + 𝑞)

3+

[1 − 6𝑝𝑞] 𝑛𝑝𝑞

(𝑝𝑒 𝑡 + 𝑞)𝑛

𝑝2 𝑞

𝑝 𝑒 −𝑡 − 𝑞

2−𝑝 9+

√𝑞

(

HIPERGEOMETRICA (𝑁, 𝑛, 𝑝) 𝑁, 𝑛 ∈ 𝑍 +

𝑛𝑝𝑞

𝑚𝑎𝑥{0, 𝑛 − 𝑁𝑞} ≤ 𝑥 ≤ min {𝑛, 𝑁𝑝}

𝑁−𝑛 𝑁−1

𝐶𝑎𝑛𝑎𝑣𝑜𝑠 𝑝𝑎𝑔. 115

𝑥 𝑒𝑛𝑡𝑒𝑟𝑜

PASCAL (𝑟, 𝑝) 𝑟 ∈ 𝑍+ POISSON () >0

(*2)

(*1)

𝑥 − 1 𝑟 𝑥−𝑟 𝑝(𝑥) = ( )𝑝 𝑞 𝑟−1 𝑥 = 𝑟, 𝑟 + 1, 𝑟 + 2, … 𝑒 −  𝑥 𝑥! 𝑥 = 0,1,2, …

𝑝(𝑥) =

𝑟 𝑝

𝑟𝑞 𝑝2

2−𝑝





√

√𝑟𝑞

3+

𝑝2 − 6𝑝 + 6 𝑟𝑞

3+

1



(

𝑟 𝑝 ) 𝑒 −𝑡 − 𝑞

𝑒 − (𝑒

𝑡 −1)

En todos los modelos con parámetro “𝑝”, se verifica 0 < 𝑝 < 1; 𝑞 = 1 − 𝑝

𝑉[𝑥] = 𝛼𝑥2

;

𝛼3 [𝑥] =

𝐸[(𝑥 − 𝐸[𝑥])3 ] 𝜎𝑥3

;

𝛼4 [𝑥] =

𝐸[(𝑥 − 𝐸[𝑥])3 ] 𝜎𝑥3

;

𝑀𝑋 [𝑡] = 𝐸[𝑒 𝑡𝑋 ]

Alejandro González Tapia - Ing. Civil en Minas

ALGUNOS MODELOS DE VARIABLES ALEATORIAS CONTINUAS (*1) MODELO (Parámetros) (*2)

FUNCIÓN DE DENSIDAD 𝑓(𝑥) = 𝑥 𝛼−1 (1 − 𝑥)𝛽−1

BETA (𝛼, 𝛽) 𝛼 > 0, 𝛽 > 0

0≤𝑥≤1 𝑟

CHI-CUADRADO (𝑟) 𝑟>0

𝑓(𝑥) = 𝑒 − 𝑥

1

1

𝑥>0



2

𝑠 𝑠−2 𝑠>2

2𝑠 2 (𝑠 + 𝑟 − 2) 𝑟(𝑠 − 2)2 (𝑠 − 4)

𝑟

𝑟 𝑟 𝑓(𝑥) = 𝐾𝐹 𝑥 2−1 (1 + 𝑥) 𝑠 𝑥>0

𝑓(𝑥) =

(*3)

(*5)

(𝑥) 𝑒 (𝑟) 𝛤 1

√2𝜋 𝜎

2𝑟

√𝑟

𝑟+𝑠 − 2

𝑡
0

𝐵𝑒𝑡𝑎: 𝐵(𝛼, 𝛽) =

;

𝛤(𝛼)𝛤(𝛽) 𝛤(𝛼 + 𝛽)

𝑟+𝑠 𝛤 ( 2 ) 𝑟 𝑟2 𝑟 𝑠 ( ) 𝛤( ) 𝛤( ) 𝑠 2 2

𝑆𝑡𝑢𝑑𝑒𝑛𝑡: 𝐾𝑡 =

𝛤[(𝑟 + 1)/2] √𝑟𝜋 𝛤(𝑟/2)

𝑉𝐿

−(𝛽𝑥)𝛼

0

𝑟 𝑟−2 𝑟>2

𝛼+𝛽 2

(𝛽 − 𝛼)2 12

Γ(1 + 𝛼 −1 )

𝑡 −1 (1 − )



𝑡0

𝑉[𝑥] = 𝛼𝑥2

2)

𝑟+1 2

𝛼≤𝑥≤𝛽

𝐾𝐹 =

𝑟

(1 − 2𝑡)−2

12 3+ 𝑟

22

−∞ < 𝑥 < ∞

UNIFORME (𝛼, 𝛽) 𝛼 0, 𝛽 > 0

CURTOSIS 𝛼4 [𝑥]

−𝑥

𝑥2 𝑓(𝑥) = 𝐾𝑡 (1 + ) 𝑟

(*5) T-STUDENT (𝑟) 𝑟>0

(*2)

𝛼𝛽 (𝛼 + 𝛽)2 (𝛼 + 𝛽 + 1)

𝑥>0

(*4) LOGNORMAL (𝜇, 𝜎 2 ) −∞ < 𝜇 < ∞; 𝜎 > 0

(*6)

𝛼 𝛼+𝛽

𝑟

𝑓(𝑥) =

NORMAL (𝜇, 𝜎 2 ) −∞ < 𝜇 < ∞; 𝜎 > 0

ASIMETRÍA 𝛼3 [𝑥]

1

𝑟−1

GAMMA (𝑟, ) 𝑟 > 0,  > 0

VARIANZA 𝑉[𝑥]

𝑥 2−1 𝑒 −2𝑥 𝑓(𝑥) = 𝑟 𝑟 22 Γ ( ) 2 𝑥>0

EXPONENCIAL () >0 (*3) F-SNEDECOR (𝑟, 𝑠) 𝑟 > 0, 𝑠 > 0

ESPERANZA 𝐸[𝑥]

𝛼4 [𝑥] =

;

𝛤(𝛼) = ∫ 𝑒−𝑥 𝑥𝛼−1 𝑑𝑥 0

(*4)

(*6)

;

𝑡≠0 (*6)

𝐸[(𝑥 − 𝐸[𝑥])3 ] 𝜎𝑥3

𝑀𝑋 [𝑡] = 𝐸[𝑒 𝑡𝑋 ]

;

∞

;

𝑒 𝑡𝛽 − 𝑒 𝑡𝛼 𝑡(𝛽 − 𝛼)

𝑟−1

𝛤(𝛼) = (𝛼 − 1)! 𝛼 𝑒𝑛𝑡𝑒𝑟𝑜 > 0

;

𝐸[𝑋 𝑟 ] = ∏(𝛼 + 𝑖)/(𝛼 + 𝛽 + 𝑖) 𝑖=0

2

2

𝐿𝑜𝑔𝑛𝑜𝑟𝑚𝑎: 𝑉𝐿 = [𝑒 (2𝜇+𝜎 ) ][𝑒 𝜎 − 1]

𝑊𝑒𝑖𝑏𝑢𝑙𝑙: 𝑉𝑊 =

𝛤(1+2𝛼−1 )−𝛤2 (1+𝛼−1 ) 𝛽2

𝐸𝐿 [𝑋 𝑟 ] = 𝑒 𝜇𝑟 𝑒 𝑟

;

2 𝜎 2/2

𝑟

;

𝐸[𝑋 𝑟 ] =

𝛤(1+𝛼) 𝛽𝑟

Alejandro González Tapia - Ing. Civil en Minas

INTERVALOS CONFIDENCIALES MÁS USUALES Intervalo Confidencial del (1 − 𝛼) 100%

Situación

Parámetro

Estimador

X v.a. normal, una m.a.

𝜎2

𝑠2

((𝑛 − 1)𝑠 2 ⁄𝜒2 (𝑛−1;1−𝛼) ,

𝜇

𝑥̅

(𝑥̅ − 𝑧1−𝛼

𝜇

𝑥̅

(𝑥̅ − 𝑡(𝑛−1;1−𝛼)

𝑝

𝑝̂

𝜎12 𝜎22

𝑠12 𝑠22

𝜇1 − 𝜇2

𝑥̅1 − 𝑥̅2

𝜇1 − 𝜇2

𝑥̅1 − 𝑥̅2

X v.a. normal, una m.a. 𝜎 2 conocido

2

𝜎 √𝑛

X v.a. normal, una m.a. 𝜎 2 desconocido

2

X v.a. Bernoulli (𝑝) m.a. de tamaño n

X v.a. normal dos m.a. independientes

(𝑛 − 1)𝑠 2⁄𝜒2 (𝑛−1;𝛼) )

2

𝑠 √𝑛

𝜎

,

𝑥̅ + 𝑧1−𝛼

,

𝑥̅ + 𝑡(𝑛−1;1−𝛼)

2

√𝑛

2

𝑝̂ 𝑞̂ (𝑝̂ − 𝑧1−𝛼 √ , 𝑛 2

(

2

) 𝑠 √𝑛

)

𝑝̂ 𝑞̂ 𝑝̂ + 𝑧1−𝛼 √ ) 𝑛 2

𝑠12 𝐹 𝛼 , 𝑠22 (𝑛2−1; 𝑛1−1;2 )

𝑠12 𝐹 𝛼 ) 𝑠22 (𝑛2−1; 𝑛1−1;1− 2 )

X v.a. normal dos m.a. independientes

1 1 𝑥̅1 − 𝑥̅2 ± 𝑧1−𝛼 𝜎 √ + 𝑛1 𝑛2 2

𝜎12 = 𝜎22 = 𝜎 2 conocida X v.a. normal dos m.a. independientes

𝑥̅1 − 𝑥̅2 ± 𝑡(𝑛

𝜎12 = 𝜎22 = 𝜎 2 desconocida

𝛼

𝑝 ;1− 2 )

Con 𝑛𝑝 = 𝑛1 + 𝑛2 − 2; X v.a. normal dos m.a. independientes

𝜇1 − 𝜇2

𝑥̅1 − 𝑥̅2

𝑠𝑝 = √

(𝑛1−1)𝑠12+(𝑛2−1)𝑠22 𝑛1+𝑛2−2

𝑠12 𝑠22 + 𝑛1 𝑛2 g.l. (ver reverso)

𝑥̅1 − 𝑥̅2 ± 𝑡(

𝜎12 ≠ 𝜎22 = 𝜎 2 desconocida

1 1 𝑠𝑝 √ + 𝑛1 𝑛2

Con

𝛼 ;1− ) 2

√

X v.a. normal dos m.a. pareadas

(𝑑̅ − 𝑡(𝑛−1;1−𝛼)

𝑠𝑑

𝑑̅ − 𝑡(𝑛−1;𝛼)

𝑠𝑑

𝜇𝑑

𝑑̅

(X,Y) ~ normal bivar.

𝜌

𝑟

𝑌 = 𝛽0 + 𝛽1 𝑥 + 𝑐

𝛽1

𝑏1

(𝑏1 − 𝑡(𝑛−2;1−𝛼)𝑠𝑏1 ,

𝑏1 + 𝑡(𝑛−2;1−𝛼) 𝑠𝑏1 )

𝛽0

𝑏0

(𝑏0 − 𝑡(𝑛−2;1−𝛼)𝑠𝑏0 ,

𝑏0 + 𝑡(𝑛−2;1−𝛼) 𝑠𝑏0 )

𝜇𝑦..𝑥

𝑏0 + 𝑏1 𝑥

(𝑥 − 𝑥̅ )2 1 𝑏0 + 𝑏1 𝑥 ± 𝑡(𝑛−2;1−𝛼) 𝑆𝑦..𝑥 √ + 𝑛 ∑(𝑥𝑖 − 𝑥̅ )2 2

𝑌

𝑏0 + 𝑏1 𝑥

(𝑥 − 𝑥̅ )2 1 𝑏0 + 𝑏1 𝑥 ± 𝑡(𝑛−2;1−𝛼)𝑆𝑦..𝑥 √1 + + 𝑛 ∑(𝑥𝑖 − 𝑥̅ )2 2

2

𝐷𝑖 = 𝑋1𝑖 − 𝑋2𝑖

√𝑛

,

2

√𝑛

)

Usar gráfico ad_hoc ∗ 2

2

∗

∈ ~ 𝑁(0, 𝜎 2 )

(∗) 𝑆 2 𝑦..𝑥 =

𝑛−1 2 (𝑠 − 𝑏12 𝑠𝑥2 ) 𝑛−2 𝑦

;

𝑠𝑏1 =

2

𝑆𝑦..𝑥 √∑(𝑥𝑖 − 𝑥̅ )2

;

2

1 𝑥̅ 2 𝑠𝑏0 = 𝑆𝑦..𝑥 √ + 𝑛 ∑(𝑥𝑖 − 𝑥̅ )2

Alejandro González Tapia - Ing. Civil en Minas

DOCIMAS MAS USUALES Situación X v.a. normal, una m.a.

X v.a. normal, una m.a. σ2 conocida

X v.a. normal, una m.a. σ2 desconocida

X ~ 𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖 (𝑝), 𝑢𝑛𝑎 m.a. X v.a normal, dos m.a.independientes

X v.a normal, dos m.a.independientes 𝜎12 , 𝜎22 , 𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠

Hip. Nula 𝜎 2 = 𝜎02

Estadística 2 (𝑛 − 1)𝑆 𝑥2 = ~ 𝑥 2 (𝑛 − 1) 𝜎02

𝜇 = 𝜇0

𝑍=

𝑥̅ − 𝜇0 ~ 𝑁 (0 ; 1 ) 𝜎 ⁄√𝑛

𝜇 = 𝜇0

𝑇=

𝑥̅ − 𝜇0 ~ 𝑡(𝑛−1) 𝑆⁄√𝑛

𝑝 = 𝑝0

𝑍=

𝜎1 2 = 𝜎22

𝐹=

𝑍=

𝜇1 − 𝜇2 = 𝜇0

𝑝̂ − 𝑝0 √𝑝0∗ 𝑞0 ⁄𝑛

𝑆12 ~𝐹(𝑛1 − 1 , 𝑛2 − 1) 𝑆22 (𝑥̅1 − 𝑥̅2 ) − 𝜇0 𝜎2 √ 1 𝑛1

X v.a normal, dos m.a.independientes 𝜎12 = 𝜎22 , 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠

𝑇=

𝜇1 − 𝜇2 = 𝜇0

~ 𝑁 (0 ; 1 )

+

~ 𝑁 (0; 1)

𝜎22 𝑛2

(𝑥̅1 − 𝑥̅2 ) − 𝜇0 𝑆𝑝 √1⁄𝑛1 + 1⁄𝑛2

~𝑡(𝑛1 + 𝑛2 − 2)

(𝑛1 −1)∗𝑆12 +(𝑛2 −1)∗𝑆22

Con 𝑆𝑝 = √

X v.a normal, dos m.a.independientes 𝜎12 ≠ 𝜎22 , 𝑑𝑒𝑠𝑐𝑜𝑛𝑜𝑐𝑖𝑑𝑎𝑠 (Problema de Behrens- Fisher)

𝑇=

𝑛1

𝛾=

X v.a normal, dos m.a. pareadas 𝐷𝑖 = 𝑋1𝑖 − 𝑋2𝑖

(𝑋 , 𝑌 )~ 𝑛𝑜𝑚𝑎𝑙 𝑏𝑖𝑣𝑎𝑟𝑖𝑎𝑑𝑎

𝑌 = 𝛽0 + 𝛽1 𝑥 + 𝜀 𝜀 ~ 𝑁 (0 ; 𝜎 2 )

𝑛−1 𝑛−2 APC/AMDA 2 𝑆𝑦.𝑥 =

( 𝑆𝑦2 − 𝑏12 ∗ 𝑆𝑥2 );

(𝑥̅1 − 𝑥̅2 ) − 𝜇0 𝑆2 √ 1

𝜇1 − 𝜇2 = 𝜇0

𝜇1 − 𝜇2 = 𝜇𝑑

𝑇=

𝜌 = 𝜌0

𝑇=

𝑛1 +𝑛2 −2

+

~ 𝑡(𝛾)

𝑆22 𝑛2

( 𝑆12 ⁄𝑛1 + 𝑆22 ⁄𝑛2 )2 2⁄ 2 (𝑆1 𝑛1 ) ⁄(𝑛1 − 1) + (𝑆22 ⁄𝑛2 )2⁄(𝑛2

(𝑥̅1 − 𝑥̅2 ) − 𝜇𝑑 ~ 𝑡(𝑛−1) 𝑆𝑑 ⁄√𝑛

(𝑟 − 𝜌0 ) ∗ √( 𝑛 − 2) √1 − 𝑟 2

𝛽1∗

𝑇=

𝑏1 − 𝛽1∗ ~ 𝑡 (𝑛 − 2) 𝑆𝑏1

𝛽0 = 𝛽0∗

𝑇=

𝑏0 − 𝛽0∗ ~ 𝑡(𝑛 − 2) 𝑆𝑏0

𝛽1 =

𝑆𝑏1 =

𝑆𝑦.𝑥 √∑(𝑥𝑖

−𝑥̅ )2

;

− 1)

1 𝑛

𝑆𝑏0 = 𝑆𝑦.𝑥 √ +

𝑥̅ 2 √∑(𝑥𝑖 −𝑥̅ )2

~ 𝑡(𝑛 − 2)

