formulario estadistica
DESCRIPTION
Formulario EstadisticaTRANSCRIPT
1
COMBIN
ATORIA
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B B B B B B B B B B B BN
interviene
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QQs
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no
si
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si no
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si no
si
Cn,r=( n r
) =n!
r!(n
−r)!
CR
n,r=( n
+r−1
r
) =(n
+r−
1)!
r!(n
−1)!
Vn,r=
n×(n
−1)
×···×
(n−r+1)
Pn=
n!
VR
n,r=
nr
PR
n1,n
2,...,n
rn
=n!
n1!×
n2!×
···×
nr!
2
Distribucion F. de densidad F. Caracterıstica Esperanza Varianza
Bernoulli B(1, p) pxq1−x x = 0, 1 q + peit p pq
Binomial B(n, p)
(nx
)pxqn−x x = 0, 1, . . . , n (q + peit)n np npq
Poisson P(λ)λx
x!e−λ x = 0, 1, . . . eλ(e
it−1) λ λ
Hipergeometrica H(n,N,A)
(Kx
)(N −An− x
)(Nn
) x = 0, 1, . . . , n nA
N= np
n(N − n)pq
N − 1
Geometrica G(p) pqx x = 0, 1, . . .p
1− qeitq
p
q
p2
Binomial Negativa BN(r, p)
(x+ r − 1
x
)prqx x = 0, 1, . . .
pr
(1− qeit)rq
pr
q
p2r
Uniforme U(a, b)1
b− aa < x < b
eibt − eiat
i(b− a)t
a+ b
2
(b− a)2
12
Normal N(µ, σ)1
σ√2π
e−1
2
(x− µ
σ
)2
x ∈ R eitµ− 1
2t2σ2
µ σ2
Exponencial Exp(λ) λe−λx x ≥ 0λ
λ− it
1
λ
1
λ2
Erlang Er(n, λ)λn
Γ(n)xn−1 e−λx x ≥ 0
(λ
λ− it
)nn
λ
n
λ2
Hipergeometrica
Binomial
Normal Poisson
n << N (n ≤ 0.10N)
p =A
N
0.10 < p < 0.90
µ = n p
σ =√
n p (1− p)
np ≥ 5, p ≤ 0.10
λ = n p
?
? ?
3
Distribucion
F.dedensidad
F.Caracterıstica
Esperanza
Varianza
Log-Normal
Log
-N(µ,σ
)1
xσ√2π
e−1 2
( Lx−µ
σ
) 2x≥
0eµ
+1 2σ2
(eσ2
−1)e
2µ+σ2
Gam
maG(p,q)
qp Γ(p)xp−1e−
qx
x≥
0
( q q−it
) pp q
p q2
WeibullW
(r,λ
)λrx
r−1e−
λxr
x≥
0λ−1/rΓ
( 1+
1 r
)λ−2/r
[ Γ
( 1+
2 r
) −Γ2
( 1+
1 r
)]
BetaB(p,q)
1
β(p,q)xp−1(1
−x)q
−1
0≤
x≥
1p
p+q
pq
(p+q)
2(p
+q+1)
Pearson
χ2 n
1
2n/2Γ( n 2
) xn/2−1e−
x/2
x≥
0(1
−2it)
−n/2
n2n
t-Studentt n
Γ
( n+1
2
)√nπΓ( n 2
)( 1+
x2 n
) −n+1
2x∈R
0(n
>1)
n
n−2
(n>
2)
F-Snedecor
Fn,m
nn/2m
m/2Γ
( n+m
2
)Γ( n 2
) Γ( m 2
)xn/2−1(m
+nx)−
n+
m
2x≥
0m
m−2
2m2(n
+m
−2)
n(m
−2)
2(m
−4)
Normal
Bidim
ension
alf(x,y)=
1
2πσ
Xσ
Y
√ 1−ρ2exp
{ −1
2(1−ρ2)
[ ( x−
µX
σX
) 2 −2ρ
( x−µ
X
σX
)( y−µ
Y
σY
) +
( y−µ
Y
σY
) 2]}
4Teorıa de la probabilidad
Probabilidad condicional
Definicion
P (S1|S) =P (S1 ∩ S)
P (S)siempre que P (S) > 0
Regla de la multiplicacion
Dados n sucesos, S1, . . . , Sn, se verifica
P
(n∩
i=1
Si
)= P (S1)P
(S2|S1
)P(S3|S1∩S2
)· · ·P
(Sn|S1∩S2∩···∩Sn−1
)Teorema de la probabilidad total
Dados un suceso A y n sucesos, S1, . . . , Sn, disjuntos dos a dos, Si∩Sj = ∅, tales quen∪
i=1
Si = E,
se verifica
P (A) =n∑
i=1
P(A|Si
)P (Si)
Teorema de Bayes
Dados un suceso A y n sucesos, S1, . . . , Sn, disjuntos dos a dos, Si∩Sj = ∅, tales quen∪
i=1
Si = E,
se verifica
P (Si|A) =P(A|Si
)P (Si)
n∑i=1
P(A|Si
)P (Si)
5Variable aleatoria unidimensional
Funcion de distribucion de una v.a. X
F (x) = P (X ≤ x) ∀x ∈ R
P (x1 < X ≤ x2) = F (x2)− F (x1) ∀x1, x2 ∈ R
Esperanza de una v.a. X
Dada una funcion T (X)
E[T (X)] =∑k
T (xk)P (X = xk) v.a. discreta
E[T (X)] =
∫ +∞
−∞T (x)f(x) dx v.a. continua
Momentos centrados en el origen de una v.a. X
mk = E[Xk] =∑n
xkn P (X = xn) v.a. discreta
mk = E[Xk] =
∫ +∞
−∞xkf(x) dx v.a. continua
Media y varianza de una v.a. X
µ = E[X] =
∑k
xk P (X = xk) v.a. discreta
∫ +∞
−∞xf(x) dx v.a. continua
σ2 = Var(X) = E[(X − µ)2]
Var(X) = E[X2]− (E[X])2 = E[X2]− µ2
Var(aX + b) = a2Var(X) ∀a, b ∈ R
Funcion caracterıstica
φ(t) = E[eitX ] =∑k
eitxkP (X = xk) v.a. discreta
φ(t) = E[eitX ] =
∫ +∞
−∞eitxf(x) dx v.a. continua
mr =1
irdrφ(t)
dtr
∣∣∣∣t=0
r = 1, 2, . . .
Cambio de variable
Dadas dos v.a. continuas relacionadas por una funcion continua y monotona Y = g(X)
fY (y) = fX
(g−1(y)
) ∣∣∣∣dxdy∣∣∣∣
6Variable aleatoria bidimensional
v.a. discreta (X,Y )
• Funcion de distribucion
F (xn, ym) = P (X ≤ xn, Y ≤ ym) =n∑
i=1
m∑j=1
P (X = xi, Y = yj)
• Distribucion marginal
FX(xn) = P (X ≤ xn, Y ≤ +∞) =
n∑i=1
∞∑j=1
P (X = xi, Y = yj) ∀xn
FY(ym) = P (X ≤ +∞, Y ≤ ym) =
∞∑i=1
m∑j=1
P (X = xi, Y = yj) ∀ym
• Probabilidad condicionada
P(X = xn|Y=ym
)=
P (X = xn, Y = ym)
P (Y = ym)
v.a. continua (X,Y )
• Funcion de distribucion
F (x, y) = P (X ≤ x, Y ≤ y) =
∫ x
−∞
∫ y
−∞f(x, y) dydx ∀x, y ∈ R
• Funcion de densidad marginal
X −→ fX(x) =
∫ +∞
−∞f(x, y) dy ∀x ∈ R
Y −→ fY(y) =
∫ +∞
−∞f(x, y) dx ∀ y ∈ R
• Funcion de densidad condicionada
X|Y=y −→ f(x|y) =f(x, y)
fY(y)
∀x ∈ R
Cambio de variable
v.a. bidimensional continua (X,Y ), con funcion de densidad conjunta
f(x, y) (x, y) ∈ Dxy
Transformacion biunıvoca
{U = u(X, Y )
V = v(X, Y )
Funcion de densidad conjunta de la v.a. bidimensional (U, V )
g(u, v) = f(x(u, v), y(u, v)) |J | (u, v) ∈ D∗uv
7
siendo
J =
∣∣∣∣∣∣∣∂x
∂u
∂x
∂v∂y
∂u
∂y
∂v
∣∣∣∣∣∣∣ =∣∣∣∣∣∣∣∣∂u
∂x
∂u
∂y∂v
∂x
∂v
∂y
∣∣∣∣∣∣∣∣−1
Jacobiano
Momentos centrados en el origen
E[XrY s] =∑i
∑j
xriy
sj P (X = xi, Y = yj) v.a. discreta
E[XrY s] =
∫ +∞
−∞
∫ +∞
−∞xrysf(x, y) dxdy v.a. continua
Media, varianza y covarianza
• v.a. discreta
µX= E[X] =
∞∑i=1
∞∑j=1
xi P (X = xi, Y = yj) =∞∑i=1
xi P (X = xi)
σ2X= Var(X) = E[(X − µ
X)2] =
∞∑i=1
(xi − µX)2 P (X = xi)
σXY
= Cov(X, Y ) = E[(X − µX)(Y − µ
Y)] =
∞∑i=1
∞∑j=1
(xi − µX)(yj − µ
Y)P (X = xi, Y = yj)
• v.a. continua
µX= E[X] =
∫ +∞
−∞
∫ +∞
−∞xf(x, y) dxdy =
∫ +∞
−∞xf
X(x) dx
σ2X= Var(X) = E[(X − µ
X)2] =
∫ +∞
−∞(x− µ
X)2f
X(x) dx
σXY
= Cov(X, Y ) = E[(X − µX)(Y − µ
Y)] =
∫ +∞
−∞
∫ +∞
−∞(x− µ
X)(y − µ
Y)f(x, y) dxdy
• Propiedades
Cov(X,Y ) = E[(X − µX)(Y − µ
Y)] = E[XY ]− E[X]E[Y ] = E[XY ]− µ
Xµ
Y
Var(aX ± bY ) = a2Var(X) + b2Var(Y )± 2abCov(X,Y )
8Distribucion Normal
Teorema de adicion
Sean X1 ≡ N(µ1, σ1), . . . , Xn ≡ N(µn, σn), n v.a. Normales independientes. Entonces, la nueva
variable aleatoria es
Y = b+ a1X1 + · · ·+ anXn ≡ N
(b+ a1µ1 + · · ·+ anµn,
√a21σ
21 + · · ·+ a2nσ
2n
)Normal estandar
X ≡ N(µ, σ) =⇒ Z =X − µ
σ≡ N(0, 1)
Teorema central del lımite
Dadas n v.a. independientes e identicamente distribuidas, {X1, . . . , Xn} tales que
E[Xk] = µ < +∞ y Var(Xk) = σ2 < +∞
Entonces, si n es suficientemente grande,
n∑k=1
Xk ≈ N(nµ, σ
√n)
Normal Bidimensional (X,Y ) ≡ NB(µX, σ
X, µ
Y, σ
Y, ρ)
• Coeficiente de correlacion lineal
ρ =Cov(X,Y )√
Var(X)√Var(Y )
=σ
XY
σXσ
Y
• Distribuciones marginales
X ≡ N(µX, σ
X) Y ≡ N(µ
Y, σ
Y)
• Distribuciones condicionadas
X|Y ≡ N(µ, σ)
µ = µ
X+ ρ
σX
σY
(y − µY)
σ = σX
√1− ρ2
Y |X ≡ N(µ, σ)
µ = µ
Y+ ρ
σY
σX
(x− µX)
σ = σY
√1− ρ2
9Teorıa de muestras
Estimacion de la mediaX ≡ N(µ, σ)
σ conocido
X − µ
σ/√n
≡ N(0, 1)
σ desconocidoX − µ
S/√n
≡ tn−1
X ≡ A(µ, σ) n > 30X − µ
S/√n
≡ N(0, 1)
Estimacion de la proporcion
X ≡ B(n, p) n > 30P − p√p(1− p)
n
≡ N(0, 1)
Estimacion de la varianza
X ≡ N(µ, σ)(n− 1)S2
σ2≡ χ2
n−1
Estimacion de la diferencia de medias
X ≡ N(µX , σX)
Y ≡ N(µY , σY )
σX , σX conocidas(X − Y )− (µX − µY )√
σ2X
nx+
σ2Y
ny
≡ N(0, 1)
σX , σY desconocidas
σ2X = σ2
Y
(X − Y )− (µX − µY )
Sp
√1
nx+
1
ny
≡ tnx+ny−2
σ2X = σ2
Y
(X − Y )− (µX − µY )√s2Xnx
+s2Yny
∼= tγ
siendo
Sp =
√(nx − 1)S2
X + (ny − 1)S2Y
nx + ny − 2γ =
(A+B)2
A2
nx − 1+
B2
ny − 1
A =S2X
nxB =
S2Y
ny
Estimacion del cociente de varianzas
X ≡ N(µX , σX) , Y ≡ N(µY , σY )S2X/σ2
X
S2Y /σ
2Y
≡ Fnx−1,ny−1
10Contrastes de bondad del ajuste
Test χ2 si no se han de estimar parametros
1. Se divide el recorrido de la distribucion poblacional en k conjuntos disjuntos o clases: A1, A2, · · · , Ak
2. Se calcula el numero ni de elementos de la muestra observados en cada clase Ai
3. Se calcula el numero ni,esp de elementos esperados en cada clase si la hipotesis H0 es cierta.
4. Se calcula el valor del estadıstico D =
k∑i=1
(ni − ni,esp)2
ni,esp= −n+
k∑i=1
n2i
ni,esp
5. D ≡ χ2k−1
Test χ2 si se han de estimar ν parametros
1. Se estiman los ν parametros a partir de la muestra empleando el criterio de maxima verosimi-
litud.
