Formulario Estadística, Ejercicios de Estadística

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SECCIÓ D'ESTADÍSTICA

Taules

19 de febrer de 2013

SECCIÓ D'ESTADÍSTICA

1 Estad´ıstica Descriptiva

Mitjana aritm`etica X¯ = (^1) n ∑ni= 1 Xi

Vari`ancia mostral S^2 = (^1) n ∑ni= 1 (Xi − X¯)^2 = (^1) n

∑ni= 1 X i^2

− X¯^2

Vari`ancia mostral corregida Sˆ^2 = (^) n−n 1 S^2 Taula 1: Principals f´ormules d’Estad´ıstica Descriptiva

2 C`alcul de probabilitats

Siguin A, B ∈ Ω i P una partici´o de Ω: P = {E 1 ,... , Ek}, aleshores tenim:

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Probabilitat condicionada

P(A/B) =

P (A ∩ B)

P(B)

Regla del Producte / Probabilitats composades

P(A 1 ∩ A 2 ∩... An) = P(A 1 ) · P(A 2 /A 1 ) · P(A 3 /A 1 ∩ A 2 ) ·... · P(An/A 1 ∩ An− 1 )

Teorema de les probabilitats totals

P(A) = P(A ∩ E 1 ) + · · · + P(A ∩ EK ) = P(A/E 1 )P(E 1 ) + · · · + P(A/Ek)P(Ek)

Teorema de Bayes

P(Ei/A) =

P(A/Ei)P(Ei) ∑kj= 1 P(A/E^ j)P(E^ j)

P(A/Ei)P(Ei) P(A/E 1 )P(E 1 ) + · · · + P(A/Ek)P(Ek)

Taula 2: Principals f´ormules del C`alcul de Probabilitats

SECCIÓ D'ESTADÍSTICA

UNIVERSITAT DE BARCELON

4 Infer`encia Estad´ıstica

A totes les taules

α 2 =^ P

Z > z^ α 2

= P

T > t^ α 2

= P

χ^2 > χ^2 α 2

= P

F > f^ α 2

Z Distribuci´o N(0,1) T Distribuci´o t de Student χ^2 Distribuci´o Khi-quadrat F Distribuci´o F de Fisher

Distribuci´o Par`ametre Interval

N( μ, σ 0 ) μ

X¯n − z α 2 √^ σ^0 n ,^ X¯n + z α 2 √^ σ^0 n

( σ 0 coneguda)

N( μ, σ ) μ

X¯n − t^ α 2 √S^ ˆ n ,^ X¯n^ +^ t^ α 2 √S^ ˆ n

( σ desconeguda) graus de llibertat = n − 1

N( μ, σ ) σ 2

S^2 n χ^2 α 2 ,^

S^2 n χ 12 − α 2

graus de llibertat = n − 1 Dues poblacions N( μ 1 , σ ) N( μ 2 , σ ) μ 1 − μ 2 ( X¯ 1 − X¯ 2 ) ± t^ α 2 · SˆT ·

1 n 1 +^

1 n 2 S^ ˆ^2 T = (n^1 −^1 )·^ Sˆ^21 +(n^2 −^1 )·^ Sˆ^22 n 1 +n 2 − 2 ( σ com´u i desconeguda ) graus de llibertat = n 1 + n 2 − 2 Dues poblacions N( μ 1 , σ 1 ) N( μ 2 , σ 2 )^ σ^ 12 σ 22

( Sˆ 2

1 /^ Sˆ^22 F α/ 2 ,^

Sˆ^21 / Sˆ^22 F 1 − α/ 2

graus de llibertat = (n 1 − 1 )/(n 2 − 1 )

Bernoulli (p) p

p ˆ ± z^ α 2

pˆ qˆ n

(asimpt`otic aproximat)

Taula 4: Principals Intervals de confian¸ca. En tots els casos es suposa la construcci´o d’un interval del ( 1 − α) · 100% de confian¸ca.

Distribuci´o Par`ametre Mida de la mostra

N( μ, σ ) μ n =

t^2 α/ 2 Sˆ^2 d^2 ( σ desconeguda) graus de llibertat = n − 1

Bernoulli (p) p n =

z^2 α/ 2 pq d^2

Taula 5: Mides de mostra necessaries per assolir una precisi´o del interval amb radi maxim d

SECCIÓ D'ESTADÍSTICA

UNIVERSITAT DE BARCELON

Contrast

H

0

Estad´

ıstic de test

Dist. de refer`

encia

H

1

Rebuig de

H

0

X

normal

coneguda

0

Z

X −

μ^0 σ

n^

Z^

N

(^0

,^1

0

0

0

|Z

|^

z

α/

2

Z^

z

α

Z^

z^ α

X

normal,

desconeguda

0

T^

X − μ^0 ˆS

n^

T^

T Stud.