UNIVERSIDAD DE SANTIAGO DE CHILE FACULTAD DE CIENCIA DEPTO. MAT y CIENCIA de la COMP AREA de ESTADISTICA

Variables Aleatorias Bidimensionales Definición: Sean (X, Y) dos variables aleatorias discretas, entonces existe: 𝑃𝑋,𝑌 (𝑥, 𝑦) = 𝑃(𝑋 = 𝑥, 𝑌 = 𝑦); ∀ (𝑥, 𝑦) ∈ 𝑅𝑒𝑐 (𝑋, 𝑌) llamada Distribución de Probabilidad Conjunta de (X, Y) o función de cuantía conjunta de (X, Y) tal que cumple con: 1. − 0 ≤ 𝑃(𝑥, 𝑦) ≤ 1 ∀(𝑥, 𝑦) ∈ 𝑅𝑒𝑐 (𝑋, 𝑌) 2. − ∑ ∑ 𝑃(𝑥, 𝑦) = 1 𝑅𝑒𝑐 𝑋 𝑅𝑒𝑐 𝑌

Definición: Sean (X,Y) dos variables aleatorias continuas, entonces existe: 𝑓𝑋,𝑌 (𝑥, 𝑦) llamada función de densidad conjunta de (X, Y), tal cumple con: 1. −𝑓(𝑥, 𝑦) ≥ 0; ∀(𝑥, 𝑦) ∈ 𝑅𝑒𝑐 (𝑋, 𝑌) 2. −

∫ 𝑓(𝑥, 𝑦) 𝑑𝑦 𝑑𝑥 = 1

∫

𝑅𝑒𝑐 𝑋 𝑅𝑒𝑐 𝑌

𝑏

3. −𝑃(𝑎 ≤ 𝑋 ≤ 𝑏; 𝑐 ≤ 𝑌 ≤ 𝑑) = ∫

𝑑

∫ 𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥

𝑋 =𝑎 𝑌 = 𝑐

Definición: Sean X e Y dos variables aleatorias discretas, tal que P(x,y) es su función de probabilidad conjunta, entonces: 𝑃(𝑋 = 𝑥) = ∑ 𝑃𝑋,𝑌 (𝑥, 𝑦) ; ∀𝑥 ∈ 𝑅𝑒𝑐(𝑋)

Distribución de probabilidad Marginal de X

𝑅𝑒𝑐 𝑌

𝑃(𝑌 = 𝑦) = ∑ 𝑃𝑋,𝑌 (𝑥, 𝑦) ; ∀𝑦 ∈ 𝑅𝑒𝑐(𝑌)

Distribución de probabilidad Marginal de Y

𝑅𝑒𝑐 𝑥

𝑃(𝑋/ 𝑦) =

𝑃𝑋,𝑌 (𝑥, 𝑦) ; ∀(𝑥, 𝑦) ∈ 𝑅𝑒𝑐(𝑋, 𝑌) 𝑃𝑌 (𝑥, 𝑦)

Distribución de probabilidad de X condicionada por Y

𝑃(𝑌/ 𝑥) =

𝑃𝑋,𝑌 (𝑥, 𝑦) ; ∀(𝑥, 𝑦) ∈ 𝑅𝑒𝑐(𝑋, 𝑌) 𝑃𝑋 (𝑥, 𝑦)

Distribución de probabilidad de Y condicionada por X

𝐸(𝑋⁄𝑌 = 𝑦𝑗 ) = ∑ 𝑥 ∙ 𝑝(𝑥 ⁄𝑦𝑗 )

Esperanza de X condicionada por Y = 𝑦𝑗

𝑅𝑒𝑐 𝑋

𝑉(𝑋⁄𝑌 = 𝑦𝑗 ) = 𝐸(𝑋 2 ⁄𝑌 = 𝑦𝑗 ) − [𝐸(𝑋⁄𝑌 = 𝑦𝑗 )]

2

Varianza de X condicionada por Y = 𝑦𝑗

𝐶𝑜𝑣(𝑋, 𝑌) = 𝐸{[𝑋 − 𝐸(𝑋)][𝑌 − 𝐸(𝑌)]} = 𝐸(𝑋𝑌) − 𝐸(𝑋)𝐸(𝑌) 𝐸(𝑋𝑌) = ∑ ∑ 𝑥𝑦 ∙ 𝑃𝑋,𝑌 (𝑥, 𝑦)

Covarianza de X, Y

Esperanza conjunta de (X, Y)

𝑅𝑒𝑐 𝑋 𝑅𝑒𝑐 𝑌

𝜌(𝑋, 𝑌) =

𝐶𝑜𝑣(𝑋, 𝑌) ; −1 ≤ 𝜌(𝑥, 𝑦) ≤ 1 Coeficiente de correlación de Pearson de (X, Y) 𝜎(𝑥) 𝜎(𝑦)

𝑃(𝑋 ≤ 𝑎⁄𝑌 = 𝑏) =

𝑃(𝑋 ≤ 𝑎, 𝑌 = 𝑏) 𝑃(𝑌 = 𝑏)

Si X e Y son Variables aleatorias Discretas

Definición: Sean X e Y dos variables aleatorias continuas, tal que f(x,y) es su función de densidad conjunta, entonces: ∫ 𝑓(𝑥, 𝑦) 𝑑𝑦 ; ∀𝑥 ∈ 𝑅𝑒𝑐 (𝑋) Función de densidad Marginal de X

𝑓(𝑥) =

𝑅𝑒𝑐 𝑌

∫ 𝑓(𝑥, 𝑦) 𝑑𝑥 ; ∀𝑥 ∈ 𝑅𝑒𝑐 (𝑌) Función de densidad Marginal de Y

𝑓(𝑦) =

𝑅𝑒𝑐 𝑋

𝑓(𝑥 ⁄𝑦) =

𝑓(𝑥, 𝑦) | 𝑓(𝑦) 𝑌 = 𝑦

;

∀𝑥 ∈ 𝑅𝑒𝑐(𝑋) Función de densidad de X condicionada por 𝑌 = 𝑦𝑗

;

∀𝑦 ∈ 𝑅𝑒𝑐(𝑌) Función de densidad de Y condicionada por 𝑋 = 𝑥𝑗

𝑗

𝑓(𝑥, 𝑦) | 𝑓(𝑥) 𝑋 = 𝑥 𝑗 Por lo tanto:

𝑓(𝑦⁄𝑥 ) =

𝑎

𝑓(𝑥, 𝑦) = 𝑓(𝑦) ∙ 𝑓(𝑥 ⁄𝑦) = 𝑓(𝑥) ∙ 𝑓(𝑦/𝑥)

𝑃(𝑋 ≤ 𝑎⁄𝑌 = 𝑏) =

∫ 𝑓(𝑥 ⁄𝑦 = 𝑏)𝑑𝑥 𝑋 = −∞

𝐸(𝑋⁄𝑌 = 𝑦𝑗 ) =

∫ 𝑥 𝑓(𝑥 ⁄𝑌 = 𝑦𝑗 ) 𝑑𝑥

Esperanza de X condicionada por 𝑌 = 𝑦𝑗

𝑅𝑒𝑐 𝑋

𝑉(𝑋⁄𝑌 = 𝑦𝑗 ) = 𝐸(𝑋 2 ⁄𝑌 = 𝑦𝑗 ) − [𝐸(𝑋⁄𝑌 = 𝑦𝑗 )] 𝐸(𝑋𝑌) =

∫

∫ 𝑥 ∙ 𝑦 ∙ 𝑓(𝑥, 𝑦) 𝑑𝑦 𝑑𝑥

2

Varianza de X condicionada por 𝑌 = 𝑦𝑗

Esperanza conjunta de (X, Y)

𝑅𝑒𝑐 𝑋 𝑅𝑒𝑐 𝑌

Propiedades de Covarianza 1. 𝐶𝑜𝑣(𝑋, 𝑋) = 𝑉(𝑋);

𝐶𝑜𝑣(𝑌, 𝑌) = 𝑉(𝑌)

2. 𝐶𝑜𝑣(𝑎𝑋 + 𝑏, 𝑐𝑌 + 𝑑) = 𝑎𝑐𝐶𝑜𝑣(𝑋, 𝑌) 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑅 3. 𝐶𝑜𝑣(𝑎𝑋 + 𝑏𝑌, 𝑐𝑋 + 𝑑𝑌) = 𝑎𝑐𝑉(𝑋) + 𝑏𝑑𝑉(𝑌) + (𝑎𝑑 + 𝑏𝑐)𝐶𝑜𝑣(𝑋, 𝑌) 𝑎, 𝑏, 𝑐, 𝑑 ∈ 𝑅 Definición: Se dice que (X,Y) son variables aleatorias Independientes si y solo si: 𝑃(𝑋 = 𝑥, 𝑌 = 𝑦) = 𝑃(𝑋 = 𝑥)𝑃(𝑌 = 𝑦) ∀(𝑥, 𝑦) ∈ 𝑅𝑒𝑐(𝑋, 𝑌) Si X e Y son v. a. Discretas 𝑓(𝑥, 𝑦) = 𝑓(𝑥)𝑓(𝑦) ∀(𝑥, 𝑦) ∈ 𝑅𝑒𝑐(𝑋, 𝑌) Si X e Y son v. a. Continuas Consecuencias: Si X e Y son variables aleatorias Independientes, entonces:  𝐸(𝑋 + 𝑌) = 𝐸(𝑋) + 𝐸(𝑌)

𝐸(𝑋𝑌) = 𝐸(𝑋)𝐸(𝑌)

 𝑉(𝑋 + 𝑌) = 𝑉(𝑋) + 𝑉(𝑌) + 2𝐶𝑜𝑣(𝑋, 𝑌) = 𝑉(𝑋) + 𝑉(𝑌)

𝐶𝑜𝑣(𝑋, 𝑌) = 0 𝜌(𝑋, 𝑌) = 0

 𝑉(𝑋 − 𝑌) = 𝑉(𝑋) + 𝑉(𝑌) − 2𝐶𝑜𝑣(𝑋, 𝑌) = 𝑉(𝑋) + 𝑉(𝑌)  𝑃(𝑋/𝑌) = 𝑃𝑋 (𝑥)

; 𝑃(𝑌/𝑋) = 𝑃𝑌 (𝑦)

 𝐸(𝑋/𝑌) = 𝐸(𝑋); 𝐸(𝑌/𝑋) = 𝐸(𝑌)  𝑋 ~ 𝑁(𝜇1 ; 𝜎12 ) y sea 𝑌 ~ 𝑁(𝜇2 ; 𝜎22 ) entonces: 𝑋 + 𝑌 ~ 𝑁(𝜇1 + 𝜇2 ; 𝜎12 + 𝜎22 ) 𝑋 − 𝑌 ~ 𝑁(𝜇1 − 𝜇2 ; 𝜎12 + 𝜎22 )