2. Se repite el proceso anterior, pero D ≡ χ2k−1−ν
Test de Kolmogorov-Smirnov
(P.M. continua y F0 completamenete especificada)
1. Se ordenan los valores muestrales x1 ≤ x2 ≤ x3 ≤ · · · ≤ xn y se calcula, para cada xh, el valor
∆n(xh) = max{|Fn(xh−1)− F0(xh)| , |Fn(xh)− F0(xh)|}
siendo Fn(x) =r
nxr ≤ x ≤ xr+1
2. El maximo de los n valores ası obtenidos es el estadıstico ∆n de Kolmogorov-Smirnov. La region
crıtica del test es de la forma ∆n > c.
Tablas de contingencia
H0 : pij = pipjB1 B2 · · · Bs Totales
A1 n11 n12 · · · n1s n1·
A2 n21 n22 · · · n2s n2·...
......
. . ....
...
Ar nr1 nr2 · · · nrs nr·
Totales n·1 n·2 · · · n·s n
χ2 =r∑
i=1
s∑j=1
(nij − ni· n·j/n)2
ni· n·j/n= n
−1 +r∑
i=1
s∑j=1
n2ij
ni· n·j/n
≡ χ2(r−1)(s−1)
11
Tablas
de
Distribuciones
12Tabla A.1: Distribucion Binomial. P (B(n, p) ≤ x) =
x∑k=0
(nk
)pk(1− p)n−k
p
n x .10 .20 .25 .30 .40 .50 .60 .70 .80 .90
1 0 .9000 .8000 .7500 .7000 .6000 .5000 .4000 .3000 .2000 .10001 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
2 0 .8100 .6400 .5625 .4900 .3600 .2500 .1600 .0900 .0400 .01001 .9900 .9600 .9375 .9100 .8400 .7500 .6400 .5100 .3600 .19002 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
3 0 .7290 .5120 .4219 .3430 .2160 .1250 .0640 .0270 .0080 .00101 .9720 .8960 .8438 .7840 .6480 .5000 .3520 .2160 .1040 .02802 .9990 .9920 .9844 .9730 .9360 .8750 .7840 .6570 .4880 .27103 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4 0 .6561 .4096 .3164 .2401 .1296 .0625 .0256 .0081 .0016 .00011 .9477 .8192 .7383 .6517 .4752 .3125 .1792 .0837 .0272 .00372 .9963 .9728 .9492 .9163 .8208 .6875 .5248 .3483 .1808 .05233 .9999 .9984 .9961 .9919 .9744 .9375 .8704 .7599 .5904 .34394 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
5 0 .5905 .3277 .2373 .1681 .0778 .0312 .0102 .0024 .0003 .00001 .9185 .7373 .6328 .5282 .3370 .1875 .0870 .0308 .0067 .00052 .9914 .9421 .8965 .8369 .6826 .5000 .3174 .1631 .0579 .00863 .9995 .9933 .9844 .9692 .9130 .8125 .6630 .4718 .2627 .08154 1.0000 .9997 .9990 .9976 .9898 .9688 .9222 .8319 .6723 .40955 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
6 0 .5314 .2621 .1780 .1176 .0467 .0156 .0041 .0007 .0001 .00001 .8857 .6554 .5339 .4202 .2333 .1094 .0410 .0109 .0016 .00012 .9841 .9011 .8306 .7443 .5443 .3438 .1792 .0705 .0170 .00133 .9987 .9830 .9624 .9295 .8208 .6562 .4557 .2557 .0989 .01594 .9999 .9984 .9954 .9891 .9590 .8906 .7667 .5798 .3446 .11435 1.0000 .9999 .9998 .9993 .9959 .9844 .9533 .8824 .7379 .46866 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
7 0 .4783 .2097 .1335 .0824 .0280 .0078 .0016 .0002 .0000 .00001 .8503 .5767 .4449 .3294 .1586 .0625 .0188 .0038 .0004 .00002 .9743 .8520 .7564 .6471 .4199 .2266 .0963 .0288 .0047 .00023 .9973 .9667 .9294 .8740 .7102 .5000 .2898 .1260 .0333 .00274 .9998 .9953 .9871 .9712 .9037 .7734 .5801 .3529 .1480 .02575 1.0000 .9996 .9987 .9962 .9812 .9375 .8414 .6706 .4233 .14976 1.0000 .9999 .9998 .9984 .9922 .9720 .9176 .7903 .52177 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
8 0 .4305 .1678 .1001 .0576 .0168 .0039 .0007 .0001 .0000 .00001 .8131 .5033 .3671 .2553 .1064 .0352 .0085 .0013 .0001 .00002 .9619 .7969 .6785 .5518 .3154 .1445 .0498 .0113 .0012 .00003 .9950 .9437 .8862 .8059 .5941 .3633 .1737 .0580 .0104 .00044 .9996 .9896 .9727 .9420 .8263 .6367 .4059 .1941 .0563 .00505 1.0000 .9988 .9958 .9887 .9502 .8555 .6846 .4482 .2031 .03816 .9999 .9996 .9987 .9915 .9648 .8936 .7447 .4967 .18697 1.0000 1.0000 .9999 .9993 .9961 .9832 .9424 .8322 .56958 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
9 0 .3874 .1342 .0751 .0404 .0101 .0020 .0003 .0000 .0000 .00001 .7748 .4362 .3003 .1960 .0705 .0195 .0038 .0004 .0000 .00002 .9470 .7382 .6007 .4628 .2318 .0898 .0250 .0043 .0003 .00003 .9917 .9144 .8343 .7297 .4826 .2539 .0994 .0253 .0031 .00014 .9991 .9804 .9511 .9012 .7334 .5000 .2666 .0988 .0196 .00095 .9999 .9969 .9900 .9747 .9006 .7461 .5174 .2703 .0856 .00836 1.0000 .9997 .9987 .9957 .9750 .9102 .7682 .5372 .2618 .05307 1.0000 .9999 .9996 .9962 .9805 .9295 .8040 .5638 .22528 1.0000 1.0000 .9997 .9980 .9899 .9596 .8658 .61269 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
10 0 .3487 .1074 .0563 .0282 .0060 .0010 .0001 .0000 .0000 .00001 .7361 .3758 .2440 .1493 .0464 .0107 .0017 .0001 .0000 .00002 .9298 .6778 .5256 .3828 .1673 .0547 .0123 .0016 .0001 .00003 .9872 .8791 .7759 .6496 .3823 .1719 .0548 .0106 .0009 .00004 .9984 .9672 .9219 .8497 .6331 .3770 .1662 .0473 .0064 .00015 .9999 .9936 .9803 .9527 .8338 .6230 .3669 .1503 .0328 .00166 1.0000 .9991 .9965 .9894 .9452 .8281 .6177 .3504 .1209 .01287 .9999 .9996 .9984 .9877 .9453 .8327 .6172 .3222 .07028 1.0000 1.0000 .9999 .9983 .9893 .9536 .8507 .6242 .26399 1.0000 .9999 .9990 .9940 .9718 .8926 .651310 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
13Tabla A.1: Distribucion Binomial (Continuacion)
p
n x .10 .20 .25 .30 .40 .50 .60 .70 .80 .90
11 0 .3138 .0859 .0422 .0198 .0036 .0005 .0000 .0000 .0000 .00001 .6974 .3221 .1971 .1130 .0302 .0059 .0007 .0000 .0000 .00002 .9104 .6174 .4552 .3127 .1189 .0327 .0059 .0006 .0000 .00003 .9815 .8389 .7133 .5696 .2963 .1133 .0293 .0043 .0002 .00004 .9972 .9496 .8854 .7897 .5328 .2744 .0994 .0216 .0020 .00005 .9997 .9883 .9657 .9218 .7535 .5000 .2465 .0782 .0117 .00036 1.0000 .9980 .9924 .9784 .9006 .7256 .4672 .2103 .0504 .00287 .9998 .9988 .9957 .9707 .8867 .7037 .4304 .1611 .01858 1.0000 .9999 .9994 .9941 .9673 .8811 .6873 .3826 .08969 1.0000 1.0000 .9993 .9941 .9698 .8870 .6779 .302610 1.0000 .9995 .9964 .9802 .9141 .686211 1.0000 1.0000 1.0000 1.0000 1.0000
12 0 .2824 .0687 .0317 .0138 .0022 .0002 .0000 .0000 .0000 .00001 .6590 .2749 .1584 .0850 .0196 .0032 .0003 .0000 .0000 .00002 .8891 .5583 .3907 .2528 .0834 .0193 .0028 .0002 .0000 .00003 .9744 .7946 .6488 .4925 .2253 .0730 .0153 .0017 .0001 .00004 .9957 .9274 .8424 .7237 .4382 .1938 .0573 .0095 .0006 .00005 .9995 .9806 .9456 .8822 .6652 .3872 .1582 .0386 .0039 .00016 .9999 .9961 .9857 .9614 .8418 .6128 .3348 .1178 .0194 .00057 1.0000 .9994 .9972 .9905 .9427 .8062 .5618 .2763 .0726 .00438 .9999 .9996 .9983 .9847 .9270 .7747 .5075 .2054 .02569 1.0000 1.0000 .9998 .9972 .9807 .9166 .7472 .4417 .110910 1.0000 .9997 .9968 .9804 .9150 .7251 .341011 1.0000 .9998 .9978 .9862 .9313 .717612 1.0000 1.0000 1.0000 1.0000 1.0000
13 0 .2542 .0550 .0238 .0097 .0013 .0001 .0000 .0000 .0000 .00001 .6213 .2336 .1267 .0637 .0126 .0017 .0001 .0000 .0000 .00002 .8661 .5017 .3326 .2025 .0579 .0112 .0013 .0001 .0000 .00003 .9658 .7473 .5843 .4206 .1686 .0461 .0078 .0007 .0000 .00004 .9935 .9009 .7940 .6543 .3530 .1334 .0321 .0040 .0002 .00005 .9991 .9700 .9198 .8346 .5744 .2905 .0977 .0182 .0012 .00006 .9999 .9930 .9757 .9376 .7712 .5000 .2288 .0624 .0070 .00017 1.0000 .9988 .9944 .9818 .9023 .7095 .4256 .1654 .0300 .00098 .9998 .9990 .9960 .9679 .8666 .6470 .3457 .0991 .00659 1.0000 .9999 .9993 .9922 .9539 .8314 .5794 .2527 .034210 1.0000 .9999 .9987 .9888 .9421 .7975 .4983 .133911 1.0000 .9999 .9983 .9874 .9363 .7664 .378712 1.0000 .9999 .9987 .9903 .9450 .745813 1.0000 1.0000 1.0000 1.0000 1.0000
14 0 .2288 .0440 .0178 .0068 .0008 .0001 .0000 .0000 .0000 .00001 .5846 .1979 .1010 .0475 .0081 .0009 .0001 .0000 .0000 .00002 .8416 .4481 .2811 .1608 .0398 .0065 .0006 .0000 .0000 .00003 .9559 .6982 .5213 .3552 .1243 .0287 .0039 .0002 .0000 .00004 .9908 .8702 .7415 .5842 .2793 .0898 .0175 .0017 .0000 .00005 .9985 .9561 .8883 .7805 .4859 .2120 .0583 .0083 .0004 .00006 .9998 .9884 .9617 .9067 .6925 .3953 .1501 .0315 .0024 .00007 1.0000 .9976 .9897 .9685 .8499 .6047 .3075 .0933 .0116 .00028 .9996 .9978 .9917 .9417 .7880 .5141 .2195 .0439 .00159 1.0000 .9997 .9983 .9825 .9102 .7207 .4158 .1298 .009210 1.0000 .9998 .9961 .9713 .8757 .6448 .3018 .044111 1.0000 .9994 .9935 .9602 .8392 .5519 .158412 1.0000 .9999 .9991 .9919 .9525 .8021 .415413 1.0000 .9999 .9992 .9932 .9560 .771214 1.0000 1.0000 1.0000 1.0000 1.0000