n^

)^

gl

0

0

0

|T

|^

t^ α

/^2

T^

t^ α

T^

t^ α

X

normal

desconeguda

(^2)

(^2 )

2

( n−

1 )

(^2) ˆ S (^2) σ 0

Khi-quadrat(n-1) gl

(^2)

(^2 )

(^2)

(^2 )

(^2)

(^2 )

2

(^2) α/

2

o

2

21 −

α /^2

2

(^2) α

2

21 −

α

X

X

2

normals

2

desconegudes,

1

2

d^

d

0

(d

T

( X

− 1 X^2

)−

d^0

ˆST^

√·

1 n^1

(^1) n^2

2 ˆS T

( n^1

− 1 )

ˆ· S 2 +( 1

n^2

− 1 )

ˆ· S 22

n^1

n^2

− 2

T^

T Stud.

(n

n

2

)^

gl

d

d

0

d^

d

0

d^

d

0

|T

|^

t^ α

/^2

T^

t^ α

T^

t^ α

X

X

2

normals

2

desconegudes,

1

2

d^

d

0

(d

T

(X

− 1 X^2

)−

d^0

ˆ( S 2 / 1

n^1

)+(

(^2) ˆ S 2 /n

) 2

T^

T Stud.

gl

ˆ( S 2 / 1

n^1

(^2) ˆ S 2 /n

) 2 2

(^2) ˆ ( S 1 /n

) 1 2 n^1 +^1

ˆ( S 2 / 2

n^2

(^2) ) n^2 +^1

d

d

0

d^

d

0

d^

d

0

|T

|^

T

′ (^) α/

2

T^

T

′ (^) α

T^

T

′ (^) α

X

X

2

normals

2 1

2 2

F

ˆS (^212) ˆS 2

F

F Fisher

(n

,^ n

)^

gl

(^2 )

(^2 )

2 1

(^2 )

2 1

(^2 )

F

f

α /^2

o

F

f

1 −

α /^2

F

f

α

F

f

1 −

α

Taula 6: Principals Contrastos d’hip`

otesis per a dades cont´

ınues

SECCIÓ D'ESTADÍSTICA

UNIVERSITAT DE BARCELON

Contrast

H

0

Estad´

ıstic de test

Dist. de refer`

encia

H

1

Rebuig de

H

0

X

Bernoulli

n^

,^

np

0

p^

p

0

Z

ˆp−

p^0 √

p^0

q^0 n

Z^

N

(^0

,^1

p

p

0

p^

p

0

p^

p

0

|Z

|^

z

α/

2

Z

z

α

Z

z^ α

X

1

Bernoulli

X

2

Bernoulli

n^1

,^

n^2

n^1

ˆp

n

2

ˆp^2

p^1

p

2

Z^

ˆp^1 − ˆp^2 √

ˆp^ ˆqn^1

ˆp^ ˆq n^2

ˆp^

n

ˆp 1

• 1 n^2

ˆp 2

n^1

n^2

Y^

N

(^0

,^1

p^1

p

2

p^1

p

2

p^1

p

2

|Z

|^

z

α/

2

Z

z

α

Z

z^ α

Condicions

Contrast

Estad´

ıstic de test

Dist. de refer`

encia

Rebuig de

H

0

X

MN

(n

,^ (

p^1

p

))k

en m´

es del 80 %

de les classes

E

≥i

H

0

:^

p^

0 p 1

p

0 ) k^

H

1

:^

p

0 p 1

p

0 ) k^

2

k ∑ i=^1

(O

−i

E

)i

2

E

i

O

:i

freq¨

u`

encia observada

E

:i

freq¨

u`

encia esperada

E

=i

n

·^

pi

2

Khi

Quadrat

k^

m

gl.

m

nombre de

par`

ametres estimats

2

(^2) α

A

1

A

s

B

1

n^11

n^1

s^

n^1

...^

...^

...^

...^

B

r^

ns

1

nrs

nr

n+

1

n+

s^

n

en m´

es del 80 %

de les caselles

E

i j^

Test de homogene¨

ıtat

Test d’independ`

encia

2

r ∑ i=^1

s ∑ j=^1

(n

i j^

E

i j

E

i j

E

i j

:^

freq¨

u`encia esperada

E

i j

n i+

n+

j n

2

Khi

Quadrat

(r

)^ ·

s^ −

)^

gl.

2

(^2) α

Correcci´

o de Yates

Test de homogene¨

ıtat

Test d’independ`

encia

2

n

n^11

n^22

− n^12

n^21

|−

n ) 2

2

n^1

n+ 2 +

n+

n 1 +^2

2

Khi

Quadrat

gl.