Profesoras: Alma Placencia Rosa Montaño Ayudante: Alejandro González

TABLA D

Valores de la función de distribución acumulativa normal estándar

𝑷 𝒁 < 𝑧 = 𝑭 𝒛; 𝟎, 𝟏 =

𝟏 𝟐𝝅

𝒕

TABLAS DE ESTADIST1CA $150.- CODIGO VENTA 381 𝐞𝐱𝐩 − 𝒕𝟐 𝟐 𝒅𝒕

−∞

Profesor: SEGUNDO KARL B. Ayudante: ALEJANDRO GONZÁLEZ 𝒛

.00

.01

.02

.03

.04

.05

.06

.07

.08

.09

-3.5 -3.4 -3.3 -3.2 -3.1 -3.0

0.0002 0.0003 0.0005 0.0007 0.0010 0.0013

0.0002 0.0003 0.0005 0.0007 0.0009 0.0013

0.0002 0.0003 0.0005 0.0006 0.0009 0.0013

0.0002 0.0003 0.0004 0.0006 0.0009 0.0012

0.0002 0.0003 0.0004 0.0006 0.0008 0.0012

0.0002 0.0003 0.0004 0.0006 0.0008 0.0011

0.0002 0.0003 0.0004 0.0006 0.0008 0.0011

0.0002 0.0003 0.0004 0.0005 0.0008 0.0011

0.0002 0.0003 0.0004 0.0005 0.0007 0.0010

0.0002 0.0002 0.0003 0.0005 0.0007 0.0010

-2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0

0.0019 0.0026 0.0035 0.0047 0.0062 0.0082 0.0107 0.0139 0.0179 0.0228

0.0018 0.0025 0.0034 0.0045 0.0060 0.0080 0.0104 0.0136 0.0174 0.0222

0.0018 0.0024 0.0033 0.0044 0.0059 0.0078 0.0102 0.0132 0.0170 0.0217

0.0017 0.0023 0.0032 0.0043 0.0057 0.0075 0.0099 0.0129 0.0166 0.0212

0.0016 0.0023 0.0031 0.0041 0.0055 0.0073 0.0096 0.0125 0.0162 0.0207

0.0016 0.0022 0.0030 0.0040 0.0054 0.0071 0.0094 0.0122 0.0158 0.0202

0.0015 0.0021 0.0029 0.0039 0.0052 0.0069 0.0091 0.0119 0.0154 0.0197

0.0015 0.0021 0.0028 0.0038 0.0051 0.0068 0.0089 0.0116 0.0150 0.0192

0.0014 0.0020 0.0027 0.0037 0.0049 0.0066 0.0087 0.0113 0.0146 0.0188

0.0014 0.0019 0.0026 0.0036 0.0048 0.0064 0.0084 0.0110 0.0143 0.0183

-1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0

0.0287 0.0359 0.0446 0.0548 0.0668 0.0808 0.0968 0.1151 0.1357 0.1587

0.0281 0.0351 0.0436 0.0537 0.0655 0.0793 0.0951 0.1131 0.1335 0.1562

0.0274 0.0344 0.0427 0.0526 0.0643 0.0778 0.0934 0.1112 0.1314 0.1539

0.0268 0.0336 0.0418 0.0516 0.0630 0.0764 0.0918 0.1093 0.1292 0.1515

0.0262 0.0329 0.0409 0.0505 0.0618 0.0749 0.0901 0.1075 0.1271 0.1492

0.0256 0.0322 0.0401 0.0495 0.0606 0.0735 0.0885 0.1056 0.1251 0.1469

0.0250 0.0314 0.0392 0.0485 0.0594 0.0721 0.0869 0.1038 0.1230 0.1446

0.0244 0.0307 0.0384 0.0475 0.0582 0.0708 0.0853 0.1020 0.1210 0.1423

0.0239 0.0301 0.0375 0.0465 0.0571 0.0694 0.0838 0.1003 0.1190 0.1401

0.0233 0.0294 0.0367 0.0455 0.0559 0.0681 0.0823 0.0985 0.1170 0.1379

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 -0.0

0.1841 0.2119 0.2420 0.2743 0.3085 0.3446 0.3821 0.4207 0.4602 0.5000

0.1814 0.2090 0.2389 0.2709 0.3050 0.3409 0.3783 0.4168 0.4562 0.4960

0.1788 0.2061 0.2358 0.2676 0.3015 0.3372 0.3745 0.4129 0.4522 0.4920

0.1762 0.2033 0.2327 0.2643 0.2981 0.3336 0.3707 0.4090 0.4483 0.4880

0.1736 0.2005 0.2297 0.2611 0.2946 0.3300 0.3669 0.4052 0.4443 0.4840

0.1711 0.1977 0.2266 0.2578 0.2912 0.3264 0.3632 0.4013 0.4404 0.4801

0.1685 0.1949 0.2236 0.2546 0.2877 0.3228 0.3594 0.3974 0.4364 0.4761

0.1660 0.1922 0.2206 0.2514 0.2843 0.3192 0.3557 0.3936 0.4325 0.4721

0.1635 0.1894 0.2177 0.2483 0.2810 0.3156 0.3520 0.3897 0.4286 0.4681

0.1611 0.1867 0.2148 0.2451 0.2776 0.3121 0.3483 0.3859 0.4247 0.4641

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159

0.5040 0.5438 0.5832 0.6217 0.6591 0.6950 0.7291 0.7611 0.7910 0.8186

0.5080 0.5478 0.5871 0.6255 0.6628 0.6985 0.7324 0.7642 0.7939 0.8212

0.5120 0.5517 0.5910 0.6293 0.6664 0.7019 0.7357 0.7673 0.7967 0.8238

0.5160 0.5557 0.5948 0.6331 0.6700 0.7054 0.7389 0.7703 0.7995 0.8264

0.5199 0.5596 0.5987 0.6368 0.6736 0.7088 0.7422 0.7734 0.8023 0.8289

0.5239 0.5636 0.6026 0.6406 0.6772 0.7123 0.7454 0.7764 0.8051 0.8315

0.5279 0.5675 0.6064 0.6443 0.6808 0.7157 0.7486 0.7794 0.8078 0.8340

0.5319 0.5714 0.6103 0.6480 0.6844 0.7190 0.7517 0.7823 0.8106 0.8365

0.5359 0.5753 0.6141 0.6517 0.6879 0.7224 0.7549 0.7852 0.8133 0.8389

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713

0.8438 0.8665 0.8869 0.9049 0.9207 0.9345 0.9463 0.9564 0.9649 0.9719

0.8461 0.8686 0.8888 0.9066 0.9222 0.9357 0.9474 0.9573 0.9656 0.9726

0.8485 0.8708 0.8907 0.9082 0.9236 0.9370 0.9484 0.9582 0.9664 0.9732

0.8508 0.8729 0.8925 0.9099 0.9251 0.9382 0.9495 0.9591 0.9671 0.9738

0.8531 0.8749 0.8944 0.9115 0.9265 0.9394 0.9505 0.9599 0.9678 0.9744

0.8554 0.8770 0.8962 0.9131 0.9279 0.9406 0.9515 0.9608 0.9686 0.9750

0.8577 0.8790 0.8980 0.9147 0.9292 0.9418 0.9525 0.9616 0.9693 0.9756

0.8599 0.8810 0.8997 0.9162 0.9306 0.9429 0.9535 0.9625 0.9699 0.9761

0.8621 0.8830 0.9015 0.9177 0.9319 0.9441 0.9545 0.9633 0.9706 0.9767

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981

0.9778 0.9826 0.9864 0.9896 0.9920 0.9940 0.9955 0.9966 0.9975 0.9982

0.9783 0.9830 0.9868 0.9898 0.9922 0.9941 0.9956 0.9967 0.9976 0.9982

0.9788 0.9834 0.9871 0.9901 0.9925 0.9943 0.9957 0.9968 0.9977 0.9983

0.9793 0.9838 0.9875 0.9904 0.9927 0.9945 0.9959 0.9969 0.9977 0.9984

0.9798 0.9842 0.9878 0.9906 0.9929 0.9946 0.9960 0.9970 0.9978 0.9984

0.9803 0.9846 0.9881 0.9909 0.9931 0.9948 0.9961 0.9971 0.9979 0.9985

0.9808 0.9850 0.9884 0.9911 0.9932 0.9949 0.9962 0.9972 0.9979 0.9985

0.9812 0.9854 0.9887 0.9913 0.9934 0.9951 0.9963 0.9973 0.9980 0.9986

0.9817 0.9857 0.9890 0.9916 0.9936 0.9952 0.9964 0.9974 0.9981 0.9986

3.0 3.1 3.2 3.3 3.4 3.5

0.9987 0.9990 0.9993 0.9995 0.9997 0.9998

0.9987 0.9991 0.9993 0.9995 0.9997 0.9998

0.9987 0.9991 0.9994 0.9995 0.9997 0.9998

0.9988 0.9991 0.9994 0.9996 0.9997 0.9998

0.9988 0.9992 0.9994 0.9996 0.9997 0.9998

0.9989 0.9992 0.9994 0.9996 0.9997 0.9998

0.9989 0.9992 0.9994 0.9996 0.9997 0.9998

0.9989 0.9992 0.9995 0.9996 0.9997 0.9998

0.9990 0.9993 0.9995 0.9996 0.9997 0.9998

0.9990 0.9993 0.9995 0.9997 0.9998 0.9998

Pág. 2 VALORES PERC ENTILES , , DE LA DIS TRIBUC IÓN CON GRADOS DE LIBERTAD

1−𝛼

0.005

0.01

0.025

0.05

0.10

0.25

0.75

0.90

0.95

0.975

0.99

0.995

1 2 3 4 5

0.010 0.072 0.207 0.412

0.020 0.115 0.297 0.554

0.001 0.051 0.216 0.484 0.831

0.004 0.103 0.352 0.711 1.145

0.016 0.211 0.584 1.064 1.610

0.102 0.575 1.213 1.923 2.675

1.329 2.773 4.108 5.385 6.626

2.706 4.605 6.251 7.779 9.236

3.841 5.991 7.815 9.488 11.071

5.024 7.378 9.348 11.143 12.833

6.635 9.210 11.345 13.277 15.086

7.879 10.597 12.838 14.860 16.750

6 7 8 9 10

0.676 0.989 1.344 1.735 2.156

0.872 1.239 1.646 2.088 2.558

1.237 1.690 2.180 2.700 3.247

1.635 2.167 2.733 3.325 3.940

2.204 2.833 3.490 4.168 4.865

3.455 4.255 5.071 5.899 6.737

7.841 9.037 10.219 11.389 12.549

10.645 12.017 13.362 14.684 15.987

12.592 14.067 15.507 16.919 18.307

14.449 16.013 17.535 19.023 20.483

16.812 18.475 20.090 21.666 23.209

18.548 20.278 21.955 23.589 25.188

11 12 13 14 15

2.603 3.074 3.565 4.075 4.601

3.053 3.571 4.107 4.660 5.229

3.816 4.404 5.009 5.629 6.262

4.575 5.226 5.892 6.571 7.261

5.578 6.304 7.042 7.790 8.547

7.584 8.438 9.299 10.165 11.037

13.701 14.845 15.984 17.117 18.245

17.275 18.549 19.812 21.064 22.307

19.675 21.026 22.362 23.685 24.996

21.920 23.337 24.736 26.119 27.488

24.725 26.217 27.688 29.141 30.578

26.757 28.299 29.819 31.319 32.801

16 17 18 19 20

5.142 5.697 6.265 6.844 7.434

5.812 6.408 7.015 7.633 8.260

6.908 7.564 8.231 8.907 9.591

7.962 8.672 9.390 10.117 10.851

9.312 10.085 10.865 11.651 12.443

11.912 12.792 13.675 14.562 15.452

19.369 20.489 21.605 22.718 23.828

23.542 24.769 25.989 27.204 28.412

26.296 27.587 28.869 30.144 31.410

28.845 30.191 31.526 32.852 34.170

32.000 33.409 34.805 36.191 37.566

34.267 35.718 37.156 38.582 39.997

21 22 23 24 25

8.034 8.643 9.260 9.886 10.520

8.897 9.542 10.196 10.856 11.524

10.283 10.982 11.689 12.401 13.120

11.591 12.338 13.091 13.848 14.611

13.240 14.042 14.848 15.659 16.473

16.344 17.240 18.137 19.037 19.939

24.935 26.039 27.141 28.241 29.339

29.615 30.813 32.007 33.196 34.382

32.671 33.924 35.172 36.415 37.652

35.479 36.781 38.076 39.364 40.646

38.932 40.289 41.638 42.980 44.314

41.401 42.796 44.181 45.559 46.928

26 27 28 29 30

11.160 11.808 12.461 13.121 13.787

12.198 12.879 13.565 14.257 14.954

13.844 14.573 15.308 16.047 16.791

15.379 16.151 16.928 17.708 18.493

17.292 18.114 18.939 19.768 20.599

20.843 21.749 22.657 23.567 24.478

30.435 31.528 32.620 33.711 34.800

35.563 36.741 37.916 39.087 40.256

38.885 40.113 41.337 42.557 43.773

41.923 43.194 44.461 45.722 46.979

45.642 46.963 48.278 49.588 50.892

48.290 49.645 50.993 52.336 53.672

31 32 33 34 35

14.458 15.134 15.815 16.501 17.192

15.655 16.362 17.074 17.789 18.509

17.539 18.291 19.047 19.806 20.569

19.281 20.072 20.867 21.664 22.465

21.434 22.271 23.110 23.952 24.797

25.390 26.304 27.219 28.136 29.054

35.887 36.973 38.058 39.141 40.223

41.422 42.585 43.745 44.903 46.059

44.985 46.194 47.400 48.602 49.802

48.232 49.480 50.725 51.966 53.203

52.191 53.486 54.776 56.061 57.342

55.003 56.328 57.648 58.964 60.275

36 37 38 39 40

17.887 18.586 19.289 19.996 20.707

19.233 19.960 20.691 21.426 22.164

21.336 22.106 22.878 23.654 24.433

23.269 24.075 24.884 25.695 26.509

25.643 26.492 27.343 28.196 29.051

29.973 30.893 31.815 32.737 33.660

41.304 42.383 43.462 44.539 45.616

47.212 48.363 49.513 50.660 51.805

50.998 52.192 53.384 54.572 55.758

54.437 55.668 56.896 58.120 59.342

58.619 59.892 61.162 62.428 63.691

61.581 62.883 64.181 65.476 66.766

41 42 43 44 45

21.421 22.138 22.859 23.584 24.311

22.906 23.650 24.398 25.148 25.901

25.215 25.999 26.785 27.575 28.366

27.326 28.144 28.965 29.787 30.612

29.907 30.765 31.625 32.487 33.350

34.585 35.510 36.436 37.363 38.291

46.692 47.766 48.840 49.913 50.985

52.949 54.090 55.230 56.369 57.505

56.942 58.124 59.304 60.481 61.656

60.561 61.777 62.990 64.201 65.410

64.950 66.206 67.459 68.710 69.957

68.053 69.336 70.616 71.893 73.166

46 47 48 49 50

25.041 25.775 26.511 27.249 27.991

26.657 27.416 28.177 28.941 29.707

29.160 29.956 30.755 31.555 32.357

31.439 32.268 33.098 33.930 34.764

34.215 35.081 35.949 36.818 37.689

39.220 40.149 41.079 42.010 42.942

52.056 53.127 54.196 55.265 56.334

58.641 59.774 60.907 62.038 63.167

62.830 64.001 65.171 66.339 67.505

66.617 67.821 69.023 70.222 71.420

71.201 72.443 73.683 74.919 76.154

74.437 75.704 76.969 78.231 79.490

51 52 53 54 55

28.735 29.481 30.230 30.981 31.735

30.475 31.246 32.018 32.793 33.570

33.162 33.968 34.776 35.586 36.398

35.600 36.437 37.276 38.116 38.958

38.560 39.433 40.308 41.183 42.060

43.874 44.808 45.741 46.676 47.610

57.401 58.468 59.534 60.600 61.665

64.295 65.422 66.548 67.673 68.796

68.669 69.832 70.993 72.153 73.311

72.616 73.810 75.002 76.192 77.380

77.386 78.616 79.843 81.069 82.292

80.747 82.001 83.253 84.502 85.749

56 57 58 59 60

32.490 33.248 34.008 34.770 35.534

34.350 35.131 35.913 36.698 37.485

37.212 38.027 38.844 39.662 40.482

39.801 40.646 41.492 42.339 43.188

42.937 43.816 44.696 45.577 46.459

48.546 49.482 50.419 51.356 52.294

62.729 63.793 64.857 65.919 66.981

69.919 71.040 72.160 73.279 74.397

74.468 75.624 76.778 77.931 79.082

78.567 79.752 80.936 82.117 83.298

83.513 84.733 85.950 87.166 88.379

86.994 88.236 89.477 90.715 91.952

0

Valores Percentiles, 1−𝛼

, de la Distribución

Pág. 3

. (Continuación).