15 0 .2059 .0352 .0134 .0047 .0005 .0000 .0000 .0000 .0000 .00001 .5490 .1671 .0802 .0353 .0052 .0005 .0000 .0000 .0000 .00002 .8159 .3980 .2361 .1268 .0271 .0037 .0003 .0000 .0000 .00003 .9444 .6482 .4613 .2969 .0905 .0176 .0019 .0001 .0000 .00004 .9873 .8358 .6865 .5155 .2173 .0592 .0093 .0007 .0000 .00005 .9977 .9389 .8516 .7216 .4032 .1509 .0338 .0037 .0001 .00006 .9997 .9819 .9434 .8689 .6098 .3036 .0950 .0152 .0008 .00007 1.0000 .9958 .9827 .9500 .7869 .5000 .2131 .0500 .0042 .00008 .9992 .9958 .9848 .9050 .6964 .3902 .1311 .0181 .00039 .9999 .9992 .9963 .9662 .8491 .5968 .2784 .0611 .002210 1.0000 .9999 .9993 .9907 .9408 .7827 .4845 .1642 .012711 1.0000 .9999 .9981 .9824 .9095 .7031 .3518 .055612 1.0000 .9997 .9963 .9729 .8732 .6020 .184113 1.0000 .9995 .9948 .9647 .8329 .451014 1.0000 .9995 .9953 .9648 .794115 1.0000 1.0000 1.0000 1.0000
14Tabla A.1: Distribucion Binomial (Continuacion)
p
n x .10 .20 .25 .30 .40 .50 .60 .70 .80 .90
16 0 .1853 .0281 .0100 .0033 .0003 .0000 .0000 .0000 .0000 .00001 .5147 .1407 .0635 .0261 .0033 .0003 .0000 .0000 .0000 .00002 .7892 .3518 .1971 .0994 .0183 .0021 .0001 .0000 .0000 .00003 .9316 .5981 .4050 .2459 .0651 .0106 .0009 .0000 .0000 .00004 .9830 .7982 .6302 .4499 .1666 .0384 .0049 .0003 .0000 .00005 .9967 .9183 .8103 .6598 .3288 .1051 .0191 .0016 .0000 .00006 .9995 .9733 .9204 .8247 .5272 .2272 .0583 .0071 .0002 .00007 .9999 .9930 .9729 .9256 .7161 .4018 .1423 .0257 .0015 .00008 1.0000 .9985 .9925 .9743 .8577 .5982 .2839 .0744 .0070 .00019 .9998 .9984 .9929 .9417 .7728 .4728 .1753 .0267 .000510 1.0000 .9997 .9984 .9809 .8949 .6712 .3402 .0817 .003311 1.0000 .9997 .9951 .9616 .8334 .5501 .2018 .017012 1.0000 .9991 .9894 .9349 .7541 .4019 .068413 1.0000 .9999 .9979 .9817 .9006 .6482 .210814 1.0000 .9997 .9967 .9739 .8593 .485315 1.0000 .9997 .9967 .9719 .814716 1.0000 1.0000 1.0000 1.0000
17 0 .1668 .0225 .0075 .0023 .0002 .0000 .0000 .0000 .0000 .00001 .4818 .1182 .0501 .0193 .0021 .0001 .0000 .0000 .0000 .00002 .7618 .3096 .1637 .0774 .0123 .0012 .0001 .0000 .0000 .00003 .9174 .5489 .3530 .2019 .0464 .0064 .0005 .0000 .0000 .00004 .9779 .7582 .5739 .3887 .1260 .0245 .0025 .0001 .0000 .00005 .9953 .8943 .7653 .5968 .2639 .0717 .0106 .0007 .0000 .00006 .9992 .9623 .8929 .7752 .4478 .1662 .0348 .0032 .0001 .00007 .9999 .9891 .9598 .8954 .6405 .3145 .0919 .0127 .0005 .00008 1.0000 .9974 .9876 .9597 .8011 .5000 .1989 .0403 .0026 .00009 .9995 .9969 .9873 .9081 .6855 .3595 .1046 .0109 .000110 .9999 .9994 .9968 .9652 .8338 .5522 .2248 .0377 .000811 1.0000 .9999 .9993 .9894 .9283 .7361 .4032 .1057 .004712 1.0000 .9999 .9975 .9755 .8740 .6113 .2418 .022113 1.0000 .9995 .9936 .9536 .7981 .4511 .082614 .9999 .9988 .9877 .9226 .6904 .238215 1.0000 .9999 .9979 .9807 .8818 .518216 1.0000 .9998 .9977 .9775 .833217 1.0000 1.0000 1.0000 1.0000
18 0 .1501 .0180 .0056 .0016 .0001 .0000 .0000 .0000 .0000 .00001 .4503 .0991 .0395 .0142 .0013 .0001 .0000 .0000 .0000 .00002 .7338 .2713 .1353 .0600 .0082 .0007 .0000 .0000 .0000 .00003 .9018 .5010 .3057 .1646 .0328 .0038 .0002 .0000 .0000 .00004 .9718 .7164 .5187 .3327 .0942 .0154 .0013 .0000 .0000 .00005 .9936 .8671 .7175 .5344 .2088 .0481 .0058 .0003 .0000 .00006 .9988 .9487 .8610 .7217 .3743 .1189 .0203 .0014 .0000 .00007 .9998 .9837 .9431 .8593 .5634 .2403 .0576 .0061 .0002 .00008 1.0000 .9957 .9807 .9404 .7368 .4073 .1347 .0210 .0009 .00009 .9991 .9946 .9790 .8653 .5927 .2632 .0596 .0043 .000010 .9998 .9988 .9939 .9424 .7597 .4366 .1407 .0163 .000211 1.0000 .9998 .9986 .9797 .8811 .6257 .2783 .0513 .001212 1.0000 .9997 .9942 .9519 .7912 .4656 .1329 .006413 1.0000 .9987 .9846 .9058 .6673 .2836 .028214 .9998 .9962 .9672 .8354 .4990 .098215 1.0000 .9993 .9918 .9400 .7287 .266216 .9999 .9987 .9858 .9009 .549717 1.0000 .9999 .9984 .9820 .849918 1.0000 1.0000 1.0000 1.0000
15Tabla A.1: Distribucion Binomial (Continuacion)
p
n x .10 .20 .25 .30 .40 .50 .60 .70 .80 .90
19 0 .1351 .0144 .0042 .0011 .0001 .0000 .0000 .0000 .0000 .00001 .4203 .0829 .0310 .0104 .0008 .0000 .0000 .0000 .0000 .00002 .7054 .2369 .1113 .0462 .0055 .0004 .0000 .0000 .0000 .00003 .8850 .4551 .2631 .1332 .0230 .0022 .0001 .0000 .0000 .00004 .9648 .6733 .4654 .2822 .0696 .0096 .0006 .0000 .0000 .00005 .9914 .8369 .6678 .4739 .1629 .0318 .0031 .0001 .0000 .00006 .9983 .9324 .8251 .6655 .3081 .0835 .0116 .0006 .0000 .00007 .9997 .9767 .9225 .8180 .4878 .1796 .0352 .0028 .0000 .00008 1.0000 .9933 .9713 .9161 .6675 .3238 .0885 .0105 .0003 .00009 .9984 .9911 .9674 .8139 .5000 .1861 .0326 .0016 .000010 .9997 .9977 .9895 .9115 .6762 .3325 .0839 .0067 .000011 1.0000 .9995 .9972 .9648 .8204 .5122 .1820 .0233 .000312 .9999 .9994 .9884 .9165 .6919 .3345 .0676 .001713 1.0000 .9999 .9969 .9682 .8371 .5261 .1631 .008614 1.0000 .9994 .9904 .9304 .7178 .3267 .035215 .9999 .9978 .9770 .8668 .5449 .115016 1.0000 .9996 .9945 .9538 .7631 .294617 1.0000 .9992 .9896 .9171 .579718 .9999 .9989 .9856 .864919 1.0000 1.0000 1.0000 1.0000
20 0 .1216 .0115 .0032 .0008 .0000 .0000 .0000 .0000 .0000 .00001 .3917 .0692 .0243 .0076 .0005 .0000 .0000 .0000 .0000 .00002 .6769 .2061 .0913 .0355 .0036 .0002 .0000 .0000 .0000 .00003 .8670 .4114 .2252 .1071 .0160 .0013 .0000 .0000 .0000 .00004 .9568 .6296 .4148 .2375 .0510 .0059 .0003 .0000 .0000 .00005 .9887 .8042 .6172 .4164 .1256 .0207 .0016 .0000 .0000 .00006 .9976 .9133 .7858 .6080 .2500 .0577 .0065 .0003 .0000 .00007 .9996 .9679 .8982 .7723 .4159 .1316 .0210 .0013 .0000 .00008 .9999 .9900 .9591 .8867 .5956 .2517 .0565 .0051 .0001 .00009 1.0000 .9974 .9861 .9520 .7553 .4119 .1275 .0171 .0006 .000010 .9994 .9961 .9829 .8725 .5881 .2447 .0480 .0026 .000011 .9999 .9991 .9949 .9435 .7483 .4044 .1133 .0100 .000112 1.0000 .9998 .9987 .9790 .8684 .5841 .2277 .0321 .000413 1.0000 .9997 .9935 .9423 .7500 .3920 .0867 .002414 1.0000 .9984 .9793 .8744 .5836 .1958 .011315 .9997 .9941 .9490 .7625 .3704 .043216 1.0000 .9987 .9840 .8929 .5886 .133017 .9998 .9964 .9645 .7939 .323118 1.0000 .9995 .9924 .9308 .608319 1.0000 .9992 .9885 .878420 1.0000 1.0000 1.0000
16
Tabla A.2: Distribucion de Poisson. P (P(λ) ≤ x) =x∑
k=0
λk
k!e−λ
λ
x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066
1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725
2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371
3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865
4 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977
5 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997
6 1.0000 1.0000 1.0000
λ
x 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067
1 0.7358 0.5578 0.4060 0.2873 0.1991 0.1359 0.0916 0.0611 0.0404
2 0.9197 0.8088 0.6767 0.5438 0.4232 0.3208 0.2381 0.1736 0.1247
3 0.9810 0.9344 0.8571 0.7576 0.6472 0.5366 0.4335 0.3423 0.2650
4 0.9963 0.9814 0.9473 0.8912 0.8153 0.7254 0.6288 0.5321 0.4405
5 0.9994 0.9955 0.9834 0.9580 0.9161 0.8576 0.7851 0.7029 0.6160
6 0.9999 0.9991 0.9955 0.9858 0.9665 0.9347 0.8893 0.8311 0.7622
7 1.0000 0.9998 0.9989 0.9958 0.9881 0.9733 0.9489 0.9134 0.8666
8 1.0000 0.9998 0.9989 0.9962 0.9901 0.9786 0.9597 0.9319
9 1.0000 0.9997 0.9989 0.9967 0.9919 0.9829 0.9682
10 0.9999 0.9997 0.9990 0.9972 0.9933 0.9863
11 1.0000 0.9999 0.9997 0.9991 0.9976 0.9945
12 1.0000 0.9999 0.9997 0.9992 0.9980