2

(^2) α

Taula 8: Principals Contrastos d’hip`

otesis per a proporcions i test khi-quadrat

SECCIÓ D'ESTADÍSTICA

5 Regressi´o

Recta de regressi´o de Y/X

Y^ ˆ = a + bX

on

b = ∑i( ∑Xi−X)(Yi−Y^ )

i(Xi−X)^2

a = Y − bX

Par`ametre Interval

β (Pendent de la recta) b ± t α/ 2

S^2 E ∑ni= 1 (Xi−X)^2

Predicci´o del valor mig de Y donat X 0 a + bX 0 ± t α/ 2

1

n +^

(X 0 −X)^2 ∑ni= 1 (Xi−X)^2

S^2 E

Predicci´o d’un valor individual de Y donat X 0 a + bX 0 ± t α/ 2

1

n +^1 +^

(X 0 −X)^2 ∑ni= 1 (Xi−X)^2

S^2 E

on S^2 E = (^) n−^12 ∑ni= 1 (Yi − Yˆi)^2 = (^) n−^12

∑ni= 1 (Yi −Y^ )^2 −^ b^ ∑ni= 1 (Xi −^ X)(Yi −Y^ )

En tots els casos es suposa la construcci´o d’un interval del ( 1 − α) · 100% de confian¸ca.

Covari`ancia entre X i Y

Sxy = 1 n ∑ni= 1 (Xi − X)(Yi −Y )

Coeficient de correlaci´o

r = SSxxySy

Coeficient de determinaci´o

r^2 = 1 −

S^2 E S^2 y

Contrast Estad´ıstic de test Dist. de refer`encia Rebuig de H 0

H 0 : ρ = 0

H 1 : ρ ̸= 0

T =

n − 2 √r

1 −r^2

T ∼ T Stud. (n − 2 ) gl |T | ≥ t α/ 2

Taula 9: An`alisi Estad´ıstic en Regressi´o

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SECCIÓ D'ESTADÍSTICA

 - 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. 6 Distribuci´o Normal N(0,1): Funci´o de distribuci´o - 0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0. 