0

0.005

0.01

0.025

0.05

0.10

0.25

0.75

0.90

0.95

0.975

0.99

1

1

1

1

1

1

1

1

1

1

1

1

0.995 1

61 62 63 64 65

36.300 37.068 37.838 38.610 39.383

38.273 39.063 39.855 40.649 41.444

41.038 42.126 42.950 43.776 44.603

44.038 44.889 45.741 46.776 47.450

47.342 48.226 49.111 49.996 50.883

53.232 54.171 55.110 56.050 56.990

68.043 69.104 70.165 71.225 72.285

75.514 76.630 77.745 78.860 79.973

80.232 81.381 82.529 83.675 84.821

84.476 85.654 86.830 88.004 89.177

89.591 90.802 92.010 93.217 94.422

93.186 94.419 95.649 96.878 98.105

66 67 68 69 70

40.158 40.935 41.713 42.494 43.275

42.240 43.038 43.838 44.639 45.442

45.431 46.261 47.092 47.924 48.758

48.305 49.162 50.020 50.879 51.739

51.770 52.659 53.548 54.438 55.329

57.931 58.872 59.814 60.756 61.698

73.344 74.403 75.461 76.519 77.577

81.085 82.197 83.308 84.418 85.527

85.965 87.108 88.250 89.391 90.531

90.349 91.519 92.689 93.856 95.023

95.626 96.828 98.028 99.228 100.425

99.330 100.554 101.776 102.996 104.215

71 72 73 74 75

44.058 44.843 45.629 46.417 47.206

46.246 47.051 47.858 48.666 49.475

49.592 50.428 51.265 52.103 52.942

52.600 53.462 54.325 55.189 56.054

56.221 57.113 58.006 58.900 59.795

62.641 63.585 64.528 65.472 66.417

78.634 79.690 80.747 81.803 82.858

86.635 87.743 88.850 89.956 91.061

91.670 92.808 93.945 95.081 96.217

96.189 97.353 98.516 99.678 100.839

101.621 102.816 104.010 105.202 106.393

105.432 106.648 107.862 109.074 110.286

76 77 78 79 80

47.997 48.788 49.582 50.376 51.172

50.286 51.097 51.910 52.725 53.540

53.782 54.623 55.466 56.309 57.153

56.920 57.786 58.654 59.522 60.391

60.690 61.586 62.483 63.380 64.278

67.362 68.307 69.252 70.198 71.145

83.913 84.968 86.022 87.077 88.130

92.166 93.270 94.374 95.476 96.578

97.351 98.484 99.617 100.749 101.879

101.999 103.158 104.316 105.473 106.629

107.583 108.771 109.958 111.144 112.329

111.495 112.704 113.911 115.117 116.321

81 82 83 84 85

51.969 52.767 53.567 54.368 55.170

54.357 55.174 55.993 56.813 57.634

57.998 58.845 59.692 60.540 61.389

61.261 62.132 63.004 63.876 64.749

65.176 66.076 66.976 67.876 68.777

72.091 73.038 73.985 74.933 75.881

89.184 90.237 91.289 92.342 93.394

97.680 98.780 99.880 100.980 102.079

103.010 104.139 105.267 106.395 107.522

107.783 108.937 110.090 111.242 112.393

113.512 114.695 115.876 117.057 118.236

117.524 118.726 119.927 121.126 122.325

86 87 88 89 90

55.973 56.777 57.582 58.389 59.196

58.456 59.279 60.103 60.928 61.754

62.239 63.089 63.941 64.793 65.647

65.623 66.498 67.373 68.249 69.126

69.679 70.581 71.484 72.387 73.291

76.829 77.777 78.726 79.675 80.625

94.449 95.497 96.548 97.599 98.650

103.177 104.275 105.372 106.469 107.565

108.648 109.773 110.898 112.022 113.145

113.544 114.693 115.841 116.989 118.136

119.414 120.591 121.767 122.942 124.116

123.522 124.718 125.913 127.106 128.299

91 92 93 94 95

60.005 60.815 61.625 62.437 63.250

62.581 63.409 64.238 65.068 65.898

66.501 67.356 68.211 69.068 69.925

70.003 70.882 71.760 72.640 73.520

74.196 75.100 76.006 76.912 77.818

81.574 82.524 83.474 84.425 85.376

99.700 100.750 101.800 102.850 103.899

108.661 109.756 110.850 111.944 113.038

114.268 115.390 116.511 117.632 118.752

119.282 120.427 121.571 122715 123.858

125.289 126.462 127.633 128.803 129.973

129.491 130.681 131.871 133.059 134.247

96 97 98 99 100

64.063 64.878 65.694 66.510 67.328

66.730 67.562 68.396 69.230 70.065

70.783 71.642 72.501 73.361 74.222

74.401 75.282 76.164 77.046 77.929

78.725 79.633 80.541 81.449 82.358

86.327 87.278 88.229 89.181 90.133

104.948 105.997 107.045 108.093 109.141

114.131 115.223 116.315 117.407 118.498

119.871 120.990 122.108 123.225 124.342

125.000 126.141 127.282 128.422 129.561

131.141 132.309 133.476 134.642 135.807

135.433 136.619 137.803 138.987 140.169

102 104 106 108 110

68.965 70.606 72.251 73.899 75.550

71.737 73.413 75.092 76.774 78.458

75.946 77.672 79.401 81.133 82.867

79.697 81.468 83.240 85.015 86.792

84.177 85.998 87.821 89.645 91.471

92.038 93.944 95.850 97.758 99.666

111.236 113.331 115.424 117.517 119.608

120.679 122.856 125.035 127.211 129.385

126.574 128.804 131.031 133.257 135.480

131.838 134.111 136.382 138.651 140.917

138.134 140.459 142.780 145.099 147.414

142.532 144.891 147.247 149.599 151.948

112 114 116 118 120

77.204 78.862 80.522 82.185 83.852

80.146 81.836 83.529 85.225 86.923

84.604 86.342 88.084 89.827 91.573

88.570 90.351 92.134 93.918 95.705

93.299 95.128 96.958 98.790 100.624

101.575 103.485 105.396 107.307 109.220

121.699 123.789 125.878 127.967 130.055

131.558 133.729 135.898 138.066 140.233

137.701 139.921 142.138 144.354 146.567

143.180 145.441 147.700 149.957 152.211

149.727 152.037 154.344 156.648 158.950

154.294 156.637 158.977 161.314 163.648

122 124 126 128 130

85.520 87.192 88.866 90.543 92.222

88.624 90.327 92.033 93.741 95.451

93.320 95.070 96.822 98.576 100.331

97.493 99.283 101.074 102.867 104.662

102.458 104.295 106.132 107.971 109.811

111.133 113.046 114.961 116.876 118.792

132.142 134.228 136.313 138.398 140.482

142.398 144.562 146.724 148.885 151.045

148.779 150.989 153.198 155.405 157.610

154.464 156.714 158.962 161.209 163.453

161.250 163.548 165.841 168.133 170.423

165.980 168.308 170.634 172.957 175.278

132 134 136 138 140

93.904 95.588 97.275 98.964 100.655

97.163 98.878 100.595 102.314 104.034

102.089 103.848 105.609 107.372 109.137

106.459 108.257 110.056 111.857 113.659

111.652 113.495 115.338 117.183 119.029

120.708 122.625 124.543 126.461 128.380

142.566 144.649 146.731 148.813 150.894

153.204 155.361 157.518 159.673 161.827

159.814 162.016 164.216 166.415 168.613

165.696 167.936 170.175 172.412 174.648

172.711 174.996 177.280 179.561 181.840

177.597 179.913 182.226 184.538 186.847

142 144 146 148 150

102.348 104.044 105.741 107.441 109.142

105.757 107.482 109.209 110.937 112.668

110.903 112.671 114.441 116.212 117.985

115.463 117.268 119.075 120.883 122.692

120.876 122.724 124.574 126.424 128.275

130.299 132.219 134.140 136.061 137.983

152.975 155.055 157.134 159.213 161.291

163.980 166.132 168.283 170.432 172.581

170.809 173.004 175.198 177.390 179.581

176.882 179.114 181.344 183.573 185.800

184.118 186.393 188.666 190.938 193.208

189.154 191.458 193.761 196.062 198.360

200 250 300 400 500

152.241 196.161 240.663 330.903 422.303

156.432 200.939 245.972 337.155 429.388

162.728 208.098 253.912 346.482 439.936

168.279 214.392 260.878 354.641 449.147

174.835 221.806 269.068 364.207 459.926

186.172 234.577 283.135 380.577 478.323

213.102 264.697 316.138 418.697 520.950

226.021 279.050 331.789 436.649 540.930

233.994 287.882 341.395 447.632 553.127

241.058 295.689 349.874 457.305 563.852

249.445 304.940 359.906 468.724 576.493

255.264 311.346 366.844 476.606 585.207

600 700 800 900 1000

514.529 607.380 700.725 794.475 888.564

522.365 615.907 709.897 804.252 898.912

534.019 628.577 723.513 818.756 914.257

544.180 639.613 735.362 831.370 927.594

556.056 652.497 749.185 846.075 943.133

576.286 674.413 772.669 871.032 969.484

622.988 724.861 826.604 928.241 1029.790

644.800 748.359 851.671 954.782 1057.724

658.094 762.661 866.911 970.904 1074.679

669.769 775.211 880.275 985.032 1089.531

683.516 789.974 895.984 1001.630 1106.969

692.982 800.131 906.786 1013.036 1118.948

0.90

3.0777 1.8856 1.6377 1.5332 1.4759

1.4398 1.4149 1.3968 1.3830 1.3722

1.3634 1.3562 1.3502 1.3450 1.3406

1.3368 1.3334 1.3304 1.3277 1.3253

1.3232 1.3212 1.3195 1.3178 1.3163

1.3150 1.3137 1.3125 1.3114 1.3104

1.3095 1.3086 1.3077 1.3070 1.3062

1.3055 1.3049 1.3042 1.3036 1.3031

1.3025 1.3020 1.3016 1.3011 1.3006

0.75

1.0000 0.8165 0.7649 0.7407 0.7267

0.7176 0.7111 0.7064 0.7027 0.6998

0.6974 0.6955 0.6938 0.6924 0.6912

0.6901 0.6692 0.6884 0.6676 0.6870

0.6864 0.6858 0.6853 0.6848 0.6844

0.6840 0.6837 0.6834 0.6830 0.6828

0.6825 0.6822 0.6820 0.6818 0.6816

0.6814 0.6812 0.6810 0.6808 0.6807

0.6805 0.6804 0.6802 0.6801 0.6800

1 2 3 4 5

6 7 8 9 10

11 12 13 14 15

16 17 18 19 20

21 22 23 24 25

26 27 28 29 30

31 32 33 34 35

36 37 38 39 40

41 42 43 44 45

1−𝛼 1−𝛼

1.6829 1.6820 1.6811 1.6802 1.6794

1.6883 1.6871 1.6860 1.6849 1.6839

1.6955 1.6939 1.6924 1.6909 1.6896

1.7056 1.7033 1.7011 1.6991 1.6973

1.7207 1.7171 l.7139 1.7109 1.7081

1.7459 1.7396 1.7341 1.7291 1.7247

1.7959 1.7823 1.7709 1.7613 1.7531

1.9432 1.8946 1.8595 1.8331 1.8125

6.3138 2.9200 2.3534 2.1318 2.0150

0.95

2.0195 2.0181 2.0167 2.0154 2.0141

2.0281 2.0262 2.0244 2.0227 2.0211

2.0395 2.0369 2.0345 2.0322 2.0301

2.0555 2.0518 2.0484 2.0452 2.0423

2.0796 2.0739 2.0687 2.0639 2.0595

2.1199 2.1098 2.1009 2.0930 2.0860

2.2010 2.1788 2.1604 2.1448 2.1315

2.4469 2.3646 2.3060 2.2622 2.2281

12.706 4.3027 3.1824 2.7764 2.5706

0.975

2.4208 2.4185 2.4163 2.4141 2.4121

2.4345 2.4314 2.4286 2.4258 2.4233

2.4528 2.4487 2.4448 2.4411 2.4377

2.4786 2.4727 2.4671 2.4620 2.4573

2.5177 2.5083 2.4999 2.4922 2.4851

2.5835 2.5669 2.5524 2.5395 2.5280

2.7181 2.6810 2.6503 2.6245 2.6025

3.1427 2.9980 2.8965 2.8214 2.7638

31.3207 6.9646 4.5407 3.7469 3.3649

0.99

2.7012 2.6981 2.6951 2.6923 2.6896

2.7195 2.7154 2.7116 2.7079 2.7045

2.7440 2.7385 2.7333 2.7284 2.7238

2.7787 2.7707 2.7633 2.7564 2.7500

2.8314 2.8188 2.8073 2.7969 2.7874

2.9208 2.8982 2.8784 2.8609 2.8453

3.1058 3.0545 3.0123 2.9768 2.9467

3.7074 3.4995 3.3554 3.2498 3.1693

63.6574 9.9248 5.8409 4.6041 4.0322

0.995

86 87 88 89 90

81 82 83 84 85

76 77 78 79 80

71 72 73 74 75

66 67 68 69 70

61 62 63 64 65

56 57 58 59 60

51 52 53 54 55

46 47 48 49 50

1−𝛼 1−𝛼

0.6774 0.6773 0.6773 0.6773 0.6772

0.6775 0.6775 0.6775 0.6774 0.6774

0.6777 0.6777 0.6776 0.6776 0.6776

0.6779 0.6779 0.6779 0.6778 0.6778

0.6782 0.6782 0.6781 0.6781 0.6780

0.6785 0.6785 0.6784 0.6783 0.6783

0.6789 0.6788 0.6787 0.6787 0.6786

0.6793 0.6792 0.6791 0.6791 0.6790

0.6799 0.6797 0.6796 0.6795 0.6794

0.75

1.2915 1.2914 1.2912 1.2911 1.2910

1.2921 1.2920 1.2918 1.2917 1.2916

1.2928 1.2926 1.2925 1.2924 1.2922

1.2936 1.2934 1.2933 1.2931 1.2929

1.2945 1.2943 1.2941 1.2939 1.2938

1.2956 1.2954 1.2951 1.2949 1.2947

1.2969 1.2966 1.2963 1.2961 1.2958

1.2984 1.2980 1.2977 1.2974 1.2971

1.3002 1.2998 1.2994 1.2991 1.2987

0.90

1.6628 1.6626 1.6624 1.6622 1.6620

1.6639 1.6636 1.6634 1.6632 1.6630

1.6652 1.6649 1.6646 1.6644 1.6641

l.6666 1.6663 1.6660 1.6657 1.6654

1.6683 1.6679 1.6676 1.6672 1.6669

1.6702 1.6698 1.6694 1.6690 1.6686

1.6725 1.6720 1.6716 1.6711 1.6706

1.6753 1.6747 1.6741 1.6736 1.6730

1.6787 1.6779 1.6772 1.6766 1.6759

0.95

1.9879 1.9876 1.9873 1.9870 1.9867

1.9897 1.9893 1.9890 1.9886 1.9883

1.9917 1.9913 1.9908 1.9905 1.9901

1.9539 1.9935 1.9930 1.9925 1.9921

1.9966 1.9960 1.9955 1.9949 1.9944

1.9996 1.9990 1.9983 1.9977 1.9971

2.0032 2.0925 2.0017 2.0010 2.0003

2.0076 2.0066 2.0057 2.0049 2.0040

2.0129 2.0117 2.0106 2.0096 2.0086

0.975

2.3705 2.3700 2.3695 2.3690 2.3685

2.3733 2.3727 2.3721 2.3716 2.3710

2.3764 2.3758 2.3751 2.3745 2.3739

2.3800 2.3793 2.3785 2.3778 2.3771

2.3842 2.3833 2.3824 2.3816 2.3808

2.3890 2.3880 2.3870 2.3860 2.3851

2.3948 2.3936 2.3924 2.3912 2.3901

2.4017 2.4002 2.3988 2.3974 2.3961

2.4102 2.4083 2.4066 2.4049 2.4033

0.99

2.6342 2.6335 2.6329 2.6322 2.6316

2.6379 2.6371 2.6364 2.6356 2.6349

2.6421 2.6412 2.6403 2.6395 2.6387

2.6469 2.6459 2.6449 2.6439 2.6430

2.6524 2.6512 2.6501 2.6490 2.6479

2.6589 2.6575 2.6561 2.6549 2.6536

2.6665 2.6649 2.6633 2.6618 2.6603

2.6757 2.6737 2.6718 2.6700 2.6682

2.6870 2.6846 2.6822 2.6800 2.6778

0.995

Pág. 4

0.90 1.2909 1.2908 1.2907 1.2906 1.2905 1.2904 1.2903 1.2902 1.2902 1.2901 1.2899 1.2897 1.2996 1.2894 1.2893 1.2892 1.2890 1.2889 1.2888 1.2886 1.2885 1.2884 1.2883 1.2882 1.2881 1.2880 1.2879 1.2878 1.2877 1.2876 1.2875 1.2875 1.2874 1.2873 1.2872 1.2858 1.2844 1.2837 1.2832 1.2830 1.2828 1.2826 1.2825 1.2824 1.2816

0.75 0.6772 0.6772 0.6771 0.6771 0.6771 0.6771 0.6770 0.6770 0.6770 0.6770 0.6769 0.6769 0.6768 0.6768 0.6767 0.6767 0.6766 0.6766 0.6766 0.6765 0.6765 0.6765 0.6764 0.6764 0.6764 0.6764 0.6763 0.6763 0.6763 0.6762 0.6762 0.6762 0.6762 0.6762 0.6761 0.6757 0.6753 0.6751 0.6750 0.6749 0.6748 0.6748 0.6748 0.6747 0.6745

91 92 93 94 95 96 97 98 99 100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 200 300 400 500 600 700 800 900 1000 ∞

1−𝛼 1−𝛼

1.6470 1.6468 1.6465 1.6464 1.6449

1.6525 1.6499 1.6487 1.6479 1.6474

1.6557 1.6555 1.6554 1.6552 1.6551

1.6565 1.6563 1.6561 1.6560 1.6558

1.6574 1.6572 1.6570 1.6568 1.6567

1.6586 1.6583 1.6581 1.6579 1.6577

1.6599 1.6596 1.6594 1.6591 1.6588

1.6609 1.6607 1.6606 1.6604 1.6602

1.6618 1.6616 1.6614 1.6612 1.6611

0.95

1.9634 1.9629 1.9626 1.9623 1.9600

1.9719 1.9679 1.9659 1.9647 1.9639

1.9768 1.9766 1.9763 1.9761 1.9759

1.9781 1.9778 1.9776 1.9773 1.9771

1.9796 1.9793 1.9790 1.9787 1.9784

1.9814 1.9810 1.9806 1.9803 1.9799

1.9835 1.9830 1.9826 1.9822 1.9818

1.9850 1.9847 1.9845 1.9842 1.9840

1.9864 1.9861 1.9858 1.9855 1.9853

0.975

2.3317 2.3310 2.3305 2.3301 2.3263

2.3451 2.3388 2.3357 2.3338 2.3326

2.3529 2.3525 2.3522 2.3518 2.3515

2.3549 2.3545 2.3541 2.3537 2.3533

2.3573 2.3568 2.3563 2.3558 2.3554

2.3601 2.3595 2.3589 2.3584 2.3578

2.3635 2.3627 2.3620 2.3614 2.3607

2.3658 2.3654 2.3650 2.3646 2.3642

2.3680 2.3676 2.3471 2.3667 2.3662

0.99

2.5829 2.5820 2.5813 2.5806 2.5758

2.6006 2.5923 2.5882 2.5857 2.5840

2.6109 2.6104 2.6099 2.6095 2.6090

2.6136 2.6130 2.6125 2.6119 2.6114

2.6167 2.6161 2.6154 2.6148 2.6142

2.6204 2.6196 2.6189 2.6181 2.6174

2.6249 2.6239 2.6230 2.6221 2.6213

2.6280 2.6275 2.6269 2.6264 2.6259

2.6309 2.6303 2.6297 2.6291 2.6286

0.995

Pág. 5

Valores Percentiles

Pág. 6 𝒑

, , de la Distribución F. (Continuación)