13 1.0000 0.9999 0.9997 0.9993
14 1.0000 0.9999 0.9998
15 1.0000 0.9999
16 1.0000
17
Tabla A.2: Distribucion de Poisson (Continuacion)
λ
x 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5
0 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001
1 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.0008
2 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.0042
3 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.0149
4 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.0403
5 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885
6 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.1649
7 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.2687
8 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.3918
9 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218
10 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6453
11 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520
12 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.8364
13 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.8981
14 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.9400
15 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.9665
16 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.9823
17 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.9911
18 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957
19 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.9980
20 1.0000 0.9999 0.9998 0.9996 0.9991
21 1.0000 0.9999 0.9998 0.9996
22 1.0000 0.9999 0.9999
23 1.0000 0.9999
24 1.0000
18
Tabla A.2: Distribucion de Poisson (Continuacion)
λ
x 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
2 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000
3 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000
4 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001
5 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003
6 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010
7 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029
8 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071
9 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154
10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304
11 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549
12 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917
13 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426
14 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081
15 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867
16 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751
17 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686
18 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622
19 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509
20 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307
21 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991
22 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551
23 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989
24 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317
25 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554
26 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718
27 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827
28 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897
29 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941
30 0.9999 0.9998 0.9994 0.9986 0.9967
31 1.0000 0.9999 0.9997 0.9993 0.9982
32 1.0000 0.9999 0.9996 0.9990
33 0.9999 0.9998 0.9995
34 1.0000 0.9999 0.9998
35 1.0000 0.9999
36 0.9999
37 1.0000
19
Tabla A.3: Distribucion Normal Estandar. P (N(0, 1) ≥ z)
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681 .4641
0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286 .4247
0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897 .3859
0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520 .3483
0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156 .3121
0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810 .2776
0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483 .2451
0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177 .2148
0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894 .1867
0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635 .1611
1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401 .1379
1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190 .1170
1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003 .0985
1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838 .0823
1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694 .0681
1.5 .0668 .0655 .0642 .0630 .0618 .0606 .0594 .0582 .0571 .0559
1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465 .0455
1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375 .0367
1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301 .0294
1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239 .0233
2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188 .0183
2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146 .0143
2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113 .0110
2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087 .0084
2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066 .0064
2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049 .0048
2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037 .0036
2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027 .0026
2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020 .0019
2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014 .0014
3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010 .0010
3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007 .0007
3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005 .0005
3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004 .0003
3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0002
20
Tabla A.4: Distribucion t-Student. P (tn ≥ a)
Probabilidades
Grados de0.40 0.25 0.15 0.10 0.05 0.025 0.01 0.005
libertad
1 0.3249 1.0000 1.9626 3.0777 6.3138 12.7062 31.8205 63.65672 0.2887 0.8165 1.3862 1.8856 2.9200 4.3027 6.9646 9.92483 0.2767 0.7649 1.2498 1.6377 2.3534 3.1824 4.5408 5.84084 0.2707 0.7407 1.1896 1.5332 2.1318 2.7764 3.7470 4.60415 0.2672 0.7267 1.1558 1.4759 2.0150 2.5706 3.3649 4.0321
6 0.2648 0.7176 1.1342 1.4398 1.9432 2.4469 3.1427 3.70747 0.2632 0.7111 1.1192 1.4149 1.8946 2.3646 2.9980 3.49958 0.2619 0.7064 1.1081 1.3968 1.8595 2.3060 2.8965 3.35549 0.2610 0.7027 1.0997 1.3830 1.8331 2.2622 2.8215 3.249810 0.2602 0.6998 1.0931 1.3722 1.8125 2.2281 2.7638 3.1693
11 0.2596 0.6974 1.0877 1.3634 1.7959 2.2010 2.7181 3.105812 0.2590 0.6955 1.0832 1.3562 1.7823 2.1788 2.6810 3.054613 0.2586 0.6938 1.0795 1.3502 1.7709 2.1604 2.6503 3.012314 0.2582 0.6924 1.0763 1.3450 1.7613 2.1448 2.6245 2.976815 0.2579 0.6912 1.0735 1.3406 1.7531 2.1314 2.6025 2.9467
16 0.2576 0.6901 1.0711 1.3368 1.7459 2.1199 2.5835 2.920817 0.2573 0.6892 1.0690 1.3334 1.7396 2.1098 2.5669 2.898218 0.2571 0.6884 1.0672 1.3304 1.7341 2.1009 2.5524 2.878419 0.2569 0.6876 1.0655 1.3277 1.7291 2.0930 2.5395 2.860920 0.2567 0.6870 1.0640 1.3253 1.7247 2.0860 2.5280 2.8453
21 0.2566 0.6864 1.0627 1.3232 1.7207 2.0796 2.5176 2.831422 0.2564 0.6858 1.0614 1.3212 1.7171 2.0739 2.5083 2.818823 0.2563 0.6853 1.0603 1.3195 1.7139 2.0687 2.4999 2.807324 0.2562 0.6848 1.0593 1.3178 1.7109 2.0639 2.4922 2.796925 0.2561 0.6844 1.0584 1.3163 1.7081 2.0595 2.4851 2.7874
26 0.2560 0.6840 1.0575 1.3150 1.7056 2.0555 2.4786 2.778727 0.2559 0.6837 1.0567 1.3137 1.7033 2.0518 2.4727 2.770728 0.2558 0.6834 1.0560 1.3125 1.7011 2.0484 2.4671 2.763329 0.2557 0.6830 1.0553 1.3114 1.6991 2.0452 2.4620 2.756430 0.2556 0.6828 1.0547 1.3104 1.6973 2.0423 2.4573 2.7500