1

2

3

4

5

6

7

8

9

10

11

12

1

.500 .750 .900 .950 .975 .990 .995

1.0000 5.8285 39.864 161.45 647.79 4052.2 16211

1.5000 7.5000 49.500 199.50 799.50 4999.5 20000

1.7092 8.1999 53.593 215.71 864.16 5403.3 21615

1.8227 8.5810 55.883 224.58 899.58 5624.6 22500

1.8937 8.8198 57.241 230.16 921.85 5763.7 23056

1.9422 8.9833 58.204 233.99 937.11 5859.0 23437

1.9774 9.1021 58.906 236.7 7 948.22 5928.3 2373 5

2.0041 9.1922 59.439 238.88 956.66 5981.1 23925

2.0250 9.2631 59.858 240.54 963.28 6011.5 24091

2.0419 9.3202 60.195 241.88 968.63 6055.8 24224

2.0558 9.3672 60.473 242.99 973.04 6083.3 24334

2.0574 9.4064 60.705 243.91 976.71 6106.3 24426

2

.500 .750 .900 .950 .975 .990 .995

.66667 2.5714 8.5263 18.513 38.506 98.503 198.50

1.0000 3.0000 9.0000 19.000 39.000 99.000 199.00

1.1349 3.1534 9.1618 19.164 39.165 99.166 199.17

1.2071 3.2320 9.2434 19.247 39.248 99.249 199.25

1.2519 3.2799 9.2926 19.296 39.298 99.299 199.30

1.2824 3.3121 9.3255 19.330 39.331 99.332 199.33

1.3045 3.3352 9.3491 19.353 39. 355 99.356 199.36

1.3213 3.3526 9.3668 19.371 39.373 99.374 199.37

1.3344 3.3661 9.3805 19.385 39.387 99.388 199.39

1.3450 3.3770 9.3916 19.396 39.398 99.399 199.40

1.3537 3.3859 9.4006 19.405 39.407 99.408 199.41

1.3610 3.3934 9.4081 19.413 39.415 99.416 199.42

3

.500 .750 .900 .950 .975 .990 .995

.58506 2.0239 5.5383 10.128 17.443 34.116 55.552

.88110 2.2798 5.4624 9.5521 16.044 30.817 49.799

1.0000 2.3555 5.3908 9.2766 15.439 29.457 47.467

1.0632 2.3901 5.3427 9.1172 15.101 28.710 46.195

1.1024 2.4095 5.3092 9.0135 14.885 28.237 45.392

1.1289 2.4218 5.2847 8.9406 14.735 27.911 44.838

1.1482 2.4302 5.2662 8.8868 14.624 27.672 44.434

1.1627 2.4364 5.2517 8.8452 14.540 27.489 44.126

1.1741 2.4410 5.2400 8.8123 14.473 27.345 43.882

1.1833 2.4447 5.2304 8.7855 14.419 27.229 43.686

1.1909 2.4476 5.2223 8.7632 14.374 27.132 43.523

1.1972 2.4500 5.2156 8.7446 14.337 27.052 43.387

4

.500 .750 .900 .950 .975 .990 .995

.54863 1.8074 4.5448 7.7086 12.218 21.198 31.333

.82843 2.0000 4.3246 6.9443 10.649 18.000 26.284

.94054 2.0467 4.1908 6.5914 9.9792 16.694 24.259

1.0000 2.0642 4.1073 6.3883 9.6045 15.977 23.155

1.0367 2.0723 4.0506 6.2560 9.3645 15.522 22.456

1.0617 2.0766 4.0098 6.1631 9.1973 15.207 21.975

1.0797 2.0790 3.9790 6.0942 9.0741 14.976 21 .622

1.0933 2.0805 3.9549 6.0410 8.9796 14.799 21.352

1.1040 2.0814 3.9357 5.9988 8.9047 14.659 21.139

1.1126 2.0820 3.9199 5.9644 8.8439 14.546 20.967

1.1196 2.0823 3.9066 5.9357 8.7933 14.452 20.824

1.1255 2.0826 3.8955 5.9117 8.751 2 14.574 20.705

5

.500 .750 .900 .950 .975 .990 .995

.52807 1.6925 4.0604 6.6079 10.007 16.258 22.785

.79877 1.8528 1.7797 5.7861 8.4336 13.274 18.314

.90715 1.8843 3.6195 5.4095 7.7636 12.060 16.530

.96456 1.8927 3.5202 5.1922 7.3879 11.392 15.556

1.0000 1.8947 3.4530 5.0503 7.1464 10.967 14.940

1.0240 1.8945 3.4045 4.9503 6.9777 10.672 14.513

1.0414 1.8935 3.3679 4.8759 6.8531 10.456 14.200

1.0545 1.8923 3.3393 4.8183 6.7572 10.289 13.961

1.0648 1.8911 3.3163 4.7725 6.6810 10.158 13.772

1.0730 1.8899 3.2974 4.7351 6.6192 10.051 13.618

1.0798 1.8987 3.2815 4.7038 6.5676 9.9623 13.490

1.0855 1.8877 3.2652 4.6777 6.5246 9.8883 13.384

6

.500 .750 .900 .950 .975 .990 .995

.51489 1.6214 3.7760 5.9874 8.8131 13.745 18.635

.77976 1.7622 3.4633 5.1433 7.2598 10.925 14.544

.88578 1.7844 3.2888 4.7571 6.5988 9.7795 12.917

.94191 1.7872 3.1808 4.5337 6.2272 9.1483 12.028

.97654 1.7852 3.1075 4.3874 5.9876 8.7459 11.464

1.0000 1.7821 3.0546 4.2839 5.8197 8.4661 11.073

1.0169 1.7789 3.0145 4.2066 5.6955 8.2600 10.786

1.0298 1.7760 2.9830 4.1468 5.5996 8.1016 10.566

1.0398 1.7733 2.9577 4.0990 5.5234 7.9761 10.391

1.0478 1.7708 2.9369 4.0600 5.4613 7.8741 10.250

1.0545 1.7686 2.9193 4.0272 5.4094 7.7891 10.132

1.0600 l.7668 2.9047 3.9999 5.3662 7.7183 10.034

7

.500 .750 .900 .950 .975 .990 .995

.50572 1.5732 3.5894 5.5914 8.0727 12.246 16.236

.76655 1.7010 3.2574 4.7374 6.5415 9.5466 12.404

.87095 1.7169 3.0741 4.3468 5.8598 8.4513 10.882

.92619 1.7157 2.9605 4.1203 5.5226 7.8467 10.050

.96026 1.7111 2.8833 3.9715 5.2852 7.4604 9.5221

.98334 1.7059 2.8274 3.8660 5.1186 7.1914 9.1554

1.0000 1.7011 2.7849 3.7870 4.9949 6.9928 8.8854

1.0126 1.6969 2.7516 3.7257 4.8994 6.8401 8.6781

1.0224 1.6931 2.7247 3.6767 4.8232 6.7188 8.5138

1.0304 1.6898 2.7025 3.6365 4.7611 6.6201 8.3803

1.0369 1.6868 2.6837 3.6028 4.7091 6.5377 8.2691

1.0423 1.6843 2.6681 3.5747 4.6658 6.4691 8.1764

8

.500 .750 .900 .950 .975 .990 .995

.49898 1.5384 3.4579 5.3177 7.5709 11.259 14.688

.75683 1.6569 3.1131 4.4590 6.0595 8.6491 11.042

.86004 1.6683 2.9238 4.0662 5.4160 7.5910 9.5965

.91464 1.6642 2.8064 3.8378 5.0526 7.0060 8.8051

.94831 1.6575 2.7265 3.6875 4.8173 6.6318 8.3018

.97111 1.6508 2.6683 3.5806 4.6517 6.3707 7.9520

.98757 1.6448 2.6241 3.5005 4.5286 6. 1776 7.6942

1.0000 1.6396 2.5893 3.4381 4.4332 6.0289 7.4960

1.0097 1.6350 2.5612 3.3881 4.3572 5.9106 7.3386

1.0175 1.6310 2.5380 3.3472 4.2951 5.8143 7.2107

1.0239 1.6274 2.5184 3.3127 4.2431 5.7338 7.1039

1.0293 1.6244 2.5020 3.2840 4.1997 5.6668 7.0149

9

.500 .750 .900 .950 .975 .990 .995

.49382 1.5121 3.3603 5.1174 7.2093 10.561 13.814

.74938 1.6236 3.0065 4.2565 5.7147 8.0215 10.107

.85168 1.6315 2.8129 3.8626 5.0781 6.9919 8.7171

.90580 1.6253 2.6927 3.6331 4.7181 6.4721 7.9559

.93916 1.6170 2.6106 3.4817 4.4844 6.0569 7.4711

.96175 1.6091 2.5509 3.3738 4.3197 5.8018 7.1338

.97805 1.6022 2.5053 3.2927 4.1971 5.6129 6.8849

.99037 1.5961 2.4694 3.2296 4.1020 5.4671 6.6933

1.0000 1.5909 2.4403 3.1789 4.0260 5.3511 6.5411

10077 1.5863 2.4163 3.1373 3.9639 5.2565 6.4171

1.0141 1.5822 2.3959 3.1022 3.9117 5.1774 6.3136

1.0194 1.5788 2.3789 3.0729 3.8682 5.1114 6.2274

10

.500 .750 .900 .950 .975 .990 .995

.48973 1.4915 3.2850 4.9646 6.9367 10.044 12.826

.74349 1.5975 2.9245 4.1028 5.4564 7.5394 9.4772

.84508 1.6028 2.7277 3.7083 4.8256 6.5523 8.0807

.89882 1.5949 2.6053 3.4780 4.4683 5.9943 7.3428

.93193 1.5853 2.5216 3.3258 4.2361 5.6363 6.8723

.95436 1.5765 2.4606 3.2172 4.0721 5.3858 6.5446

.97054 1.5688 2.4140 3.1355 3.9498 5.2001 6.3025

.98276 1.5621 2.3772 3.0717 3.8549 5.0567 6.1159

.99232 1.5563 2.3473 3.0204 3.7790 4.9424 5.9676

1.0000 1.5513 2.3226 2.9782 3.7168 4.8492 5.8467

1.0063 1.5468 2.3016 2.9426 3.6645 4.7710 5.7456

1.0116 1.5430 2.2841 2.9130 3.6209 4.7059 5.6613

11

.500 .750 .900 .950 .975 .990 .995

.48644 1.4749 3.2252 4.8443 6.7241 9.6460 12.226

.73872 1.5767 2.8595 3.9823 5.2555 7.2057 8.9122

.83973 1.5798 2.6602 3.5674 4.6300 6.2167 7.6004

.89316 1.5704 2.5362 3.3567 4.2751 5.6683 6.8809

.92608 1.5598 2.4512 3.2039 4.0440 5.3160 6.4217

.94837 1.5502 2.3891 3.0946 3.8807 5.0692 6.1015

.96445 1.5418 2.3416 3.0123 3.7586 4.8861 5.8648

.97661 1.5346 2.3040 2.9480 3.6638 4.7445 5.6821

.98610 1.3284 2.2735 2.8962 3.5879 4.6315 5.5368

.99373 1.5230 2.2482 2.8536 3.5257 4.5393 5.4182

.99999 1.5181 2.2267 2.8176 3.4733 4.4619 5.3190

1.0052 1.5140 2.2087 2.7876 3.4296 4.3974 5.2363

12

.500 .750 .900 .950 .975 .990 .995

.48369 1.4613 3.1765 4.7472 6.5538 9.3302 11 .754

.73477 1.5595 2.8065 3.8853 5.0959 6.9265 8.5095

.83530 1.5609 2.6055 3.4903 4.4742 5.9526 7.2258

.88848 1.5503 2.4801 3.2592 4.1212 5.4119 6.5211

.92124 1.5389 2.3940 3.1059 3.8911 5.0643 6.0711

.94342 1.5286 2.3310 2.9961 3.7283 4.8206 5.7570

.95943 1 .5197 2.2828 2.9134 3.6065 4.6395 5.5245

.97152 1.5120 2.2446 2.8486 3.5118 4.4994 5.3451

.98097 1.5054 2.2135 2.7964 3.4358 4.3875 5.2021

.98856 1.4996 2.1878 2.7534 3.3736 4.2961 5.0855

.99480 1.4945 2.1658 2.7170 3.3211 4.2193 4.9888

1.0000 1.4902 2.1474 2.6866 3.2773 4.1553 4.9063

Valores Percentiles

, de la Distribución F. (Continuación)

Pág. 7

1

2

3

4

5

6

7

8

9

10

11

12

13

.500 .750 .900 .950 .975 .990 .995

.48141 1.4500 3.1362 4.6672 6.4143 9.0738 11.374

.73145 1.5452 2.7632 3.8056 4.9653 6.7010 8.1865

.83159 1.5451 2.5603 3.4105 4.3472 5.7394 6.9257

.88454 1.5336 2.4337 3.1791 3.9959 5.2053 6.2335

.91718 1.5214 2.3467 3.0254 3.7667 4.8616 5.7910

.93926 1.5105 2.2830 2.9153 3.6043 4.6204 5.4819

.9.5520 1.5011 2.2341 2.8321 3.4827 4.4410 5.2529

.96724 1.4931 2.1953 2.7669 3.3880 4.3021 5.0761

.97665 1.4861 2.1638 2.7144 3.3120 4.1911 4.9351

.98421 1.4801 2.1376 2.6710 3.2497 4.1003 4.8199

.99042 1.4746 2.1152 2.6343 3.1971 4.0239 4.7234

.99560 1.4701 2.0966 2.6037 3.1532 3.9603 4.6429

14

.500 .750 .900 .950 .975 .990 .995

.4 7944 1.4403 3.1022 4.6001 6 .2979 8 .8616 11.060

.72862 1.5331 2.7265 3.7389 4.8567 6.5149 7.9216

.82842 1.5317 2.5222 3.3439 4.2417 5.5639 6.6803

.88119 1.5194 2.3947 3.1122 3.8919 5.0334 5.9984

.91371 1.5066 2.3069 2.9582 3.6634 4.6950 5.5623

.93573 1.4952 2.2426 2.8477 3.5014 4.4558 5.2574

.95161 1.4854 2.1931 2.7642 3.3799 4.2779 5.0313

.96360 1.4770 2.1539 2.6987 3.2853 4.1399 4.8566

.97298 1.4697 2.1220 2.6458 3.2093 4.0297 4.7173

.98051 1.4634 2.0954 2.6021 3.1469 3.9394 4.6034

.98670 1.4577 2.0727 2.5651 3.0941 3.8634 4.5078

.99186 1.4530 2.0537 2.5342 3.0501 3.8001 4.4281

15

.500 .750 .900 .950 .975 .990 .995

.47775 1.4321 3.0732 4.5431 6 .1995 8.6831 10.798

.72619 1.5227 2.6952 3.6823 4.7650 6.3589 7.7008

.82569 1.5202 2.4898 3.2874 4.1528 5.4170 6.4760

.87830 1.5071 2.3614 3.0556 3.8043 4.8932 5.8029

.91073 1.4938 2.2730 2.9013 3.5764 4.5556 5.3721

.93267 1.4820 2.2081 2.7905 3.4147 4.3183 5.0708

.94850 1.4718 2.1582 2.7066 3.2934 4.1415 4.8473

.96046 1.4631 2.1185 2.6408 3.1987 4.0045 4.6743

.96981 1.4556 2.0862 2.5876 3.1227 3.8948 4.5364

.97732 1.4491 2.0593 2.5437 3.0602 3.8049 4.4236

.98349 1.4432 2.0363 2.5064 3.0073 3.7292 4.3288

.98863 1.4383 2.0171 2.4753 2.9633 3.6662 4.2498

16

.500 .750 .900 .950 .975 .990 .995

.47628 1.4249 3.0481 4.4940 6.1151 8.5310 10.575

.72406 1.5137 2.6682 3.6337 4.6867 6.2262 7.5138

.82330 1.5103 2.4618 3.2389 4.0768 5.2922 6.3034

.87578 1.4965 2.3327 3.0069 3.7294 4.7726 5.6378

.90812 1.4827 2.2438 2.8524 3.5021 4.4374 5.2117

.93001 1.4705 2.1783 2.7413 3.3406 4.2016 4.9134

.94580 1.4601 2.1280 2.6572 3.2194 4.0259 4.6920

.95773 1.4511 2.0880 2.5911 3.1248 3.8896 4.5207

.96705 1.4433 2.0553 2.5377 3.0488 3.7804 4.3838

.97454 1.4366 2.0281 2.4935 2.9862 3.6909 4.2719

.98069 1.4309 2.0048 2.4560 2.9332 3.6155 4.1778

.98582 1.4255 1.9854 2.4247 2.8590 3.5527 4.0994

17

.500 .750 .900 .950 .975 .990 .995

.4799 1.4386 3.0262 4.4513 6.0420 8.3997 10.384

.72219 1.5057 2.6446 3.5915 4.6189 6.1121 7.3536

.82121 1.5015 2.4374 3.1968 4.0112 5.1850 6.1556

.87357 1.4873 2.3077 2.9647 3.6648 4.6690 5.4967

.90584 1.4730 2.2103 2.8100 3.4379 4.3359 5.0746

.92767 1.4605 2.1524 2.6987 3.2767 4.1015 4.7789

.94342 1.4497 2.1017 2.6143 3.1556 3.9267 4.5594

.95532 1.4405 2.0613 2.5480 3.0610 3.7910 4.3893

.96462 1.4325 2.0284 2.4943 2.9849 3.6822 4.2535

.97209 1.4256 2.0009 2.4499 2.9222 3.5931 4.1423

.97823 1.4194 1.9773 2.4122 2.8691 3.5179 4.0488

.98334 1.4142 1.9577 2.3807 2.8249 3.4552 3.9709

18

.500 .750 .900 .950 .975 .990 .995

.47385 1.4130 3.0070 4.4139 5.9781 8.2854 10.218

.72053 1.4988 2.6239 3.5597 4.5597 6.0129 7.2148

.81936 1.4938 2.4160 3.1599 3.9539 5.0919 6.0277

.87161 1.4790 2.2858 2.9277 3.6083 4.5790 5.3746

.90381 1.4644 2.1958 2.7729 3.3820 4.2479 4.9560

.92560 1.4516 2.1296 2.6613 3.2209 4.0146 4.6627

.94132 1.4406 2.0785 2.5767 3.0999 3.8406 4.4448

.95319 1.4312 2.0379 2.5102 3.0053 3.7054 4.2759

.96247 1.4230 2.0047 2.4563 2.9291 3.5971 4.1410

.96993 l.4159 1.9770 2.4117 2.8664 3.5082 4.0305

.97606 1.4095 1.9532 2.3737 2.8132 3.4331 3.9374

.98116 1.4042 1.9333 2.3421 2.7689 3.3706 3.8599

19

.500 .750 .900 .950 .975 .990 .995

.47284 1.4081 2.9899 4.3808 5.9216 8.1850 10.073

.71906 1.4925 2.6056 3.5219 4.5075 5.9259 7.0935

.81771 1.4870 2.3970 3.1274 3.9034 5.0103 5.9161

.86987 1.4717 2.2663 2.8951 3.5587 4.5003 5.2681

.90200 1.4568 2.1760 2.7401 3.3327 4.1708 4.8526

.92375 1.4437 2.1094 2.6283 3.1718 3.9386 4.5614

.93944 1.4325 2.0580 2.5435 3.0509 3.7653 4.3448

.95129 1.4228 2.0171 2.4768 2.9563 3.6305 4.1770

.96056 1.4145 1.9836 2.4227 2.8800 3.5225 4.0428

.96800 1.4073 1.9557 2.3779 2.8173 3.4338 3.9329

.97411 1.4008 1.9317 2.3598 2.7640 3.3589 3.8403

.97920 1.3953 1.9117 2.3080 2.7186 3.2965 3.7631

20

.500 .750 .900 .950 .975 .990 .995

.47192 1.4037 2.9747 4.3513 3.8715 8.0960 9.9439

.71773 1.4870 2.5893 3.4928 4.4613 5.8489 6.9865

.81621 1.4808 2.3801 3.0984 3.8587 4.9382 5.8177

.86830 1.4652 2.2489 2.8661 3.5147 4.4307 5.1743

.90038 1.4500 2.1582 2.7109 3.2891 4.1027 4.7616

.92210 1.4366 2.0913 2.5990 3.1283 3.8714 4.4721

.93776 1.4252 2.0397 2.5140 3.0074 3.6987 4.2569

.94959 1.4153 1.9985 2.4471 2.9128 3.5644 4.0900

.95884 1.4069 1.9649 2.3928 2.8365 3.4567 3.9564

.96626 1.3995 1.9367 2.3479 2.7737 3.3682 3.8470

.97237 1.3928 1.9125 2.3096 2.7203 3.2934 3.7548

.97746 1.3873 1.8924 2.2776 2.6758 3.2311 3.6779

21

.500 .750 .900 .950 .975 .990 .995

.47108 1.3997 2.9609 4.3248 5.8266 8.0166 9.8295

.71653 1.4820 2.5746 3.4668 4.4199 5.7804 6.8914

.81487 1.4753 2.3649 3.0725 3.8188 4.8740 5.7304

.86688 1.4593 2.2333 2.8401 3.4754 4.3688 5.0911

.89891 1.4438 2.1423 2.6848 3.2501 4.0421 4.6808

.92060 1.4302 2.0751 2.5727 3.0895 3.8117 4.3931

.93624 1.4186 2.0232 2.4876 2.9686 3.6396 4.1789

.94805 1.4086 1.9819 2.4205 2.8740 3.3056 4.0128

.95728 1.4000 1.9480 2.3661 2.7977 3.3981 3.9799

.96470 1.3925 1.9197 2.3210 2.7348 3.3098 3.7709

.97079 1.3857 1.8953 2.2825 2.6813 3.2351 3.6790

.97587 1.3801 1.8750 2.2504 2.6368 3.1729 3.6024

22

.500 .750 .900 .950 .975 .990 .995

.47033 1.3961 2.9486 4.3009 5.7863 7.9454 9.7271

.71545 1.4774 2.5613 3.4434 4.3828 5.7190 6.8064

.61365 1.4703 2.3512 3.0491 3.7829 4.8166 5.6524

.86559 1.4540 2.2193 2.8167 3.4401 4.3134 5.0168

.89759 1.4382 2.1279 2.6613 3.2152 3.9880 4.6088

.91924 1.4244 2.0605 2.5491 3.0546 3.7583 4.3225

.93486 1.4126 2.0084 2.4638 2.9338 3.5867 4.1094

.94665 1.4025 1.9668 2.3965 2.8392 5.4530 3.9440

.95588 1.3937 1.9327 2.3419 2.7628 3.3458 3.8116

.96328 1.3861 1.9043 2.2967 2.6998 3.2576 3.7030

.96937 1.3792 1.8798 2.2580 2.6463 3.1830 3.6114

.97444 1.3735 1.8593 2.2258 2.6017 3.1209 3.5350

23

.500 .750 .900 .950 .975 .990 .995

.46965 1.3928 2.9374 4.2793 5.7498 7.8811 9.6348

.71446 1.4733 2.5493 3.4221 4.3492 5.6637 6.7300

.81255 1.4657 2.3387 3.0280 3.7505 4.7649 5.5823

.86442 1.4491 2.2065 2.7955 3.4083 4.2635 4.9500

.89638 1.4331 2.1149 2.6400 3.1835 3.9392 4.5441

.91800 1.4191 2.0472 2.5277 3.0232 3.7102 4.2591

.93360 1.4072 1.9949 2.4422 2.9024 3.5390 4.0469

.94538 1.3969 1.9531 2.3748 2.8077 3.4057 3.8822

.95459 1.3880 1.9189 2.3201 2.7313 3.2986 3.7502

.96199 1.3803 1.8903 2.2747 2.6682 3.2106 3.6420

.96807 1.3733 1.8656 2.2359 2.6146 3.1361 3.5506

.97313 1.3675 1.8450 2.2036 2.5699 3.0740 3.4745

24

.500 .750 .900 .950 .975 .990 .995

.46902 1.3898 2.9271 4.2597 5.7167 7.8229 9.5513

.71356 1.4695 2.5383 3.4028 4.3187 5.6136 6.6609

.81153 1.4615 2.3274 3.0088 3.7211 4.7181 5.5190

.86335 1.4447 2.1949 2.7763 3.3194 4.2184 4.8898

.89527 1.4285 2.1030 2.6207 3.1548 3.8951 4.4857

.91687 1.4143 2.0351 2.5082 2.9946 3.6667 4.2019

.93245 1.4022 1.9826 2.4226 2.8738 3.4959 3.9905

.94422 1.3918 1.9407 2.3551 2.7791 3.3629 3.8264

.95342 1.3826 1.9063 2.3002 2.7027 3.2560 3.6949

.96081 1.3750 1.8775 2.2547 2.6396 3.1681 3.5870

.96688 1.3680 1.8526 2.2158 2.5859 3.0936 3.4959

.97194 1.3621 1.8319 2.1834 2.5412 3.0316 3.4199

𝒑

Valores Percentiles

Pág. 8

, de la Distribución F. (Continuación)

5

6

7

8

9

10

11

12

.86236 1.4406 2.1843 2.7587 3.3530 4.1774 4.8351

.89425 1.4242 2.0922 2.6030 3.1287 3.8550 4.4327

.91583 1.4099 2.0241 2.4904 2.9685 3.6272 4.1500

.93140 1.3976 1.9714 2.4047 2.8478 3.4568 3.9394

.94315 1.3871 1.9292 2.3371 2.7531 3.3239 3.7758

.95234 1.3780 1.8947 2.2821 2.6766 3.2172 3.6447

.95972 1.3701 1.8658 2.2365 2.6135 3.1294 3.5370

.96579 1.3630 1.8409 2.1974 2.5597 3.0551 3.4461

.97084 1.3570 1.8200 2.1649 2.5149 2.9931 3.3704

.80975 1.4542 2.3075 2.9751 3.6697 4.6366 5.4091

.86145 1.4368 2.1745 2.7426 3.3289 4.1400 4.7852

.89331 1.4203 2.0822 2.5868 3.1048 3.8183 4.3844

.91487 1.4058 2.0139 2.4741 2.9447 3.5911 4.1027

.93042 1.3935 1.9610 2.3883 2.8240 3.4210 3.8928

.94217 1.3828 1.9188 2.3205 2.7293 3.2884 3.7797

.95135 1.3737 1.8841 2.2655 2.6528 3.1818 3.5989

.95872 1.3656 1.8550 2.2197 2.5895 3.0941 3.4916

.96478 1.3584 1.8299 2.1805 2.5357 3.0198 3.4008

.96983 1.3524 1.8090 2.1479 2.4909 2.9579 3.3252

.71124 1.4600 2.5106 3.3541 4.2421 5.4881 6.4885

.80894 1.4510 2.2987 2.9604 3.6472 4.6009 5.3611

.86061 1.4334 2.1655 2.7278 3.3067 4.1056 4.7396

.89244 1.4166 2.0730 2.5719 3.0828 3.7848 4.3402

.91399 1.4021 2.0045 2.4591 2.9228 3.5580 4.0594

.92952 1.3896 1.9515 2.3732 2.8021 3.3082 3.8501

.94126 1.3788 1.9091 2.3053 2.7074 3.2558 3.6875

.95044 1.3696 1.8743 2.2501 2.6309 3.1494 3.5571

.95779 1.3615 1.8451 2.2043 2.5676 3.0618 3.4499

.96385 1.3542 1.8199 2.1650 2.5137 2.9875 3.3594

.96889 1.3481 1.7989 2.1323 2.4688 2.9256 3.2839

.46697 1.3800 2.8939 4.1960 5.6096 7.6336 9.2838

.71059 1.4573 2.5028 3.3404 4.2205 5.4529 6.4403

.80820 1.4480 2.2906 2.9467 3.6264 4.5681 5.3170

.85983 1.4302 2.1571 2.7141 3.2863 4.0740 4.6977

.89164 1.4133 2.0645 2.5581 3.0625 3.7539 4.2996

.91317 1.3986 1.9959 2.4453 2.9027 3.5276 4.0197

.92869 1.3860 1.9427 2.3593 2.7820 3.3581 3.8110

.94041 1.3752 1.9001 2.2913 2.6872 3.2259 3.6487

.94958 1.3658 1.8652 2.2360 2.6106 3.1195 3.5186

.95694 1.3576 1.8359 2.1900 2.5473 3.0320 3.4117

.96298 1.3502 1.8106 2.1507 2.4934 2.9578 3.3213

.96802 1.3441 2.7895 2.1179 2.4484 2.8959 3.2460

29

.500 .750 .900 .950 .975 .990 .995

.46654 1.3780 2.8671 4.1830 5.5878 7.3976 9.2297

.70999 1.4547 2.4995 3.3277 4.2006 5.4205 6.3958

.80733 1.4452 2.2831 2.9340 3.6072 4.5378 5.2764

.85911 1.4272 2.1494 2.7014 3.2674 4.0449 4.6591

.89089 1.4102 2.0566 2.5454 3.0438 3.7254 4.2622

.91741 1.3953 1.9878 2.4324 2.8840 3.4995 3.9830

.92791 1.3826 1.9345 2.3463 2.7633 3.3302 3.7749

.93963 1.3717 1.8918 2.2782 2.6686 3.1982 3.6130

.94879 1.3623 1.8568 2.2229 2.5919 3.0920 3.4832

.95614 1.3541 1.8274 2.1768 2.5286 3.0045 3.3765

.96218 1.3468 1.8020 2.1374 2.4745 2.9303 3.2863

.96722 1.3404 1.7808 2.1045 2.4295 2.8685 3.2111

30

.500 .750 .900 .950 .975 .990 .995

.46616 1.3761 2.8807 4.1709 5.5675 7.5625 9.1797

.70941 1.4524 2.4887 3.3158 4.1821 5.3903 6.3547

.80689 1.4426 2.2761 2.9223 3.5894 4.5097 5.2388

.85844 1.4244 2.1422 2.6896 3.2499 4.0179 4.6233

.89019 1.4073 2.0492 2.5336 3.0265 3.6990 4.2276

.91169 1.3923 l.9803 2.4205 2.8667 3.4735 3.9492

.92719 1.3795 1.9269 2.3343 2.7460 3.3045 3.7416

.93889 1.3685 1.8841 2.2662 2.6513 3.1726 35801

.94805 1.3590 1.8490 2.2107 2.5746 3.0665 3.4503

.95540 1.3507 1.8195 2.1646 2.5112 2.9791 3.3440

.96144 1.3432 1.7940 2.1251 2.4571 2.9049 3.2538

.96647 1.3369 1.7727 2.0921 2.4120 2.8431 3.1787

40

.500 .750 .900 .950 .975 .990 .995

.46330 1.3626 2.8354 4.0848 5.4239 7.3141 8.8278

.70531 1.4355 2.4404 3.2317 4.0510 5.1785 6.0664

.80228 1 .4239 2.2261 2.8387 3.4633 4.3126 4.9759

.85357 1.4045 2.0909 2.6060 3.1261 3.8283 4.3738

.08516 1.3863 1.9960 2.4495 2.9037 3.5138 3.9860

.90654 1.3706 1.9269 2.3359 2.7444 3.2910 3.7129

.92197 1.3571 1.8725 2.2490 2.6238 3.1238 3.5088

.93361 1.3455 1.8289 2.1802 2.5289 2.9930 3.3498

.94272 1.3354 1.7929 2.1240 2.4519 2.8876 3.2270

.95003 1.3266 1.7627 2.0772 2.3862 2.8005 3.1167

.95604 1.3186 1.7365 2.0370 2.3337 2.7265 3.0275

.96104 l.3119 l.7146 2.0035 2.2882 2.6648 2.9531

48

.500 .750 .900 .950 .975 .990 .995

.46192 1.3560 2.8134 4.0430 5.3548 7.1956 8.6612

.70327 1.4272 7.4168 3.1911 3.9882 5.0780 5.9307

.79999 1.4147 2.2018 2.7984 3.4029 4.2193 4.8525

.85115 1.3947 2.0660 2.5656 3.0669 3.7387 4.2569

.88267 1.3760 1.9713 2.4089 2.8450 3.4264 3.8730

.90399 1.3599 1.9008 2.2950 2.6859 3.2049 3.6024

.91938 1.3460 1.8460 2.2078 2.5653 3.0384 3.4000

.93100 1.3341 1.8019 2.1386 2.4703 2.9082 3.2421

.94008 1.3237 1.7655 2.0821 2.3932 2.8031 3.1132

.94737 1.3146 1.7349 2.0349 2.3292 2.7162 3.0105

.95336 1.3063 1.7082 1.9943 2.2744 2.6421 2.9216

.95835 1.2995 1.6860 1.9605 2.2287 2.5805 2,8475

60

.500 .750 .900 .950 .975 .990 .995

.46053 1.3493 2.7914 4.0012 5.2857 7.0771 8.4946

.70122 1.4188 2.3933 3.1504 3.9253 4.9774 5.7950

.79770 1.4055 2.1774 2.7581 3.3425 4.1259 4.7290

.84873 1.3848 2.0410 2.5252 3.0077 3.6491 4.1399

.88017 1.3657 1.9457 2.3683 2.7863 3.3389 3.7600

.90144 1.3491 1.8747 2.2540 2.6274 3.1187 3.4918

.91679 1.3349 1.8194 2.1665 2.5068 2.9530 3.2911

.92838 1.3226 1.7748 2.0970 2.4117 2.8233 3.1344

.93743 1.3119 1.7380 2.0401 2.3344 2.7185 3.0083

.94471 1.3026 1.7070 1.9926 2.2702 2.6318 2.9042

.95068 1.2941 1.6799 1.9516 2.2151 2.5578 7.8157

.95566 1.2870 1.6574 1.9174 2.1692 2.4961 2.7419

80

.500 .750 .900 .950 .975 .990 .995

.45914 1.3428 2.7696 3.9607 5.2191 6.964l 8.3368

.69920 1.4106 2.3703 3.1111 3.8650 4.8820 5.6672

.79542 1.3964 2.1537 2.7192 3.2848 4.0375 4.6132

.84633 1.3751 2.0167 2.4862 2.9510 3.5644 4.0303

.87769 1.3555 1.9208 2.3292 2.7302 3.2562 3.6541

.89891 1.3385 1.8493 2.2145 2.5714 3.0373 3.3884

.91422 1.3239 1.7935 2.1266 2.4508 2.8724 3.1893

.92578 1.3113 1.7484 2.0567 2.3556 2.7431 3.0337

.93481 1.3003 1.7112 1.9995 2.2781 2.6386 2.9083

.94207 1.2907 1.6797 1.9516 2.2136 2.5520 2.8047

.94803 1.2819 1.6522 1.9101 2.1582 2.4779 2.7164

.95299 1.2745 1.6293 1.8756 2.1120 2.4162 2.6429

.500 .750 .900 120 .950 .975 .990 .995

.45774 1.3362 2.7478 3.9201 3.1524 6.8510 8.1790

.69717 1.4024 2.3473 3.0718 3.8046 4.7865 5.5393

.79314 1.3873 2.1300 2.6802 3.2270 3.9491 4.4973

.84392 1.3654 1.9923 2.4472 2.8943 3.4796 3.9207

.87521 1.3453 1.8959 2.2900 2.6740 3.1735 3.5482

.89637 1.3278 1.8238 2.1750 2.5154 2.9559 3.2849

.91164 1.3128 1.7675 2.0867 2.3948 2.7918 3.0874

.92318 1.2999 1.7220 2.0164 2.2994 2.6629 2.9330

.93218 1.2886 1.6843 1.9588 2.2217 2.5586 2.8083

.93943 1.2787 1.6524 1.9105 2.1570 2.4721 2.7052

.94937 1.2696 1.6245 1.8686 2.1013 2.3980 2.6172

.95032 1.2621 1.6012 1.8337 2.0548 2.3363 2.5439

.500 .750 .900 .950 .975 .990 .995

.45494 1.3233 2.7055 3.8415 5.0239 6.6349 7.8794

.69313 1.3863 2.3026 2.9957 3.6889 4.6052 5.2983

.78866 1.3694 2.0838 2.6049 3.1161 3.7816 4.2794

.83918 1.3463 1.9449 2.3719 2.7858 3.3192 3.7151

.87029 1.3251 1.8473 2.2141 2.5663 3.0173 3.3499

.89135 1.3068 1.7741 2.0986 2.4082 2.8020 3.0913

.90654 1.2910 1.7167 2.0096 2.2875 2.6393 2.8968

.91802 1.2774 1.6702 1.9384 2.1918 2.5113 2.7444

.92698 1.2654 1.6315 1.8799 2.1136 2.4073 2.6210

.93418 1.2549 1.5987 1.8307 2.0483 2.3209 2.5188

.94010 1.2452 1.5698 1.7879 1.9918 2.2467 2.4313

.94503 1.2371 1.5458 1.7522 1.9447 2.1846 2.3583

1

2

3

25

.500 .750 .900 .950 .975 .990 .995

.46844 1.3870 2,9177 4.2417 5.6864 7.7698 9.4753

.71272 1.4661 2.5283 3.3852 4.2909 5.5680 6.5982

.81061 1.4577 2.3170 2.9912 3.6943 4.6755 5.4615

26

.500 .750 .900 .950 .975 .990 .995

.46793 1.3845 2.9091 4.2252 5.6586 7.7213 9.4059

.71195 1.4629 2.5191 3.3690 4.2655 5.5263 6.5409

27

.500 .750 .900 .950 .975 .990 .995

.46744 1.3822 2.9012 4.2100 5.6331 7.6767 9.3423

28

.500 .750 .900 .950 .975 .990 .995

𝒑

∞

4

Valores Percentiles

Pág. 9

, de la Distribución F. (Continuación)