35 0.2553 0.6816 1.0520 1.3062 1.6896 2.0301 2.4377 2.723840 0.2550 0.6807 1.0500 1.3031 1.6839 2.0211 2.4233 2.704545 0.2549 0.6800 1.0485 1.3006 1.6794 2.0141 2.4121 2.689650 0.2547 0.6794 1.0473 1.2987 1.6759 2.0086 2.4033 2.677860 0.2545 0.6786 1.0455 1.2958 1.6706 2.0003 2.3901 2.6603
70 0.2543 0.6780 1.0442 1.2938 1.6669 1.9944 2.3808 2.647980 0.2542 0.6776 1.0432 1.2922 1.6641 1.9901 2.3739 2.638790 0.2541 0.6772 1.0424 1.2910 1.6620 1.9867 2.3685 2.6316100 0.2540 0.6770 1.0418 1.2901 1.6602 1.9840 2.3642 2.6259120 0.2539 0.6765 1.0409 1.2886 1.6577 1.9799 2.3578 2.6174
150 0.2538 0.6761 1.0400 1.2872 1.6551 1.9759 2.3515 2.6090200 0.2537 0.6757 1.0391 1.2858 1.6525 1.9719 2.3451 2.6006300 0.2536 0.6753 1.0382 1.2844 1.6499 1.9679 2.3388 2.5923∞ 0.2533 0.6745 1.0364 1.2816 1.6449 1.9600 2.3263 2.5758
21Tabla
A.5:Distribuconχ2 n.P(χ
2 n≥
a)
Probabilidades
Gradosde
0.99
0.975
0.95
0.90
0.75
0.50
0.25
0.10
0.05
0.025
0.01
libertad
11.571∗
9.821∗
39.320∗
0.016
0.102
0.455
1.323
2.706
3.841
5.024
6.635
20.020
0.051
0.103
0.211
0.575
1.386
2.773
4.605
5.991
7.378
9.210
30.115
0.216
0.352
0.584
1.213
2.366
4.108
6.252
7.815
9.349
11.346
40.297
0.484
0.711
1.064
1.923
3.357
5.385
7.779
9.488
11.143
13.277
50.554
0.831
1.145
1.610
2.675
4.351
6.626
9.236
11.070
12.832
15.086
60.872
1.237
1.635
2.204
3.455
5.348
7.841
10.645
12.592
14.449
16.812
71.239
1.690
2.167
2.833
4.255
6.346
9.037
12.017
14.067
16.013
18.475
81.646
2.180
2.733
3.490
5.071
7.344
10.219
13.362
15.507
17.535
20.090
92.088
2.700
3.325
4.168
5.899
8.343
11.389
14.684
16.919
19.023
21.666
10
2.558
3.247
3.940
4.865
6.737
9.342
12.549
15.987
18.307
20.483
23.209
11
3.053
3.816
4.575
5.578
7.584
10.341
13.701
17.275
19.675
21.920
24.725
12
3.571
4.404
5.226
6.304
8.438
11.340
14.845
18.549
21.026
23.337
26.217
13
4.107
5.009
5.892
7.041
9.299
12.340
15.984
19.812
22.362
24.712
27.688
14
4.660
5.629
6.571
7.790
10.165
13.339
17.117
21.064
23.685
26.119
29.141
15
5.229
6.262
7.261
8.547
11.037
14.339
18.245
22.307
24.996
27.488
30.578
16
5.812
6.908
7.962
9.312
11.912
15.338
19.369
23.542
26.296
28.845
32.000
17
6.408
7.564
8.672
10.085
12.792
16.338
20.489
24.769
27.587
30.191
33.409
18
7.015
8.231
9.390
10.865
13.675
17.338
21.605
25.989
28.869
31.526
34.805
19
7.633
8.907
10.117
11.651
14.562
18.338
22.718
27.204
30.144
32.852
36.191
20
8.260
9.591
10.851
12.443
15.452
19.337
23.828
28.412
31.410
34.170
37.566
21
8.897
10.283
11.591
13.240
16.344
20.337
24.935
29.615
32.671
35.479
38.932
22
9.542
10.982
12.338
14.041
17.240
21.337
26.039
30.813
33.924
36.781
40.289
23
10.196
11.689
13.091
14.848
18.137
22.337
27.141
32.007
35.172
38.076
41.638
24
10.856
12.401
13.848
15.659
19.037
23.337
28.241
33.196
36.415
39.364
42.980
25
11.524
13.120
14.611
16.473
19.939
24.337
29.339
34.382
37.652
40.646
44.314
26
12.198
13.844
15.379
17.292
20.843
25.336
30.435
35.563
38.885
41.923
45.642
27
12.879
14.573
16.151
18.114
21.749
26.336
31.528
36.741
40.113
43.194
46.963
28
13.565
15.308
16.928
18.939
22.657
27.336
32.620
37.916
41.329
44.461
48.278
29
14.256
16.047
17.708
19.768
23.567
28.336
33.711
39.087
42.557
45.722
49.588
30
14.954
16.791
18.493
20.599
24.478
29.336
34.800
40.256
43.773
46.979
50.892
40
22.164
24.433
26.509
29.050
33.660
39.335
45.616
51.805
55.758
59.342
63.691
50
29.707
32.357
34.764
37.689
42.942
49.335
56.334
63.167
67.505
71.420
76.154
60
37.485
40.482
43.188
46.459
52.294
59.335
66.981
74.397
79.082
83.298
88.379
70
45.442
48.758
51.739
55.329
61.698
69.334
77.577
85.527
90.531
95.023
100.425
80
53.540
57.153
60.391
64.278
71.144
70.334
88.130
96.578
101.879
106.629
112.329
90
61.754
65.647
69.126
73.291
80.625
89.334
98.650
107.565
113.145
118.136
124.116
100
70.065
74.222
77.929
82.358
90.133
99.334
109.141
118.498
124.342
129.561
135.807
∗Dividir
entre1000
22
Tabla
A.6.1:Distribucion
Fnm.P(F
nm≥
a)=
0.25
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
15.83
7.50
8.20
8.58
8.82
8.98
9.10
9.19
9.26
9.32
9.41
9.49
9.58
9.63
9.67
9.71
9.76
9.80
9.85
22.57
3.00
3.15
3.23
3.28
3.31
3.34
3.35
3.37
3.38
3.39
3.41
3.43
3.43
3.44
3.45
3.46
3.47
3.48
32.02
2.28
2.36
2.39
2.41
2.42
2.43
2.44
2.44
2.44
2.45
2.46
2.46
2.46
2.47
2.47
2.47
2.47
2.47
41.81
2.00
2.05
2.06
2.07
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
2.08
51.69
1.85
1.88
1.89
1.89
1.89
1.89
1.89
1.89
1.89
1.89
1.89
1.88
1.88
1.88
1.88
1.87
1.87
1.87
61.62
1.76
1.78
1.79
1.79
1.78
1.78
1.78
1.77
1.77
1.77
1.76
1.76
1.75
1.75
1.75
1.74
1.74
1.74
71.57
1.70
1.72
1.72
1.71
1.71
1.70
1.70
1.69
1.69
1.68
1.68
1.67
1.67
1.66
1.66
1.65
1.65
1.65
81.54
1.66
1.67
1.66
1.66
1.65
1.64
1.64
1.63
1.63
1.62
1.62
1.61
1.60
1.60
1.59
1.59
1.58
1.58
91.51
1.62
1.63
1.63
1.62
1.61
1.60
1.60
1.59
1.59
1.58
1.57
1.56
1.56
1.55
1.54
1.54
1.53
1.53
10
1.49
1.60
1.60
1.59
1.59
1.58
1.57
1.56
1.56
1.55
1.54
1.53
1.52
1.52
1.51
1.51
1.50
1.49
1.48
11
1.47
1.58
1.58
1.57
1.56
1.55
1.54
1.53
1.53
1.52
1.51
1.50
1.49
1.49
1.48
1.47
1.47
1.46
1.45
12
1.46
1.56
1.56
1.55
1.54
1.53
1.52
1.51
1.51
1.50
1.49
1.48
1.47
1.46
1.45
1.45
1.44
1.43
1.42
13
1.45
1.55
1.55
1.53
1.52
1.51
1.50
1.49
1.49
1.48
1.47
1.46
1.45
1.44
1.43
1.42
1.42
1.41
1.40
14
1.44
1.53
1.53
1.52
1.51
1.50
1.49
1.48
1.47
1.46
1.45
1.44
1.43
1.42
1.41
1.41
1.40
1.39
1.38
15
1.43
1.52
1.52
1.51
1.49
1.48
1.47
1.46
1.46
1.45
1.44
1.43
1.41
1.41
1.40
1.39
1.38
1.37
1.36
16
1.42
1.51
1.51
1.50
1.48
1.47
1.46
1.45
1.44
1.44
1.43
1.41
1.40
1.39
1.38
1.37
1.36
1.35
1.34
17
1.42
1.51
1.50
1.49
1.47
1.46
1.45
1.44
1.43
1.43
1.41
1.40
1.39
1.38
1.37
1.36
1.35
1.34
1.33
18
1.41
1.50
1.49
1.48
1.46
1.45
1.44
1.43
1.42
1.42
1.40
1.39
1.38
1.37
1.36
1.35
1.34
1.33
1.32
19
1.41
1.49
1.49
1.47
1.46
1.44
1.43
1.42
1.41
1.41
1.40
1.38
1.37
1.36
1.35
1.34
1.33
1.32
1.30
20
1.40
1.49
1.48
1.47
1.45
1.44
1.43
1.42
1.41
1.40
1.39
1.37
1.36
1.35
1.34
1.33
1.32
1.31
1.29
21
1.40
1.48
1.48
1.46
1.44
1.43
1.42
1.41
1.40
1.39
1.38
1.37
1.35
1.34
1.33
1.32
1.31
1.30
1.28
22
1.40
1.48
1.47
1.45
1.44
1.42
1.41
1.40
1.39
1.39
1.37
1.36
1.34
1.33
1.32
1.31
1.30
1.29
1.28
23
1.39
1.47
1.47
1.45
1.43
1.42
1.41
1.40
1.39
1.38
1.37
1.35
1.34
1.33
1.32
1.31
1.30
1.28
1.27
24
1.39
1.47
1.46
1.44
1.43
1.41
1.40
1.39
1.38
1.38
1.36
1.35
1.33
1.32
1.31
1.30
1.29
1.28
1.26
25
1.39
1.47
1.46
1.44
1.42
1.41
1.40
1.39
1.38
1.37
1.36
1.34
1.33
1.32
1.31
1.29
1.28
1.27
1.25
26
1.38
1.46
1.45
1.44
1.42
1.41
1.39
1.38
1.37
1.37
1.35
1.34
1.32
1.31
1.30
1.29
1.28
1.26
1.25
27
1.38
1.46
1.45
1.43
1.42
1.40
1.39
1.38
1.37
1.36
1.35
1.33
1.32
1.31
1.30
1.28
1.27
1.26
1.24
28
1.38
1.46
1.45
1.43
1.41
1.40
1.39
1.38
1.37
1.36
1.34
1.33
1.31
1.30
1.29
1.28
1.27
1.25
1.24
29
1.38
1.45
1.45
1.43
1.41
1.40
1.38
1.37
1.36
1.35
1.34
1.32
1.31
1.30
1.29
1.27
1.26
1.25
1.23
30
1.38
1.45
1.44
1.42
1.41
1.39
1.38
1.37
1.36
1.35
1.34
1.32
1.30
1.29
1.28
1.27
1.26
1.24
1.23
40
1.36
1.44
1.42
1.40
1.39
1.37
1.36
1.35
1.34
1.33
1.31
1.30
1.28
1.26
1.25
1.24
1.22
1.21
1.19
60
1.35
1.42
1.41
1.38
1.37
1.35
1.33
1.32
1.31
1.30
1.29
1.27
1.25
1.24
1.22
1.21
1.19
1.17
1.15
120
1.34
1.40
1.39
1.37
1.35
1.33
1.31
1.30
1.29
1.28
1.26
1.24
1.22
1.21
1.19
1.18
1.16
1.13
1.10
∞1.32
1.39
1.37
1.35
1.33
1.31
1.29
1.28
1.27
1.25
1.24
1.22
1.19
1.18
1.16
1.14
1.12
1.08
1.00
23
Tabla
A.6.2:Distribucion
Fnm.P(F
nm≥
a)=
0.10
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
139.86
49.50
53.59
55.83
57.24
58.20
58.91
59.44
59.86
60.19
60.71
61.22
61.74
62.00
62.26
62.53
62.79
63.06
63.33
28.53
9.00
9.16
9.24
9.29
9.33
9.35
9.37
9.38
9.39
9.41
9.42
9.44
9.45
9.46
9.47
9.47
9.48
9.49
35.54
5.46
5.39
5.34
5.31
5.28
5.27
5.25
5.24
5.23
5.22
5.20
5.18
5.18
5.17
5.16
5.15
5.14
5.13
44.54
4.32
4.19
4.11
4.05
4.01
3.98
3.95
3.94
3.92
3.90
3.87
3.84
3.83
3.82
3.80
3.79
3.78
3.76
54.06
3.78
3.62
3.52
3.45
3.40
3.37
3.34
3.32
3.30
3.27
3.24
3.21
3.19
3.17
3.16
3.14
3.12
3.10
63.78
3.46
3.29
3.18
3.11
3.05
3.01
2.98
2.96
2.94
2.90
2.87
2.84
2.82
2.80
2.78
2.76
2.74
2.72
73.59
3.26
3.07
2.96
2.88
2.83
2.78
2.75
2.72
2.70
2.67
2.63
2.59
2.58
2.56
2.54
2.51
2.49
2.47
83.46
3.11
2.92
2.81
2.73
2.67
2.62
2.59
2.56
2.54
2.50
2.46
2.42
2.40
2.38
2.36
2.34
2.32
2.29
93.36
3.01
2.81
2.69
2.61
2.55
2.51
2.47
2.44
2.42
2.38
2.34
2.30
2.28
2.25
2.23
2.21
2.18
2.16
10
3.29
2.92
2.73
2.61
2.52
2.46
2.41
2.38
2.35
2.32
2.28
2.24
2.20
2.18
2.16
2.13
2.11
2.08
2.06
11
3.23
2.86
2.66
2.54
2.45
2.39
2.34
2.30
2.27
2.25
2.21
2.17
2.12
2.10
2.08
2.05
2.03
2.00
1.97
12
3.18
2.81
2.61
2.48
2.39
2.33
2.28
2.24
2.21
2.19
2.15
2.10
2.06
2.04
2.01
1.99
1.96
1.93
1.90
13
3.14
2.76
2.56
2.43
2.35
2.28
2.23
2.20
2.16
2.14
2.10
2.05
2.01
1.98
1.96
1.93
1.90
1.88
1.85
14
3.10
2.73
2.52
2.39
2.31
2.24
2.19
2.15
2.12
2.10
2.05
2.01
1.96
1.94
1.91
1.89
1.86
1.83
1.80
15
3.07
2.70
2.49
2.36
2.27
2.21
2.16
2.12
2.09
2.06
2.02
1.97
1.92
1.90
1.87
1.85
1.82
1.79
1.76
16
3.05
2.67
2.46
2.33
2.24
2.18
2.13
2.09
2.06
2.03
1.99
1.94
1.89
1.87
1.84
1.81
1.78
1.75
1.72
17
3.03
2.64
2.44
2.31
2.22
2.15
2.10
2.06
2.03
2.00
1.96
1.91
1.86
1.84
1.81
1.78
1.75
1.72
1.69
18
3.01
2.62
2.42
2.29
2.20
2.13
2.08
2.04
2.00
1.98
1.93
1.89
1.84
1.81
1.78
1.75
1.72
1.69
1.66
19
2.99
2.61
2.40
2.27
2.18
2.11
2.06
2.02
1.98
1.96
1.91
1.86
1.81
1.79
1.76
1.73
1.70
1.67
1.63
20
2.97
2.59
2.38
2.25
2.16
2.09
2.04
2.00
1.96
1.94
1.89
1.84
1.79
1.77
1.74
1.71
1.68
1.64
1.61
21
2.96
2.57
2.36
2.23
2.14
2.08
2.02
1.98
1.95
1.92
1.87
1.83
1.78
1.75
1.72
1.69
1.66
1.62
1.59
22
2.95
2.56
2.35
2.22
2.13
2.06
2.01
1.97
1.93
1.90
1.86
1.81
1.76
1.73
1.70
1.67
1.64
1.60
1.57
23
2.94
2.55
2.34
2.21
2.11
2.05
1.99
1.95
1.92
1.89
1.84
1.80
1.74
1.72
1.69
1.66
1.62
1.59
1.55
24
2.93
2.54
2.33
2.19
2.10
2.04
1.98
1.94
1.91
1.88
1.83
1.78
1.73
1.70
1.67
1.64
1.61
1.57
1.53
25
2.92
2.53
2.32
2.18
2.09
2.02
1.97
1.93
1.89
1.87
1.82
1.77
1.72
1.69
1.66
1.63
1.59
1.56
1.52
26
2.91
2.52
2.31
2.17
2.08
2.01
1.96
1.92
1.88
1.86
1.81
1.76
1.71
1.68
1.65
1.61
1.58
1.54
1.50
27
2.90
2.51
2.30
2.17
2.07
2.00
1.95
1.91
1.87
1.85
1.80
1.75
1.70
1.67
1.64
1.60
1.57
1.53
1.49
28
2.89
2.50
2.29
2.16
2.06
2.00
1.94
1.90
1.87
1.84
1.79
1.74
1.69
1.66
1.63
1.59
1.56
1.52
1.48
29
2.89
2.50
2.28
2.15
2.06
1.99
1.93
1.89
1.86
1.83
1.78
1.73
1.68
1.65
1.62
1.58
1.55
1.51
1.47
30
2.88
2.49
2.28
2.14
2.05
1.98
1.93
1.88
1.85
1.82
1.77
1.72
1.67
1.64
1.61
1.57
1.54
1.50
1.46
40
2.84
2.44
2.23
2.09
2.00
1.93
1.87
1.83
1.79
1.76
1.71
1.66
1.61
1.57
1.54
1.51
1.47
1.42
1.38
60
2.79
2.39
2.18
2.04
1.95
1.87
1.82
1.77
1.74
1.71
1.66
1.60
1.54
1.51
1.48
1.44
1.40
1.35
1.29
120
2.75
2.35
2.13
1.99
1.90
1.82
1.77
1.72
1.68
1.65
1.60
1.55
1.48
1.45
1.41
1.37
1.32
1.26
1.19
∞2.71
2.30
2.08
1.94
1.85
1.77
1.72
1.67
1.63
1.60
1.55
1.49
1.42
1.38
1.34
1.30
1.24
1.17
1.00
24
Tabla
A.6.3:Distribucion
Fnm.P(F
nm≥
a)=
0.05
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
1161.40
199.50
215.70
224.60
230.20
234.00
236.80
238.90
240.50
241.90
243.90
245.90
248.00
249.10
250.10
251.10
252.20
253.30
254.30
218.51
19.00
19.16
19.25
19.30
19.33
19.35
19.37
19.39
19.40
19.41
19.43
19.45
19.45
19.46
19.47
19.48
19.49
19.50
310.13
9.55
9.28
9.12
9.01
8.94
8.89
8.85
8.81
8.79
8.75
8.70
8.66
8.64
8.62
8.59
8.57
8.55
8.53
47.71
6.94
6.59
6.39
6.26
6.16
6.09
6.04
6.00
5.96
5.91
5.86
5.80
5.77
5.75
5.72
5.69
5.66
5.63
56.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
4.74
4.68
4.62
4.56
4.53
4.50
4.46
4.43
4.40
4.36
65.99
5.14
4.76
4.53
4.39
4.28
4.21
4.15
4.10
4.06
4.00
3.94
3.87
3.84
3.81
3.77
3.74
3.70
3.67
75.59
4.74
4.35
4.12
3.97
3.87
3.79
3.73
3.68
3.64
3.57
3.51
3.44
3.41
3.38
3.34
3.30
3.27
3.23
85.32
4.46
4.07
3.84
3.69
3.58
3.50
3.44
3.39
3.35
3.28
3.22
3.15
3.12
3.08
3.04
3.01
2.97
2.93
95.12
4.26
3.86
3.63
3.48
3.37
3.29
3.23
3.18
3.14
3.07
3.01
2.94
2.90
2.86
2.83
2.79
2.75
2.71
10
4.96
4.10
3.71
3.48
3.33
3.22
3.14
3.07
3.02
2.98
2.91
2.85
2.77
2.74
2.70
2.66
2.62
2.58
2.54
11
4.84
3.98
3.59
3.36
3.20
3.09
3.01
2.95
2.90
2.85
2.79
2.72
2.65
2.61
2.57
2.53
2.49
2.45
2.40
12
4.75
3.89
3.49
3.26
3.11
3.00
2.91
2.85
2.80
2.75
2.69
2.62
2.54
2.51
2.47
2.43
2.38
2.34
2.30
13
4.67
3.81
3.41
3.18
3.03
2.92
2.83
2.77
2.71
2.67
2.60
2.53
2.46
2.42
2.38
2.34
2.30
2.25
2.21
14
4.60
3.74
3.34
3.11
2.96
2.85
2.76
2.70
2.65
2.60
2.53
2.46
2.39
2.35
2.31
2.27
2.22
2.18
2.13
15
4.54
3.68
3.29
3.06
2.90
2.79
2.71
2.64
2.59
2.54
2.48
2.40
2.33
2.29
2.25
2.20
2.16
2.11
2.07
16
4.49
3.63
3.24
3.01
2.85
2.74
2.66
2.59
2.54
2.49
2.42
2.35
2.28
2.24
2.19
2.15
2.11
2.06
2.01
17
4.45
3.59
3.20
2.96
2.81
2.70
2.61
2.55
2.49
2.45
2.38
2.31
2.23
2.19
2.15
2.10
2.06
2.01
1.96
18
4.41
3.55
3.16
2.93
2.77
2.66
2.58
2.51
2.46
2.41
2.34
2.27
2.19
2.15
2.11
2.06
2.02
1.97
1.92
19
4.38
3.52
3.13
2.90
2.74
2.63
2.54
2.48
2.42
2.38
2.31
2.23
2.16
2.11
2.07
2.03
1.98
1.93
1.88
20
4.35
3.49
3.10
2.87
2.71
2.60
2.51
2.45
2.39
2.35
2.28
2.20
2.12
2.08
2.04
1.99
1.95
1.90
1.84
21
4.32
3.47
3.07
2.84
2.68
2.57
2.49
2.42
2.37
2.32
2.25
2.18
2.10
2.05
2.01
1.96
1.92
1.87
1.81
22
4.30
3.44
3.05
2.82
2.66
2.55
2.46
2.40
2.34
2.30
2.23
2.15
2.07
2.03
1.98
1.94
1.89
1.84
1.78
23
4.28
3.42
3.03
2.80
2.64
2.53
2.44
2.37
2.32
2.27
2.20
2.13
2.05
2.01
1.96
1.91
1.86
1.81
1.76
24
4.26
3.40
3.01
2.78
2.62
2.51
2.42
2.36
2.30
2.25
2.18
2.11
2.03
1.98
1.94
1.89
1.84
1.79
1.73
25
4.24
3.39
2.99
2.76
2.60
2.49
2.40
2.34
2.28
2.24
2.16
2.09
2.01
1.96
1.92
1.87
1.82
1.77
1.71
26
4.23
3.37
2.98
2.74
2.59
2.47
2.39
2.32
2.27
2.22
2.15
2.07
1.99
1.95
1.90
1.85
1.80
1.75
1.69
27
4.21
3.35
2.96
2.73
2.57
2.46
2.37
2.31
2.25
2.20
2.13
2.06
1.97
1.93
1.88
1.84
1.79
1.73
1.67
28
4.20
3.34
2.95
2.71
2.56
2.45
2.36
2.29
2.24
2.19
2.12
2.04
1.96
1.91
1.87
1.82
1.77
1.71
1.65
29
4.18
3.33
2.93
2.70
2.55
2.43
2.35
2.28
2.22
2.18
2.10
2.03
1.94
1.90
1.85
1.81
1.75
1.70
1.64
30
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
2.16
2.09
2.01
1.93
1.89
1.84
1.79
1.74
1.68
1.62
40
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
2.08
2.00
1.92
1.84
1.79
1.74
1.69
1.64
1.58
1.51
60
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
1.99
1.92
1.84
1.75
1.70
1.65
1.59
1.53
1.47
1.39
120
3.92
3.07
2.68
2.45
2.29
2.17
2.09
2.02
1.96
1.91
1.83
1.75
1.66
1.61
1.55
1.50
1.43
1.35
1.25
∞3.84
3.00
2.60
2.37
2.21
2.10
2.01
1.94
1.88
1.83
1.75
1.67
1.57
1.52
1.46
1.39
1.32
1.22
1.00
25
Tabla
A.6.4:Distribucion
Fnm.P(F
nm≥
a)=
0.025
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
1647.80
799.50
864.20
899.60
921.80
937.10
948.20
956.70
963.30
968.60
976.70
984.90
993.10
997.20
1001.00
1006.00
1010.00
1014.00
1018.00
238.51
39.00
39.17
39.25
39.30
39.33
39.36
39.37
39.39
39.40
39.41
39.43
39.45
39.46
39.46
39.47
39.48
39.49
39.50
317.44
16.04
15.44
15.10
14.88
14.73
14.62
14.54
14.47
14.42
14.34
14.25
14.17
14.12
14.08
14.04
13.99
13.95
13.90
412.22
10.65
9.98
9.60
9.36
9.20
9.07
8.98
8.90
8.84
8.75
8.66
8.56
8.51
8.46
8.41
8.36
8.31
8.26
510.01
8.43
7.76
7.39
7.15
6.98
6.85
6.76
6.68
6.62
6.52
6.43
6.33
6.28
6.23
6.18
6.12
6.07
6.02
68.81
7.26
6.60
6.23
5.99
5.82
5.70
5.60
5.52
5.46
5.37
5.27
5.17
5.12
5.07
5.01
4.96
4.90
4.85
78.07
6.54