13

14

15

18

20

24

30

40

48

60

120

∞

1

.500 .750 .900 .950 .975 .990 .995

2.0773 9.4399 60.903 244.69 979.85 6123.9 24504

2.0858 9.4685 61.073 245.37 982.54 6142.7 24572

2.0931 9.4934 61.220 245.95 984.87 6157.3 24630

2.1104 9.5520 61.567 247.32 990.36 6191.6 24767

2.1190 9.5813 61.740 248.01 993.10 6208.7 24836

2.1321 9.6255 62.002 249.05 997.25 6234.6 24940

2.1452 9.6698 62.265 250.09 1001.4 6260.7 25044

2.1584 9.7144 62.529 251. 14 1005.6 6286.8 25148

2.1650 9.7368 62.662 251.67 1007.7 6299.9 25201

2.1716 9.7591 62.794 252.20 1009.8 6313.0 25253

2.1848 9.8041 63.061 253.25 1014.0 6339.4 25359

2.1981 9.8492 63.328 254.32 1018.3 6366.0 22463

2

.500 .750 .900 .950 .975 .990 .995

1.3672 3.3997 9.4145 19.419 39.421 99.422 199.42

1.3725 3.4051 9.4200 19.424 39.426 99.427 199.43

1.3771 3.4098 9.4247 19.429 39.431 99.432 199.43

1.3879 3.4208 9.4358 19.440 39.442 99.443 199.44

1.3993 3.4263 9.4413 19.446 39.448 99.449 199.45

1.4014 3.4365 9.6496 19.454 39.456 99.458 199.46

1.4096 3.4428 9.4579 19.462 39.465 99.466 199.47

1.4178 3.4511 9.4662 19.471 39.473 99.474 199.47

1.4220 3.4553 9.4705 19.475 39.477 99.478 199.47

1.4261 3.4594 9.4746 19.479 39.481 99.483 199.48

1.4344 3.4677 9.4829 19.487 39.490 99.491 199.49

1.4427 3.4761 9.4913 19.496 39.495 99.499 199.51

3

.500 .750 .900 .950 .975 .990 .995

1.2025 2.4520 5.2097 8.7286 14.305 26.983 43.271

1.2071 2.4537 5.2047 8.7148 14.277 26.923 43.171

1.2111 2.4552 5.2003 8.7029 14.253 26.872 43.085

1.2205 2.4585 5.1898 8.6744 14.196 26.751 47.880

1.2252 2.4602 5.1845 8.6602 14.167 26.690 42.778

1.2322 2.4626 5.1764 8.6385 14.124 26.598 42.622

1.2393 2.4650 5.1681 8.6166 14.081 26.505 42.466

1.2464 2.4674 5.1597 8.5944 14.037 26.41 1 42.308

1.2500 2.4686 5.1555 8.5832 14.015 26.364 42.229

1.2536 2.4697 5.1512 8.5720 13.992 26.316 42.149

1.2608 2.4720 5.1425 8.5494 13.947 26.221 41.989

1.2680 2.4742 5.1337 8.5265 13.902 26.125 41.829

4

.500 .750 .900 .950 .975 .990 .995

1.1305 2.0827 3.8553 5.8910 8.7148 14.306 20.602

1.1349 2.0828 3.8765 5.8732 8.6836 14.248 20.514

1.1386 2.0829 3.8689 5.8578 8.6565 14.198 20.438

1.1473 2.0828 3.8525 5.8209 8.5921 14.079 20.257

1.1517 2.0828 3.8443 5.8025 8.5599 14.020 20.167

1.1583 2.0827 3.8310 5.7744 8.5109 13.929 20.030

1.1649 2.0825 3.8174 5.7459 8.4613 13.838 19.892

1.1716 2.0821 3.8036 5.7170 8.4111 13.745 19.752

1.1749 2.0819 3.7966 5.7024 8.3858 13.699 19.682

1.1782 2.0817 3.7896 5.6878 8.3604 13.652 19.611

1.1849 2.0812 3.7753 5.6581 8.3092 13.558 19.468

1.1916 2.0806 3.7607 5.6281 8.2573 13.463 19.325

5

.500 .750 .900 .950 .975 .990 .995

1.0903 1.8867 3.2566 4.6550 6.4873 9.8244 13.292

1.0944 1.8858 3.2466 4.6356 6.4554 9.7697 13.214

1.0980 1.8851 3.2380 4.6188 6.4277 9.7222. 13.146

1.1064 1.8830 3.2171 4.5783 6.3616 9.6092 12.984

1.1106 1.8820 3.2067 4.5581 6.3285 9.5527 12.903

1.1170 1.8802 3.1905 4.5272 6.2780 9.4665 12.780

1.1234 1.8784 3.1741 4.4957 6.2269 9.3793 12.656

1.1297 1.8763 3.1573 4.4638 6.1751 9.2912 12.530

1.1329 1.8753 3.1488 4.4476 6.1488 9.2466 12.466

1.1361 1.8742 3.1402 4.4314 6.1225 9.2020 12.402

1.1426 1.8719 3.1228 4.3984 6.0693 9.1118 12.274

1.1490 1.8694 3.1050 4.3650 6.0153 9.0204 12.144

6

.500 .750 .900 .950 .975 .990 .995

1.0647 1.7650 2.8918 3.9761 5.3287 7.6570 9.9494

1.0687 1.7634 2.8808 3.9558 5.2966 7.6045 9.8769

1.0722 1.7621 2.8712 3.9381 3.2687 7.5590 9.8140

1.0804 1.7586 2.8479 3.8955 5.2018 7.4502 9.6639

1.0845 1.7569 2.8363 3.8742 5.1684 7.3938 9.5888

1.0907 1.7540 2.8183 3.8415 5.1172 7.3127 9.4741

1.0969 1.7510 2.8000 3.8082 5.0652 7.2285 9.3583

1.1031 1.7477 2.7812 3.7743 5.0125 7.1432 9.2408

1.1062 1.7460 2.7716 3.7571 4.9857 7.1000 9.1814

1.1093 1.7443 2.7620 3.7398 4.9589 7.0568 9.1219

1.1156 1.7407 2.7423 3.7047 4.9045 6.9690 9.0015

1.1219 1.7368 2.7222 3.6688 4.8491 6.8501 8.8793

7

.500 .750 .900 .950 .975 .990 .995

1.0469 1.6819 2.6543 3.5501 4.6281 6.4096 8.0962

1.0509 1.6799 2.6425 3.5291 4.5958 6.3585 8.0274

1.0543 1.6781 2.6322 3.5108 4.5678 6.3143 7.9678

1.0624 1.6735 2.6072 3.4666 4.5004 6.2004 7.8253

1.0664 1.6712 2.5947 3.4445 4.4667 6.1554 7.7540

1.0724 1.6675 2.5753 3.4105 4.4150 6.0743 7.6450

1.0785 1.6635 2.5555 3.3758 4.3624 5.9921 7.5345

1.0846 1.6593 2.5351 3.3404 4.3089 5.9084 7.4225

1.0877 1.6571 2.5247 3.3224 4.2817 5.8660 7.3657

1.0908 1.6548 2.5142 3.3043 4.2544 5.8236 7.3088

1.0969 1.6502 2.4928 3.2674 4.1989 5.7372 7.1933

1.1031 1.6452 2.4708 3.2298 4.1423 5.6495 7.0760

8

.500 .750 .900 .950 .975 .990 .995

1.00339 1.6216 2.4875 3.2588 4.1618 5.6085 6.9377

1.0378 1.6191 2.4750 3.2371 4.1293 5.5584 6.8716

1.0412 1.6170 2.4642 3.2184 4.1012 5.5151 6.8143

1.0491 1.6115 2.4378 3.1730 4.0334 5.4111 6.6769

1.0531 1.6088 2.4246 3.1503 3.9995 5.3591 6.6082

1.0591 1.6043 2.4041 3.1152 3.9472 5.2793 6.5029

1.0651 1.5996 2.3830 3.0794 3.8940 5.1981 6.3961

1.0711 1.5945 2.3614 3.0428 3.8398 5.1136 6.2875

1.0741 1.5919 2.3503 3.0241 3.8121 5.0736 6.2324

1.0771 1.5692 2.3391 3.0053 3.7844 5.0316 6.1772

1.0832 1.5836 2.3162 2.9669 3.7279 4.9460 6.0649

1.0893 1.5777 2.2926 2.9276 3.6702 4.8588 5.9505

9

.500 .750 .900 .950 .975 .990 .995

1.0239 1.5756 2.3638 3.0472 3.8302 5.0540 6.1524

1.0278 1.5729 2.3508 3.0252 3.7976 5.0048 6.0882

1.0311 1.5705 2.3396 3.0061 3.7694 4.9621 6.0325

1.0390 1.5642 2.3121 2.9597 3.7011 4.8594 5.8987

1.0429 1.5611 2.2983 2.9365 3.6669 4.8080 5.8318

1.0489 1.5560 2.2768 2.9005 3.6142 4.7290 5.7292

1.0548 1.5506 2.2547 2.8637 3.5604 4.6486 5.6248

1.0608 1.5450 2.2320 2.8259 3.5055 4.5667 5.5186

1.0638 1.5420 2.2203 2.8066 3.4774 4.5249 5.4645

1.0667 1.5389 2.2085 2.7872 3.4493 4.4831 5.4104

1.0727 1.5325 2.1843 2.7475 3.3918 4.3978 5.3001

1.0788 1.5257 2.1592 2.7067 3.3329 4.3105 5.1875

10

.500 .750 .900 .950 .975 .990 .995

1.0161 1.5395 2.2685 2.8868 3.5827 4.6491 5.5880

1.0199 1.5364 2.2551 2.8644 3.5500 4.6004 5.5252

1.0232 1.5338 2.2435 2.8450 3.5217 4.5582 5.4707

1.0310 1.5269 2.2150 2.7977 3.4530 4.4563 5.3396

1.0349 1.5235 2.2007 2.7740 3.4186 4.4054 5.2740

1.0408 1.5179 2.1784 2.7372 3.3654 4.3269 5.1732

1.0467 1.5119 2.1554 2.6996 3.3110 4.2469 5.0705

1.0526 1.5056 2.1317 2.6609 3.2554 4.1653 4.9659

1.0556 1.5023 2.1195 2.6410 3.2269 4.1236 4.9126

1.0585 1.4990 2.1072 2.6211 3.1984 4.0819 4.8592

1.0645 1.4919 2.0818 2.5801 3.1399 3.9965 4.7501

1.0705 1.4843 2.0554 2.5379 3.0798 3.9090 4.6385

11

.500 .750 .900 .950 .975 .990 .995

1.0097 1.5102 2.1927 2.7611 3.3913 4.3411 5.1642

1.0135 1.5069 2.1790 2.7383 3.3584 4.2928 5.1024

1.0168 1.5041 2.1671 2.7186 3.3299 4.2509 5.0489

1.0245 1.4967 2.1377 2.6705 3.2607 4.1496 4.9198

1.0284 1.4930 2.1230 2.6464 3.2261 4.0990 4.8352

1.0343 1.4869 2.1000 2.6090 3.1725 4.0209 4.7557

1.0401 1.4805 2.0762 2.5705 3.1176 3.9411 4.0543

1.0460 1.4737 2.0516 2.5309 3.0613 3.8596 4.5508

1.0490 1.4701 2.0389 2.5105 3.0324 3.8179 4.4979

1.0519 1.4664 2.0261 2.4901 3.0035 3.7761 4.4450

1.0578 1.4587 1.9997 2.4480 2.9441 3.6904 4.3367

1.0637 1.4504 1.9721 2.4045 2.8828 3.6025 4.2256

12

.500 .750 .900 .950 .975 .990 .995

1.0044 1.4861 2.1311 2.6598 3.2388 4.0993 4.8352

1.0082 1.4826 2.1170 2.6358 3.2038 4.0512 4.7742

1.0115 1.4796 2.1049 2.6169 3.1772 4.0096 4.7214

1.0192 1.4717 2.0748 2.5680 3.1076 3.9088 4.5937

1.0231 1.4678 2.0597 2.5436 3.0728 3.8584 4.5299

1.0289 1.4613 2.0360 2.5055 3.0187 3.7805 4.4315

1.0347 1.4544 2.0115 2.4663 2.9633 3.7008 4.3309

1.0405 1.4471 1.9861 2.4259 2.9063 3.6192 4.2282

1.0435 1.4432 1.9729 2.4051 2.8771 3.5774 4.1756

1.0464 1.4393 1.9597 2.3842 2.8478 3.5355 4.1229

1.0523 1.4310 1.9323 2.3410 2.7874 3.4494 4.0149

1.0582 l.4221 1.9036 2.2962 2.7249 3.3608 3.9039

𝒑

Valores Percentiles

Pág. 10

, de la Distribución F. (Continuación)