5.89
5.52
5.29
5.12
4.99
4.90
4.82
4.76
4.67
4.57
4.47
4.41
4.36
4.31
4.25
4.20
4.14
87.57
6.06
5.42
5.05
4.82
4.65
4.53
4.43
4.36
4.30
4.20
4.10
4.00
3.95
3.89
3.84
3.78
3.73
3.67
97.21
5.71
5.08
4.72
4.48
4.32
4.20
4.10
4.03
3.96
3.87
3.77
3.67
3.61
3.56
3.51
3.45
3.39
3.33
10
6.94
5.46
4.83
4.47
4.24
4.07
3.95
3.85
3.78
3.72
3.62
3.52
3.42
3.37
3.31
3.26
3.20
3.14
3.08
11
6.72
5.26
4.63
4.28
4.04
3.88
3.76
3.66
3.59
3.53
3.43
3.33
3.23
3.17
3.12
3.06
3.00
2.94
2.88
12
6.55
5.10
4.47
4.12
3.89
3.73
3.61
3.51
3.44
3.37
3.28
3.18
3.07
3.02
2.96
2.91
2.85
2.79
2.72
13
6.41
4.97
4.35
4.00
3.77
3.60
3.48
3.39
3.31
3.25
3.15
3.05
2.95
2.89
2.84
2.78
2.72
2.66
2.60
14
6.30
4.86
4.24
3.89
3.66
3.50
3.38
3.29
3.21
3.15
3.05
2.95
2.84
2.79
2.73
2.67
2.61
2.55
2.49
15
6.20
4.77
4.15
3.80
3.58
3.41
3.29
3.20
3.12
3.06
2.96
2.86
2.76
2.70
2.64
2.59
2.52
2.46
2.40
16
6.12
4.69
4.08
3.73
3.50
3.34
3.22
3.12
3.05
2.99
2.89
2.79
2.68
2.63
2.57
2.51
2.45
2.38
2.32
17
6.04
4.62
4.01
3.66
3.44
3.28
3.16
3.06
2.98
2.92
2.82
2.72
2.62
2.56
2.50
2.44
2.38
2.32
2.25
18
5.98
4.56
3.95
3.61
3.38
3.22
3.10
3.01
2.93
2.87
2.77
2.67
2.56
2.50
2.44
2.38
2.32
2.26
2.19
19
5.92
4.51
3.90
3.56
3.33
3.17
3.05
2.96
2.88
2.82
2.72
2.62
2.51
2.45
2.39
2.33
2.27
2.20
2.13
20
5.87
4.46
3.86
3.51
3.29
3.13
3.01
2.91
2.84
2.77
2.68
2.57
2.46
2.41
2.35
2.29
2.22
2.16
2.09
21
5.83
4.42
3.82
3.48
3.25
3.09
2.97
2.87
2.80
2.73
2.64
2.53
2.42
2.37
2.31
2.25
2.18
2.11
2.04
22
5.79
4.38
3.78
3.44
3.22
3.05
2.93
2.84
2.76
2.70
2.60
2.50
2.39
2.33
2.27
2.21
2.14
2.08
2.00
23
5.75
4.35
3.75
3.41
3.18
3.02
2.90
2.81
2.73
2.67
2.57
2.47
2.36
2.30
2.24
2.18
2.11
2.04
1.97
24
5.72
4.32
3.72
3.38
3.15
2.99
2.87
2.78
2.70
2.64
2.54
2.44
2.33
2.27
2.21
2.15
2.08
2.01
1.94
25
5.69
4.29
3.69
3.35
3.13
2.97
2.85
2.75
2.68
2.61
2.51
2.41
2.30
2.24
2.18
2.12
2.05
1.98
1.91
26
5.66
4.27
3.67
3.33
3.10
2.94
2.82
2.73
2.65
2.59
2.49
2.39
2.28
2.22
2.16
2.09
2.03
1.95
1.88
27
5.63
4.24
3.65
3.31
3.08
2.92
2.80
2.71
2.63
2.57
2.47
2.36
2.25
2.19
2.13
2.07
2.00
1.93
1.85
28
5.61
4.22
3.63
3.29
3.06
2.90
2.78
2.69
2.61
2.55
2.45
2.34
2.23
2.17
2.11
2.05
1.98
1.91
1.83
29
5.59
4.20
3.61
3.27
3.04
2.88
2.76
2.67
2.59
2.53
2.43
2.32
2.21
2.15
2.09
2.03
1.96
1.89
1.81
30
5.57
4.18
3.59
3.25
3.03
2.87
2.75
2.65
2.57
2.51
2.41
2.31
2.20
2.14
2.07
2.01
1.94
1.87
1.79
40
5.42
4.05
3.46
3.13
2.90
2.74
2.62
2.53
2.45
2.39
2.29
2.18
2.07
2.01
1.94
1.88
1.80
1.72
1.64
60
5.29
3.93
3.34
3.01
2.79
2.63
2.51
2.41
2.33
2.27
2.17
2.06
1.94
1.88
1.82
1.74
1.67
1.58
1.68
120
5.15
3.80
3.23
2.89
2.67
2.52
2.39
2.30
2.22
2.16
2.05
1.94
1.82
1.76
1.69
1.61
1.53
1.43
1.31
∞5.02
3.69
3.12
2.79
2.57
2.41
2.29
2.19
2.11
2.05
1.94
1.83
1.71
1.64
1.57
1.48
1.39
1.27
1.00
26
Tabla
A.6.5:Distribucion
Fnm.P(F
nm≥
a)=
0.01
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
14052.19
4999.50
5403.00
5625.00
5764.00
5859.00
5928.00
5982.00
6022.00
6056.00
6106.00
6157.00
6209.00
6235.00
6261.00
6287.00
6313.00
6399.00
6366.00
298.50
99.00
99.17
99.25
99.30
99.33
99.36
99.37
99.39
99.40
99.42
99.43
99.45
99.46
99.47
99.47
99.48
99.49
99.00
334.12
30.82
29.46
28.71
28.24
27.91
27.67
27.49
27.35
27.23
27.05
26.87
26.69
26.60
26.50
26.41
26.32
26.22
26.13
421.20
18.00
16.69
15.98
15.52
15.21
14.98
14.80
14.66
14.55
14.37
14.20
14.02
13.93
13.84
13.75
13.65
13.56
13.46
516.26
13.27
12.06
11.39
10.97
10.67
10.46
10.29
10.16
10.05
9.89
9.72
9.55
9.47
9.38
9.29
9.20
9.11
9.02
613.75
10.92
9.78
9.15
8.75
8.47
8.26
8.10
7.98
7.87
7.72
7.56
7.40
7.31
7.23
7.14
7.06
6.97
6.88
712.25
9.55
8.45
7.85
7.46
7.19
6.99
6.84
6.72
6.62
6.47
6.31
6.16
6.07
5.99
5.91
5.82
5.74
5.65
811.26
8.65
7.59
7.01
6.63
6.37
6.18
6.03
5.91
5.81
5.67
5.52
5.36
5.28
5.20
5.12
5.03
4.95
4.86
910.56
8.02
6.99
6.42
6.06
5.80
5.61
5.47
5.35
5.26
5.11
4.96
4.81
4.73
4.65
4.57
4.48
4.40
4.31
10
10.04
7.56
6.55
5.99
5.64
5.39
5.20
5.06
4.94
4.85
4.71
4.56
4.41
4.33
4.25
4.17
4.08
4.00
3.91
11
9.65
7.21
6.22
5.67
5.32
5.07
4.89
4.74
4.63
4.54
4.40
4.25
4.10
4.02
3.94
3.86
3.78
3.69
3.60
12
9.33
6.93
5.95
5.41
5.06
4.82
4.64
4.50
4.39
4.30
4.16
4.01
3.86
3.78
3.70
3.62
3.54
3.45
3.36
13
9.07
6.70
5.74
5.21
4.86
4.62
4.44
4.30
4.19
4.10
3.96
3.82
3.66
3.59
3.51
3.43
3.34
3.25
3.17
14
8.86
6.51
5.56
5.04
4.69
4.46
4.28
4.14
4.03
3.94
3.80
3.66
3.51
3.43
3.35
3.27
3.18
3.09
3.00
15
8.68
6.36
5.42
4.89
4.56
4.32
4.14
4.00
3.89
3.80
3.67
3.52
3.37
3.29
3.21
3.13
3.05
2.96
2.87
16
8.53
6.23
5.29
4.77
4.44
4.20
4.03
3.89
3.78
3.69
3.55
3.41
3.26
3.18
3.10
3.02
2.93
2.84
2.75
17
8.40
6.11
5.19
4.67
4.34
4.10
3.93
3.79
3.68
3.59
3.46
3.31
3.16
3.08
3.00
2.92
2.83
2.75
2.65
18
8.29
6.01
5.09
4.58
4.25
4.01
3.84
3.71
3.60
3.51
3.37
3.23
3.08
3.00
2.92
2.84
2.75
2.66
2.57
19
8.18
5.93
5.01
4.50
4.17
3.94
3.77
3.63
3.52
3.43
3.30
3.15
3.00
2.92
2.84
2.76
2.67
2.58
2.49
20
8.10
5.85
4.94
4.43
4.10
3.87
3.70
3.56
3.46
3.37
3.23
3.09
2.94
2.86
2.78
2.69
2.61
2.52
2.42
21
8.02
5.78
4.87
4.37
4.04
3.81
3.64
3.51
3.40
3.31
3.17
3.03
2.88
2.80
2.72
2.64
2.55
2.46
2.36
22
7.95
5.72
4.82
4.31
3.99
3.76
3.59
3.45
3.35
3.26
3.12
2.98
2.83
2.75
2.67
2.58
2.50
2.40
2.31
23
7.88
5.66
4.76
4.26
3.94
3.71
3.54
3.41
3.30
3.21
3.07
2.93
2.78
2.70
2.62
2.54
2.45
2.35
2.26
24
7.82
5.61
4.72
4.22
3.90
3.67
3.50
3.36
3.26
3.17
3.03
2.89
2.74
2.66
2.58
2.49
2.40
2.31
2.21
25
7.77
5.57
4.68
4.18
3.85
3.63
3.46
3.32
3.22
3.13
2.99
2.85
2.70
2.62
2.54
2.45
2.36
2.27
2.17
26
7.72
5.53
4.64
4.14
3.82
3.59
3.42
3.29
3.18
3.09
2.96
2.81
2.66
2.58
2.50
2.42
2.33
2.23
2.13
27
7.68
5.49
4.60
4.11
3.78
3.56
3.39
3.26
3.15
3.06
2.93
2.78
2.63
2.55
2.47
2.38
2.29
2.20
2.10
28
7.64
5.45
4.57
4.07
3.75
3.53
3.36
3.23
3.12
3.03
2.90
2.75
2.60
2.52
2.44
2.35
2.26
2.17
2.06
29
7.60
5.42
4.54
4.04
3.73
3.50
3.33
3.20
3.09
3.00
2.87
2.73
2.57
2.49
2.41
2.33
2.23
2.14
2.03
30
7.56
5.39
4.51
4.02
3.70
3.47
3.30
3.17
3.07
2.98
2.84
2.70
2.55
2.47
2.39
2.30
2.21
2.11
2.01
40
7.31
5.18
4.31
3.83
3.51
3.29
3.12
2.99
2.89
2.80
2.66
2.52
2.37
2.29
2.20
2.11
2.02
1.92
1.80
60
7.08
4.98
4.13
3.65
3.34
3.12
2.95
2.82
2.72
2.63
2.50
2.35
2.20
2.12
2.03
1.94
1.84
1.73
1.60
120
6.85
4.79
3.95
3.48
3.17
2.96
2.79
2.66
2.56
2.47
2.34
2.19
2.03
1.95
1.86
1.76
1.66
1.53
1.38
∞6.63
4.61
3.78
3.32
3.02
2.80
2.64
2.51
2.41
2.32
2.18
2.04
1.88
1.79
1.70
1.59
1.47
1.32
1.00
27
Tabla
A.6.6:Distribucion
Fnm.P(F
nm≥
a)=
0.005
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
1162.11∗
200.00∗
216.15∗
225.00∗
230.56∗
234.37∗
237.15∗
239.25∗
240.91∗
242.24∗
244.26∗
246.30∗
248.36∗
249.40∗
250.44∗
251.48∗
252.53∗
253.59∗
254.65∗
2198.50
199.00
199.17
199.25
199.30
199.33
199.36
199.37
199.39
199.40
199.42
199.43
199.45
199.46
199.47
199.47
199.48
199.49
199.50
355.55
49.80
47.47
46.19
45.39
44.84
44.43
44.13
43.88
43.69
43.39
43.08
42.78
42.62
42.47
42.31
42.15
41.99
41.83
431.33
26.28
24.26
23.15
22.46
21.97
21.62
21.35
21.14
20.97
20.70
20.44
20.17
20.03
19.89
19.75
19.61
19.47
19.32
522.78
18.31
16.53
15.56
14.94
14.51
14.20
13.96
13.77
13.62
13.38
13.15
12.90
12.78
12.66
12.53
12.40
12.27
12.14
618.63
14.54
12.92
12.03
11.46
11.07
10.79
10.57
10.39
10.25
10.03
9.81
9.59
9.47
9.36
9.24
9.12
9.00
8.88
716.24
12.40
10.88
10.05
9.52
9.16
8.89
8.68
8.51
8.38
8.18
7.97
7.75
7.64
7.53
7.42
7.31
7.19
7.08
814.69
11.04
9.60
8.81
8.30
7.95
7.69
7.50
7.34
7.21
7.01
6.81
6.61
6.50
6.40
6.29
6.18
6.06
5.95
913.61
10.11
8.72
7.96
7.47
7.13
6.88
6.69
6.54
6.42
6.23
6.03
5.83
5.73
5.62
5.52
5.41
5.30
5.19
10
12.83
9.43
8.08
7.34
6.87
6.54
6.30
6.12
5.97
5.85
5.66
5.47
5.27
5.17
5.07
4.97
4.86
4.75
4.64
11
12.23
8.91
7.60
6.88