13

14

15

18

20

24

30

40

48

60

120

∞

13

.500 .750 .900 .950 .975 .990 .995

1.0000 1.4658 2.0799 2.5765 3.1145 3.9046 4.5726

1.0038 1.4622 2.0656 2.5583 3.0814 3.8568 4.5123

1.0071 1.4590 2.0532 2.5331 3.0527 3.8154 4.4600

1.0148 1.4507 2.0224 2.4836 2.9827 3.7149 4.3333

1.0186 1.4465 2.0070 2.4589 2.9477 3.6646 4.2703

1.0243 1.4397 1.9827 2.4202 2.8932 3.5868 4.1726

1.0301 1.4924 1.9576 2.3803 2.8373 3.5070 4.0727

1.0360 1.4247 1.9315 2.3397 2.7797 3.4253 3.9704

1.0389 1.4206 1.9179 2.9179 2.7501 3.3833 3.9180

1.0418 1.4164 1.9043 2.2966 2.7204 3.3413 3.8655

1.0476 1.4075 1.8759 2.2524 2.6990 3.2348 3.7577

1.0935 1.3980 1.8462 2.2064 2.5955 3.1654 3.8485

14

.500 .750 .900 .950 .975 .990 .995

.99626 1.4485 2.0367 2.5063 3.0113 3.7446 4.3584

1.0000 1.4447 2.0221 2.4833 2.9781 3.6970 4.2986

1.0033 1.4414 2.0095 2.4630 2.9493 3.6557 4.2468

1.0109 1.4327 1.9782 2.4129 2.8789 3.5554 4.1213

1.0147 1.4284 1.9625 2.3879 2.8437 3.5052 4.0585

1.0205 1.4212 1.9377 2.3487 2.7888 3.4274 3.9614

1.0263 1.4136 1.9119 2.3082 2.7324 3.3476 3.8619

1.0321 1.4055 1.8852 2.2664 2.6747 3.2656 3.7600

1.0350 1.4011 1.8712 2.2447 2.6442 3.2235 3.7077

1.0379 1.3967 1.8572 2.2230 2.6142 3.1813 3.6553

1.0437 1.3874 1.8280 2.3778 2.5519 3.0942 3.5473

1.0495 1.3772 1.7973 2.1307 2.4872 3.0040 3.4359

15

.500 .750 .900 .950 .975 .990 .995

.99300 1.4337 1.9998 2.4477 2.9244 3.6108 4.1806

.99675 1.4297 1.9850 2.4240 2.8910 3.5633 4.1212

1.0000 1.4263 1.9722 2.4033 2.8623 3.5222 4.0698

1.0076 1.4172 1.9403 2.3528 2.7913 3.4220 3.9450

1.0114 1.4127 1.9243 2.3275 2.7559 3.9789 3.8826

1.1720 1.4052 1.8990 2.2678 2.7006 3.2940 3.7859

1.0229 1.3973 1.8728 2.2468 2.6437 3.2141 3.6867

1.0287 1.3888 1.8454 2.2065 2.5850 3.1319 3.5850

1.0316 1.3847 1.8311 2.1022 2.5546 3.0895 3.5327

1.0345 1.3796 1.8168 2.1601 2.5242 3.0471 3.4803

1.0403 1.3698 1.7867 2.1141 2.4611 2.9505 3.3722

1.0461 1.3591 1.7551 2.0656 2.3953 2.8684 3.2602

16

.500 .750 .900 .950 .975 .990 .995

.99018 1.4207 1.9679 2.3968 2.8500 3.4500 4.0306

.99392 1.4166 1.9529 2.3729 2.8165 3.4500 3.9716

.99716 1.4130 1.9399 2.3522 2.7875 3.4089 3.9205

1.0048 1.4037 1.9075 2.3011 2.7164 3.3088 3.7963

1.0086 1.3990 1.8913 2.2756 2.6808 3.2588 3.7342

1.0143 1.3913 1.8656 2.2354 2.6252 3.1808 3.6378

1.0200 1.3630 1.8388 2.1938 2.5678 3.1007 3.5388

1.0258 1.3742 1.8108 2.1507 2.5085 3.0182 3.4372

1.0287 1.3694 1.7962 2.1283 2.4778 2.9756 3.3848

1.0315 1.3646 1.7816 2.1058 2.4471 2.9330 3.3324

1.0373 1.3543 1.7507 2.0589 2.3831 2.8447 3.7240

1.0431 1.3432 1.7182 2.0096 2.3163 2.7528 3.1115

17

.500 .750 .900 .950 .975 .990 .995

.98769 1.4093 1.9400 2.3526 2.7857 3.4000 3.9024

.99143 1.4051 1.9248 2.3286 2.7521 3.3527 3.8438

.99466 1.4014 1.9117 2.3077 2.7230 3.3117 3.7929

1.0022 1.3917 1.8788 2.2562 2.6515 3.2116 3.6692

1.0060 1.3869 1.8624 2.2304 2.6158 3.1615 3.6073

1.0117 1.3790 1.6362 2.1898 2.5598 3.0835 3.5112

1.0174 1.3704 1.8090 2.1477 2.5021 3.0032 3.4124

1.0232 1.3613 1.7805 2.1040 2.4422 2.9205 3.3107

1.0261 1.3564 1.7656 2.0812 2.4112 2.8777 3.2483

1.0289 1.3514 1.7506 2.0584 2.3801 2.8348 3.2058

1.0347 1.3406 1.7191 2.0107 2.3157 2.7459 3.0971

1.0403 1.3290 1.6456 1.9604 2.2474 2.6530 2.9839

18

.500 .750 .900 .950 .975 .990 .995

.98550 1.3992 1.9154 2.3138 2.7296 3.3153 3.7917

.98922 1.3948 1.9001 2.2896 2.6959 3.2682 3.7333

.99245 1.3911 1.8868 2.2686 2.6667 3.2273 3.6827

1.0000 1.3812 1.8533 2.2166 2.5949 3.1272 3.5594

1.0038 1.3762 1.8368 2.1906 2.5590 3.0771 3.4977

1.0095 1.3680 1.8103 2.1497 2.5027 2.9990 3.4017

1.0152 1.3592 1.7827 2.1071 2.4445 2.9185 3.3030

1.0209 1.3497 1.7537 2.0629 2.3842 2.8354 3.2014

1.0238 1.3446 1.7385 2.0398 2.3528 2.7924 3.1488

1.0267 1.3395 1.7232 2.0166 2.3214 2.7493 3.0962

1.0324 1.3284 1.6910 1.9681 2.2558 2.6597 2.9871

1.0382 1.3262 1.6567 1.9168 2.1869 2.5660 2.8732

19

.500 .750 .900 .950 .975 .990 .995

.95354 1.3901 1.8936 2.2796 2.6602 3.2414 3.6952

.98725 1.3857 1.8781 2.2552 2.6464 3.1942 3.6370

.99047 1.3819 1.8647 2.2341 2.6171 3.1593 3.5866

.99802 1.9717 1.8310 2.1817 2.5450 3.0532 3.4633

1.0018 1.3666 1.8142 2.1555 2.5089 3.0031 3.4020

1.0075 1.3582 1.7873 2.1141 2.4523 2.9249 3.3062

1.0132 1.3492 1.7592 2.0712 2.3937 2.8442 3.2075

1.0189 1.3394 1.7298 2.0264 2.3329 2.7608 3.1058

1.0218 1.3342 1.7143 2.0030 2.3012 2.7173 3.0531

1.0246 1.3289 1.6988 1.9796 2.2693 2.6742 3.0004

1.0304 1.3174 1.6659 1.9302 2.2032 2.5839 2.8908

1.0361 1.3048 1.6308 1.8780 2.1333 2.4893 2.7762

20

.500 .750 .900 .950 .975 .990 .995

.98178 1.3820 1.8741 2.2490 2.6363 3.1761 3.6102

.98549 1.3775 1.8585 2.2245 2.6024 3.1289 3.5523

.98870 1.3736 1.5449 2.2033 2.5731 3.0880 3.5020

.99623 1.3632 1.8108 2.1506 2.5007 2.9878 3.3792

1.0000 1.3580 1.7938 2.1242 2.4645 2.9377 3.3178

1.0057 1.3494 1.7667 2.0825 2.4076 2.8594 3.2220

1.0114 1.3401 1.7382 2.0391 2.3486 2.7785 3.1234

1.0171 1.3301 1.7083 1.9938 2.2873 2.6947 3.0215

1.0200 1.3247 1.6926 1.9701 2.2554 2.6512 2.9687

1.0228 1.3193 1.6768 1.9464 2.2234 2.6077 2.9159

1.0285 1.3074 1.6433 1.8963 2.1562 2.5168 2.8058

1.0343 1.2943 1.6074 1.8432 2.0853 2.4212 2.6904

21

.500 .750 .900 .950 .975 .990 .995

.98019 1.3747 1.8566 2.2217 2.5972 3.1179 3.5349

.98389 1.3701 1.6409 2.1970 2.5632 3.0708 3.4771

.98710 1.3661 1.8272 2.1757 2.5338 3.0299 3.4270

.99462 1.3555 1.7928 2.1226 2.4611 2.9297 3.3044

.99838 1.3502 1.7756 2.0960 2.4247 2.8796 3.2431

1.0040 1.3414 1.7481 2.0540 2.3675 2.8011 3.1474

1.0097 1.3319 1.7193 2.0102 2.3082 2.7200 3.0488

1.0154 1.3217 1.6890 1.9645 2.2465 2.6359 2.9467

1.0183 1.3161 1.6730 1.9495 2.2142 2.5922 2.8938

1.0211 1.3105 1.6569 1.9165 2.1819 2.5484 2.8408

1.0268 1.2983 1.6228 1.8657 2.1141 2.4568 2.7302

1.0326 1.2848 1.5862 1.8127 2.0422 2.3603 2.6140

22

.500 .750 .900 .950 .975 .990 .995

.97875 1.3680 1.8408 2.1970 2.5620 3.0659 3.4677

.98245 1.3634 1.8249 2.1722 2.5279 3.0188 3.4100

.98565 1.3593 1.8111 2.3508 2.4984 2.9780 3.3600

.99316 1.3485 1.7764 2.0974 2.4255 2.8776 3.2376

.99692 1.3431 1.7590 2.0707 2.3890 2.8274 3.1764

1.0026 1.3341 1.7312 2.0283 2.3315 2.7488 3.0807

1.0082 1.3245 1.7021 1.9842 2.2718 2.6675 2.9821

1.0139 1.3140 1.6714 1.9380 2.2097 2.5831 2.8799

1.0168 1.3083 1.6552 1.9138 2.1772 2.5391 2.8268

1.0196 1.3025 1.6389 1.8895 2.1446 2.4951 2.7736

1.0253 1.2900 1.6042 1.8380 2.0760 2.4029 2.6625

1.0311 1.2761 1.5668 1.7831 2.0032 2.3055 2.5455

23

.500 .750 .900 .950 .975 .990 .995

.97744 1.3620 1.8263 2.1746 2.5301 3.0190 3.4073

.98113 1.3572 1.8103 2.1497 2.4960 2.9719 3.3498

.98433 1.3531 1.7964 2.1282 2.4665 2.9311 3.2999

.99183 1.3422 1.7614 2.0745 2.3933 2.8307 3.1776

.99558 1.3366 1.7439 2.0476 2.3567 2.7805 3.1165

1.0012 1.3275 1.7159 2.0050 2.2989 2.7017 3.0208

1.0069 1.3176 1.6864 1.9605 2.2389 2.6202 2.9221

1.0126 1.3069 1.6554 1.9139 2.1763 2.5355 2.8198

1.0155 1.3011 1.6389 1.8894 2.1435 2.4913 2.7665

1.0183 1.2952 1.6224 1.8649 2.1107 2.4471 2.7132

1.0240 1.2824 1.5871 1.8128 2.0415 2.3542 2.6016

1.0297 1.2681 1.5490 1.7570 1.9677 2.2559 2.4837

24

.500 .750 .900 .950 .975 .990 .995

.97624 1.3564 1.8131 2.1543 2.5013 2.9766 3.3529

.97993 1.3516 1.7970 2.1293 2.4671 2.9295 3.2954

.98312 1.3474 1.7031 2.1077 2.4374 2.8887 3.2456

.99061 1.3363 1.7478 2.0537 2.3640 2.7882 3.1235

.99436 1.3307 1.7302 2.0267 2.3273 2.7380 3.0624

1.0000 1.3214 1.7019 1.9838 2.2693 2.6591 2.9667

1.0057 1.3113 1.6721 1.9390 2.2090 2.5773 2.8679

1.0113 1.3004 1.6407 1.8920 2.1460 2.4923 2.7654

1.0142 1.2945 1.6240 1.8672 2.1130 2.4479 2.7120

1.0170 1.2885 1.6073 1.8424 2.0799 2.4035 2.6585

1.0227 1.2754 1.5715 1.7897 2.0099 2.3099 2.5463

1.0284 1.2607 1.5327 1.7331 1.9353 2.2107 2.4276

𝒑

Valores Percentiles

Pág. 11

, de la Distribución F. (Continuación)

13

14

15

18

20

24

30

40

48

60

120

∞

25

.500 .750 .900 .950 .975 .990 .995

.97514 1.3513 1.8011 2.1357 2.4749 2.9381 3.3034

.97882 1.3464 1.7849 2.1106 2.4407 2.8910 3.2460

.98201 1.3422 1.7708 2.0889 2.4110 2.8502 3.1963

.98950 1.3309 1.7353 2.0346 2.3373 2.7496 3.0743

.99324 1.3252 1.7175 2.0075 2.3005 2.6993 3.0133

.99887 1.3158 1.6890 1.9643 2.2422 2.6203 2.9176

1.0045 1.3056 1.6589 1.9192 2.1816 2.9383 2.8187

1.0102 1.2945 1.6272 1.8718 2.1183 2.4530 2.7160

1.0131 1.2884 1.6103 1,8468 1.0850 2.4084 2.6624

1.0159 1.2823 1.5934 1.8217 2.0517 2.3637 2.6088

1.0219 1.2689 1.5570 1.7684 1.9811 2.2695 2.4960

1.0273 1.2538 1.5176 1.7110 1.9055 2.1694 2.3765

26

.500 .750 .900 .950 .975 .990 .995

.97412 1.3466 1.7900 2.1186 2.4508 2.9029 3.2584

.97780 1.3417 1.7737 2.0934 2.4165 2.8558 3.2011

.98099 1.3374 1.7596 2.0716 2.3667 2.8150 3.1515

.98846 1.3259 1.7238 2.0171 2.3128 2.7143 3.0295

.99220 1.3202 1.7059 1.9898 2.2759 2.6640 2.9685

.99783 1.3106 1.6771 1.9464 2.2174 2.5848 1.8728

1.0035 1.3002 1.6468 1.9010 2.1565 2.5026 2.7738

1.0091 1.2886 1.6147 1.8533 2.0928 2.4170 2.6709

1.0120 1.2827 1.5876 1.8280 2.0593 2.3722 2.6171

1.0148 1.2765 1.5805 1.8027 2.0257 2.3273 2.5633

1.0205 1.2628 1.5437 1.7488 1.9545 2.2325 2.4501

1.0262 1.2474 1.5086 1.6906 1.8781 2.1315 2.3297

27

.500 .750 .900 .950 .975 .990 .995

.97318 1.3423 1.7798 2.1029 2.4286 2.8706 3.2172

.97685 1.3372 1.7634 2.0777 2.3942 2.8235 3.1600

.98004 1.3329 1.7492 2.0558 2.3644 2.7827 3.1104

.98751 1.3213 1.7131 2.0010 2.2903 2.6820 2.9885

.99125 1.3155 1.6951 1.9736 2.2533 2.6316 2.9275

.99687 1.3058 1.6662 1.9299 2.1946 2.5522 2.8328

1.0025 1.2953 1.6356 1.8842 2.1334 2.4699 2.7327

1.0082 1.2838 1.6032 1.8361 2.0693 2.3840 2.6296

1.0110 1.2775 1.5859 1.8106 2.0356 2.3389 2.5757

1.0138 1.2712 1.5686 1.7851 2.0018 2.2938 2.5217

1.0195 1.2572 1.5313 1.7307 1.9299 2.1984 2.4078

1.0252 1.2414 1.4906 1.8717 1.8527 2.0965 2.2867

28

.500 .750 .900 .950 .975 .990 .995

.97231 1.3382 1.7703 2.0884 2.4082 2.8409 3.1793

.97598 1.3332 1.7538 2.0630 2.3737 2.7938 3.1222

.97917 1.3288 1.7395 2.0411 2.3438 2.7530 3.0727

.98663 1.3171 1.7033 1.9861 2.2695 2.6521 2.9508

.99036 1.3112 1.6852 1.9586 2.2324 2.6017 2.8899

.99598 1.3013 1.6560 1.9147 2.1735 2.5223 2.7941

1.0016 1.2906 1.6252 1.8687 2.1121 2.4397 2.6949

1.0073 1.2790 1.5925 1.8203 2.0477 2.3535 2.5916

1.0101 1.2726 1.5750 1.7946 2.0137 2.3082 2.5875

1.0129 1.2662 1.5575 1.7689 1.9796 2.2629 2.4834

1.0286 1.2519 1.5198 1.7138 1.9072 2.1670 2.3689

1.0243 1.2358 1.4754 1.6541 1.8291 2.0642 2.2469

29

.500 .750 .900 .950 .975 .990 .995

.97180 1.3344 1.7615 2.0749 2.3892 2.8135 3.1445

.97517 1.3293 1.7449 2.0495 2.3547 2.7664 3.0874

.97835 1.3249 1.7306 2.0275 2.3248 2.7256 3.0379

.98581 1.3130 1.6941 1.9722 2.2503 2.6247 2.9160

.98954 1.3071 1.6759 1.9446 2.2131 2.5742 2.8551

.99515 1.2971 1.6465 1.9005 2.1540 2.4946 2.7594

1.0008 1.2863 1.6155 1.8543 2.0926 2.4118 2.6601

1.0064 1.2745 1.5825 1.8055 2.0276 2.3253 2.5565

1.0093 1.2680 1.5649 1.7796 1.9934 2.2799 2.5022

1.0121 1.2615 1.5472 1.7537 1.9591 2.2344 2.4479

1.0177 1.2470 1.5090 1.6981 1.8691 2.1378 2.3330

1.0234 1.2306 1.4670 1.6377 1.6072 2.0342 2.2102

30

.500 .750 .900 .950 .975 .990 .995

.97075 1.3309 1.7533 2.0624 2.3717 2.7881 3.1122

.97441 1.3258 1.7367 2.0369 2.3371 2.7410 3.0551

.97759 1.3213 1.7223 2.0148 2.3072 2.7002 3.0057

.98504 1.3093 1.6856 1.9594 2.2325 2.5992 2.8839

.98877 1.3033 1.6673 1.9317 2.1952 2.5487 2.8230

.99438 1.2933 1.6377 1.8874 2.1359 2.4689 2.7272

1.0000 1.2823 1.6065 1.8409 2.0739 2.3860 2.6278

1.0056 1.2703 1.5732 1.7918 2.0089 2.2992 2.5241

1.0085 1.2637 1.5554 1.7657 1.9745 2.2536 2,4996

1.0113 1.2571 1.5376 1.7396 1.9400 2.2079 2.4151

1.0170 1.2424 1.4989 1.6835 1.8664 2.1107 2.7997

1.0226 1.2256 1.4564 1.6223 1.7867 2.0062 2.1760

40

.500 .750 .900 .950 .975 .990 .995

.96530 1.3055 1.6945 1.9731 2.2473 2.6097 2.8869

.96895 1.3000 1.6773 1.9471 2.2123 2.5625 2.8302

.97952 1.2823 1.6624 1.9245 2.1819 2.5216 2.7811

.97952 1.2823 1.6243 1.8674 2.1058 2.4198 2.6593

.98323 1.2758 1.6052 1.8389 2.0677 2.3689 2.5984

.98880 1.2649 1.5741 1.7929 2.0069 2.2880 2.5020

.99440 1.2529 1.9411 1.7444 1.9429 2.2034 2.4015

1.0000 1.2397 1.5056 1.6928 1.8752 2.1142 2.2958

1.0028 1.2323 1.4864 1.6651 1.8390 2.0668 2.2398

1.0056 1.2249 1.4672 1.6373 1.8028 2.0194 2.1838

1.0113 1.2080 1.4248 1.5766 1.7742 1.9172 2.0635

1.0169 1.1803 1.3769 1.5089 1.6871 1.8047 1.9318

48

.500 .750 .900 .950 .975 .990 .995

.96260 1.2928 1.6656 1.9297 2.1875 2.5293 2.7815

.96624 1.2871 1.6481 1.9033 2.1522 2.4780 2.7249

.96939 1.2822 1.6329 1.8005 2.1216 2.4370 2.6758

.97678 1.2687 1.5939 1.8225 2.0446 2.3346 2.5538

.98048 1.2620 1.5744 1.7935 2.0061 2.2834 2.4928

.98604 1.7505 1.5424 1.7465 1.9443 2.2017 2.3959

.99162 1.2379 1.5083 1.6968 1,8791 2.1160 2.2945

.99721 1.2239 1.4715 1.6436 1.8096 2.0251 2.1874

1.0000 1.2160 1.4513 1.6147 1.7722 1.9765 2.1302

1.0028 1.2081 1.4312 1.5858 1.7348 1.9279 2.0730

1.0085 1.1898 1.3862 1.5220 1.6526 1.8218 1.9488

1.0141 1.1679 1.3342 1.4491 1.5597 1.7027 1.8102

60

.500 .750 .900 .950 .975 .990 .995

.95990 1.2801 1.6366 1.8862 2.1277 2.4408 2.6760

.96352 1.2742 1.6188 1.8595 2.0921 2.3934 2.6195

.96667 1.2691 1.6034 1.8364 2.0613 2.3523 2.5705

.97404 1.2551 1.3635 1.7775 1.9834 2.2493 2.4483

.97773 1.2481 1.5435 1.7480 1.9445 2.1978 2.3872

.98328 1.2361 1.5107 1.7001 1.8817 2.1154 2.2898

.98884 1.2229 1.4755 1.6491 1.8152 2.0285 2.1874

.99441 1.2081 1.4373 1.5943 1.7440 1.9360 2.0789

.99721 1.1997 1.4163 1.5643 1.7054 1.8862 2.0706

1.0000 1.1912 1.3952 1.5343 1.6668 1.8363 1.9622

1.0056 1.1715 1.3676 1.4673 1.5810 1.7263 1.8341

1.0112 1.1474 1.2915 1.3893 1.4822 1.6006 1.6885

80

.500 .750 .900 .950 .975 .990 .995

.95722 1.2674 1.6081 1.8440 2.0701 2.3607 2.5770

.96084 1.7613 1.5899 1.8169 2.0343 2.3131 2.5205

.96398 1.2560 1.5742 1.7935 2.0037 2.2719 2.4716

.97133 1.2414 1.5333 1.7374 1.9242 2.1681 2.3490

.97501 1.2341 1.5128 1.7034 1.8847 2.1162 2.2877

.98054 1.2215 1.4790 1.6543 1.8707 2.0327 2.1894

.98609 1.2075 1.4475 1.6017 1.7526 1.9443 2.0857

.99164 1.1917 1.4025 1.5448 1.6791 1.8494 1.9749

.99443 1.1025 1.3801 1.5132 1.6387 1.7977 1.9147

.99722 1.1734 1.3578 1.4817 1.5984 1.7460 1.8546

1.0028 1.1515 1.3061 1.4096 1.5069 1.6297 1.7198

1.0084 1.1231 1.2471 1.3216 1.3963 1.4906 1.5598

120

.500 .750 .900 .950 .975 .990 .995

.95454 1.2547 1.5796 1.8017 2.0126 2.2806 2.4781

.95815 1.2483 1.5611 1.7743 1.9764 2.2329 2.4216

.96128 1.2428 1.5430 1.7505 1.9450 2.1915 2.3727

.96861 1.2276 1.5031 1.6893 1.8649 2.0869 2.2496

.97228 1.2200 1.4821 1.6587 1.8249 2.0346 2.1881

.97780 1.2068 1.4472 1.6084 1.7597 1.9500 2.0890

.98333 1.1921 1.4094 1.5543 1.6899 1.8600 1.9839

.98887 1.1752 1.3676 1.4952 1.6141 1.7628 1.8709

.99165 1.1654 1.3440 1.4621 1.5720 1.7093 1.8089

.99443 1.1555 1.3203 1.4290 1.5299 1.6557 1.7469

1.0000 1.1314 1.2646 1.3519 1.4328 1.5330 1.6099

1.0056 1.0987 1.1926 1.2539 1.3104 1.3805 1.4311

∞

.500 .750 .900 .950 .975 .990 .995

.94922 1.2291 1.5232 1.7192 1.9016 2.1285 2.2923

.95282 1.2222 1.5039 1.6909 1.8646 2.0803 2.2358

.95593 1.2163 1.4871 1.6664 1.8376 2.0385 2.1868

.96327 1.1997 1.4428 1.6025 1.7499 1.9317 2.0621

.96687 1.1914 1.4206 1.5705 1.7085 1.8783 1.9998

.97236 1.1767 1.3832 1.5173 1.6402 1.7908 1.8983

.97787 1.1600 1.3419 1.4591 1.5660 1.6964 1.7891

.98339 1.1404 1.2951 1.3940 1.4835 1.5923 1.6691

.98615 1.1284 1.2676 1.3560 1.4359 1.5327 1.6008

.98891 1.1164 1.2400 1.3180 1.3883 1.4730 1.5325

.99445 1.0838 1.1686 1.2214 1.2684 1.3246 1.3637

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

𝒑