6.42
6.10
5.86
5.68
5.54
5.42
5.24
5.05
4.86
4.76
4.65
4.55
4.44
4.34
4.23
12
11.75
8.51
7.23
6.52
6.07
5.76
5.52
5.35
5.20
5.09
4.91
4.72
4.53
4.43
4.33
4.23
4.12
4.01
3.90
13
11.37
8.19
6.93
6.23
5.79
5.48
5.25
5.08
4.94
4.82
4.64
4.46
4.27
4.17
4.07
3.97
3.87
3.76
3.65
14
11.06
7.92
6.68
6.00
5.56
5.26
5.03
4.86
4.72
4.60
4.43
4.25
4.06
3.96
3.86
3.76
3.66
3.55
3.44
15
10.80
7.70
6.48
5.80
5.37
5.07
4.85
4.67
4.54
4.42
4.25
4.07
3.88
3.79
3.69
3.58
3.48
3.37
3.26
16
10.58
7.51
6.30
5.64
5.21
4.91
4.69
4.52
4.38
4.27
4.10
3.92
3.73
3.64
3.54
3.44
3.33
3.22
3.11
17
10.38
7.35
6.16
5.50
5.07
4.78
4.56
4.39
4.25
4.14
3.97
3.79
3.61
3.51
3.41
3.31
3.21
3.10
2.98
18
10.22
7.21
6.03
5.37
4.96
4.66
4.44
4.28
4.14
4.03
3.86
3.68
3.50
3.40
3.30
3.20
3.10
2.99
2.87
19
10.07
7.09
5.92
5.27
4.85
4.56
4.34
4.18
4.04
3.93
3.76
3.59
3.40
3.31
3.21
3.11
3.00
2.89
2.78
20
9.94
6.99
5.82
5.17
4.76
4.47
4.26
4.09
3.96
3.85
3.68
3.50
3.32
3.22
3.12
3.02
2.92
2.81
2.69
21
9.83
6.89
5.73
5.09
4.68
4.39
4.18
4.01
3.88
3.77
3.60
3.43
3.24
3.15
3.05
2.95
2.84
2.73
2.61
22
9.73
6.81
5.65
5.02
4.61
4.32
4.11
3.94
3.81
3.70
3.54
3.36
3.18
3.08
2.98
2.88
2.77
2.66
2.55
23
9.63
6.73
5.58
4.95
4.54
4.26
4.05
3.88
3.75
3.64
3.47
3.30
3.12
3.02
2.92
2.82
2.71
2.60
2.48
24
9.55
6.66
5.52
4.89
4.49
4.20
3.99
3.83
3.69
3.59
3.42
3.25
3.06
2.97
2.87
2.77
2.66
2.55
2.43
25
9.48
6.60
5.46
4.84
4.43
4.15
3.94
3.78
3.64
3.54
3.37
3.20
3.01
2.92
2.82
2.72
2.61
2.50
2.38
26
9.41
6.54
5.41
4.79
4.38
4.10
3.89
3.73
3.60
3.49
3.33
3.15
2.97
2.87
2.77
2.67
2.56
2.45
2.33
27
9.34
6.49
5.36
4.74
4.34
4.06
3.85
3.69
3.56
3.45
3.28
3.11
2.93
2.83
2.73
2.63
2.52
2.41
2.29
28
9.28
6.44
5.32
4.70
4.30
4.02
3.81
3.65
3.52
3.41
3.25
3.07
2.89
2.79
2.69
2.59
2.48
2.37
2.25
29
9.23
6.40
5.28
4.66
4.26
3.98
3.77
3.61
3.48
3.38
3.21
3.04
2.86
2.76
2.66
2.56
2.45
2.33
2.21
30
9.18
6.35
5.24
4.62
4.23
3.95
3.74
3.58
3.45
3.34
3.18
3.01
2.82
2.73
2.63
2.52
2.42
2.30
2.18
40
8.83
6.07
4.98
4.37
3.99
3.71
3.51
3.35
3.22
3.12
2.95
2.78
2.60
2.50
2.40
2.30
2.18
2.06
1.93
60
8.49
5.79
4.73
4.14
3.76
3.49
3.29
3.13
3.01
2.90
2.74
2.57
2.39
2.29
2.19
2.08
1.96
1.83
1.69
120
8.18
5.54
4.50
3.92
3.55
3.28
3.09
2.93
2.81
2.71
2.54
2.37
2.19
2.09
1.98
1.87
1.75
1.61
1.43
∞7.88
5.30
4.28
3.72
3.35
3.09
2.90
2.74
2.62
2.52
2.36
2.19
2.00
1.90
1.79
1.67
1.53
1.36
1.00
*Muliplicarpor
100
28
Tabla
A.6.7:Distribucion
Fnm.P(F
nm≥
a)=
0.001
Gra
dosde
Gra
dosdellibertad
delnum
era
dor(n
)
libertad
del
denomin
ador(m
)1
23
45
67
89
10
12
15
20
24
30
40
60
120
∞
14053∗
5000∗
5404∗
5625∗
5764∗
5859∗
5929∗
5981∗
6023∗
6056∗
6107∗
6158∗
6209∗
6235∗
6261∗
6287∗
6313∗
6340∗
6366∗
2998.50
999.00
999.20
999.20
999.30
999.30
999.40
999.40
999.40
999.40
999.40
999.40
999.40
999.50
999.50
999.50
999.50
999.50
999.50
3167.00
148.50
141.10
137.10
134.60
132.80
131.60
130.60
129.90
129.20
128.30
127.40
126.40
125.90
125.40
125.00
124.50
124.00
123.50
474.14
61.25
56.18
53.44
51.71
50.53
49.66
49.00
48.47
48.05
47.41
46.76
46.10
45.77
45.43
45.09
44.75
44.40
44.05
547.18
37.12
33.20
31.09
29.75
28.83
28.16
27.65
27.24
26.92
26.42
25.91
25.39
25.13
24.87
24.60
24.33
24.06
23.79
635.51
27.00
23.70
21.92
20.80
20.03
19.46
19.03
18.69
18.41
17.99
17.56
17.12
16.90
16.67
16.44
16.21
15.98
15.75
729.25
21.69
18.77
17.20
16.21
15.52
15.02
14.63
14.33
14.08
13.71
13.32
12.93
12.73
12.53
12.33
12.12
11.91
11.70
825.41
18.49
15.83
14.39
13.48
12.86
12.40
12.05
11.77
11.54
11.19
10.84
10.48
10.30
10.11
9.92
9.73
9.53
9.33
922.86
16.39
13.90
12.56
11.71
11.13
10.70
10.37
10.11
9.89
9.57
9.24
8.90
8.72
8.55
8.37
8.19
8.00
7.81
10
21.04
14.91
12.55
11.28
10.48
9.93
9.52
9.20
8.96
8.75
8.45
8.13
7.80
7.64
7.47
7.30
7.12
6.94
6.76
11
19.69
13.81
11.56
10.35
9.58
9.05
8.66
8.35
8.12
7.92
7.63
7.32
7.01
6.85
6.68
6.52
6.35
6.17
6.00
12
18.64
12.97
10.80
9.63
8.89
8.38
8.00
7.71
7.48
7.29
7.00
6.71
6.40
6.25
6.09
5.93
5.76
5.59
5.42
13
17.82
12.31
10.21
9.07
8.35
7.86
7.49
7.21
6.98
6.80
6.52
6.23
5.93
5.78
5.63
5.47
5.30
5.14
4.97
14
17.14
11.78
9.73
8.62
7.92
7.44
7.08
6.80
6.58
6.40
6.13
5.85
5.56
5.41
5.25
5.10
4.94
4.77
4.60
15
16.59
11.34
9.34
8.25
7.57
7.09
6.74
6.47
6.26
6.08
5.81
5.54
5.25
5.10
4.95
4.80
4.64
4.47
4.31
16
16.12
10.97
9.01
7.94
7.27
6.80
6.46
6.19
5.98
5.81
5.55
5.27
4.99
4.85
4.70
4.54
4.39
4.23
4.06
17
15.72
10.66
8.73
7.68
7.02
6.56
6.22
5.96
5.75
5.58
5.32
5.05
4.78
4.63
4.48
4.33
4.18
4.02
3.85
18
15.38
10.39
8.49
7.46
6.81
6.35
6.02
5.76
5.56
5.39
5.13
4.87
4.59
4.45
4.30
4.15
4.00
3.84
3.67
19
15.08
10.16
8.28
7.27
6.62
6.18
5.85
5.59
5.39
5.22
4.97
4.70
4.43
4.29
4.14
3.99
3.84
3.68
3.51
20
14.82
9.95
8.10
7.10
6.46
6.02
5.69
5.44
5.24
5.08
4.82
4.56
4.29
4.15
4.00
3.86
3.70
3.54
3.38
21
14.59
9.77
7.94
6.95
6.32
5.88
5.56
5.31
5.11
4.95
4.70
4.44
4.17
4.03
3.88
3.74
3.58
3.42
3.26
22
14.38
9.61
7.80
6.81
6.19
5.76
5.44
5.19
4.99
4.83
4.58
4.33
4.06
3.92
3.78
3.63
3.48
3.32
3.15
23
14.20
9.47
7.67
6.70
6.08
5.65
5.33
5.09
4.89
4.73
4.48
4.23
3.96
3.82
3.68
3.53
3.38
3.22
3.05
24
14.03
9.34
7.55
6.59
5.98
5.55
5.23
4.99
4.80
4.64
4.39
4.14
3.87
3.74
3.59
3.45
3.29
3.14
2.97
25
13.88
9.22
7.45
6.49
5.89
5.46
5.15
4.91
4.71
4.56
4.31
4.06
3.79
3.66
3.52
3.37
3.22
3.06
2.89
26
13.74
9.12
7.36
6.41
5.80
5.38
5.07
4.83
4.64
4.48
4.24
3.99
3.72
3.59
3.44
3.30
3.15
2.99
2.82
27
13.61
9.02
7.27
6.33
5.73
5.31
5.00
4.76
4.57
4.41
4.17
3.92
3.66
3.52
3.38
3.23
3.08
2.92
2.75
28
13.50
8.93
7.19
6.25
5.66
5.24
4.93
4.69
4.50
4.35
4.11
3.86
3.60
3.46
3.32
3.18
3.02
2.86
2.69
29
13.39
8.85
7.12
6.19
5.59
5.18
4.87
4.64
4.45
4.29
4.05
3.80
3.54
3.41
3.27
3.12
2.97
2.81
2.64
30
13.29
8.77
7.05
6.12
5.53
5.12
4.82
4.58
4.39
4.24
4.00
3.75
3.49
3.36
3.22
3.07
2.92
2.76
2.59
40
12.61
8.25
6.59
5.70
5.13
4.73
4.44
4.21
4.02
3.87
3.64
3.40
3.14
3.01
2.87
2.73
2.57
2.41
2.23
60
11.97
7.77
6.17
5.31
4.76
4.37
4.09
3.86
3.69
3.54
3.32
3.08
2.83
2.69
2.55
2.41
2.25
2.08
1.89
120
11.38
7.32
5.78
4.95
4.42
4.04
3.77
3.55
3.38
3.24
3.02
2.78
2.53
2.40
2.26
2.11
1.95
1.76
1.54
∞10.83
6.91
5.42
4.62
4.10
3.74
3.47
3.27
3.10
2.96
2.74
2.51
2.27
2.13
1.99
1.84
1.66
1.45
1.00
*Muliplicarpor
100
29
Tabla A.7: Distribucion del Estadıstico ∆n de Kolmogorov-Smirnov. P (∆n > x) = p
p 0.2 0.1 0.05 0.02 0.01
n
2 0.684 0.776 0.842 0.900 0.929
3 0.565 0.636 0.708 0.785 0.829
4 0.493 0.565 0.624 0.689 0.734
5 0.447 0.509 0.563 0.627 0.669
6 0.410 0.468 0.519 0.577 0.617
7 0.381 0.436 0.483 0.538 0.576
8 0.358 0.410 0.454 0.507 0.542
9 0.339 0.387 0.430 0.480 0.513
10 0.323 0.369 0.409 0.457 0.489
11 0.308 0.352 0.391 0.437 0.468
12 0.296 0.338 0.375 0.419 0.449
13 0.285 0.325 0.361 0.404 0.432
14 0.275 0.314 0.349 0.390 0.418
15 0.266 0.304 0.338 0.377 0.404
16 0.258 0.295 0.327 0.366 0.392
17 0.250 0.286 0.318 0.355 0.381
18 0.244 0.279 0.309 0.346 0.371
19 0.237 0.271 0.301 0.337 0.361
20 0.232 0.265 0.294 0.329 0.352
21 0.226 0.259 0.287 0.321 0.344
22 0.221 0.253 0.281 0.314 0.337
23 0.216 0.247 0.275 0.307 0.330
24 0.212 0.242 0.269 0.301 0.323
25 0.208 0.238 0.264 0.295 0.317
26 0.204 0.233 0.259 0.290 0.311
27 0.200 0.229 0.254 0.284 0.305
28 0.197 0.225 0.250 0.279 0.300
29 0.193 0.221 0.246 0.275 0.295
30 0.190 0.218 0.242 0.270 0.290
31 0.187 0.214 0.238 0.266 0.285
32 0.184 0.211 0.234 0.262 0.281
33 0.182 0.208 0.231 0.258 0.277
34 0.179 0.205 0.227 0.254 0.273
35 0.177 0.202 0.224 0.251 0.269
36 0.174 0.199 0.221 0.247 0.265
37 0.172 0.196 0.218 0.244 0.262
38 0.170 0.194 0.215 0.241 0.258
39 0.168 0.191 0.213 0.238 0.255
40 0.165 0.189 0.210 0.235 0.252
n > 40 1.07/√n 1.22/
√n 1.36/
√n 1.52/
√n 1.63/
√n