• 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.
• 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.
• 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.
• 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.
• 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.
• 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.
• 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.
• 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.
• 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.
  • 1 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.
• 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.
• 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.
• 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.
• 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.
• 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.
• 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.
• 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.
• 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.
• 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.
  • 2 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.
• 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.
• 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.
• 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.
• 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.
• 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.
• 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.
• 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.
• 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.
• 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0. - x 3 3.1 3.2 3.3 3.4 3.5 3.6 3.8 Taula 10: Funci´o de distribuci´o de la distribuci´o N(0,1)
• F(x) 0.99865 0.99903 0.99931 0.99952 0.99966 0.99977 0.99984 0.99993 0. - x 2.576 2.326 2.17 2.054 1.96 1.645 1.44 1.282 1. Taula 11: Taula per grans valors de x - F(x) 0.995 0.990 0.985 0.980 0.975 0.950 0.925 0.900 0.
• gl \ F(x) 0.6 0.7 0.75 0.8 0.85 0.9 0.925 0.95 0.975 0.99 0.995 0. 7 Distribuci´o T de Student - 3 0.277 0.584 0.765 0.978 1.250 1.638 1.924 2.353 3.182 4.541 5.841 12. - 4 0.271 0.569 0.741 0.941 1.190 1.533 1.778 2.132 2.776 3.747 4.604 8. - 5 0.267 0.559 0.727 0.920 1.156 1.476 1.699 2.015 2.571 3.365 4.032 6. - 6 0.265 0.553 0.718 0.906 1.134 1.440 1.650 1.943 2.447 3.143 3.707 5. - 7 0.263 0.549 0.711 0.896 1.119 1.415 1.617 1.895 2.365 2.998 3.499 5. - 8 0.262 0.546 0.706 0.889 1.108 1.397 1.592 1.860 2.306 2.896 3.355 5. - 9 0.261 0.543 0.703 0.883 1.100 1.383 1.574 1.833 2.262 2.821 3.250 4. - 10 0.260 0.542 0.700 0.879 1.093 1.372 1.559 1.812 2.228 2.764 3.169 4. - 11 0.260 0.540 0.697 0.876 1.088 1.363 1.548 1.796 2.201 2.718 3.106 4. - 12 0.259 0.539 0.695 0.873 1.083 1.356 1.538 1.782 2.179 2.681 3.055 4. - 13 0.259 0.538 0.694 0.870 1.079 1.350 1.530 1.771 2.160 2.650 3.012 4. - 14 0.258 0.537 0.692 0.868 1.076 1.345 1.523 1.761 2.145 2.624 2.977 4. - 15 0.258 0.536 0.691 0.866 1.074 1.341 1.517 1.753 2.131 2.602 2.947 4. - 16 0.258 0.535 0.690 0.865 1.071 1.337 1.512 1.746 2.120 2.583 2.921 4. - 17 0.257 0.534 0.689 0.863 1.069 1.333 1.508 1.740 2.110 2.567 2.898 3. - 18 0.257 0.534 0.688 0.862 1.067 1.330 1.504 1.734 2.101 2.552 2.878 3. - 19 0.257 0.533 0.688 0.861 1.066 1.328 1.500 1.729 2.093 2.539 2.861 3. - 20 0.257 0.533 0.687 0.860 1.064 1.325 1.497 1.725 2.086 2.528 2.845 3. - 21 0.257 0.532 0.686 0.859 1.063 1.323 1.494 1.721 2.080 2.518 2.831 3. - 22 0.256 0.532 0.686 0.858 1.061 1.321 1.492 1.717 2.074 2.508 2.819 3. - 23 0.256 0.532 0.685 0.858 1.060 1.319 1.489 1.714 2.069 2.500 2.807 3. - 24 0.256 0.531 0.685 0.857 1.059 1.318 1.487 1.711 2.064 2.492 2.797 3. - 25 0.256 0.531 0.684 0.856 1.058 1.316 1.485 1.708 2.060 2.485 2.787 3. - 26 0.256 0.531 0.684 0.856 1.058 1.315 1.483 1.706 2.056 2.479 2.779 3. - 27 0.256 0.531 0.684 0.855 1.057 1.314 1.482 1.703 2.052 2.473 2.771 3. - 28 0.256 0.530 0.683 0.855 1.056 1.313 1.480 1.701 2.048 2.467 2.763 3. - 29 0.256 0.530 0.683 0.854 1.055 1.311 1.479 1.699 2.045 2.462 2.756 3. - 30 0.256 0.530 0.683 0.854 1.055 1.310 1.477 1.697 2.042 2.457 2.750 3. - 40 0.255 0.529 0.681 0.851 1.050 1.303 1.468 1.684 2.021 2.423 2.704 3. - 50 0.255 0.528 0.679 0.849 1.047 1.299 1.462 1.676 2.009 2.403 2.678 3. - 60 0.254 0.527 0.679 0.848 1.045 1.296 1.458 1.671 2.000 2.390 2.660 3. - 100 0.254 0.526 0.677 0.845 1.042 1.290 1.451 1.660 1.984 2.364 2.626 3.
  • 10000 0.253 0.524 0.675 0.842 1.036 1.282 1.440 1.645 1.960 2.327 2.576 3.
    • 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. SECCIÓ D'ESTADÍSTICA
  • 0 0.5000 0.5039 0.5078 0.5116 0.5155 0.5194 0.5233 0.5271 0.5310 0.
• 0.1 0.5387 0.5426 0.5464 0.5503 0.5541 0.5580 0.5618 0.5656 0.5694 0.
• 0.2 0.5770 0.5808 0.5846 0.5884 0.5921 0.5959 0.5996 0.6034 0.6071 0.
• 0.3 0.6145 0.6182 0.6219 0.6255 0.6292 0.6328 0.6364 0.6400 0.6436 0.
• 0.4 0.6508 0.6543 0.6578 0.6613 0.6648 0.6683 0.6718 0.6752 0.6787 0.
• 0.5 0.6855 0.6888 0.6922 0.6955 0.6988 0.7021 0.7054 0.7087 0.7119 0.
• 0.6 0.7183 0.7215 0.7247 0.7278 0.7309 0.7340 0.7371 0.7402 0.7432 0.
• 0.7 0.7492 0.7522 0.7551 0.7580 0.7609 0.7638 0.7667 0.7695 0.7723 0.
• 0.8 0.7778 0.7806 0.7833 0.7860 0.7887 0.7913 0.7939 0.7965 0.7991 0.
• 0.9 0.8042 0.8067 0.8092 0.8117 0.8141 0.8165 0.8189 0.8213 0.8237 0.
  • 1 0.8283 0.8306 0.8328 0.8351 0.8373 0.8395 0.8416 0.8438 0.8459 0.
• 1.1 0.8501 0.8521 0.8541 0.8562 0.8581 0.8601 0.8621 0.8640 0.8659 0.
• 1.2 0.8696 0.8715 0.8733 0.8751 0.8768 0.8786 0.8803 0.8820 0.8837 0.
• 1.3 0.8870 0.8887 0.8903 0.8919 0.8935 0.8950 0.8965 0.8981 0.8996 0.
• 1.4 0.9025 0.9039 0.9053 0.9067 0.9081 0.9095 0.9109 0.9122 0.9135 0.
• 1.5 0.9161 0.9173 0.9186 0.9198 0.9210 0.9222 0.9234 0.9246 0.9257 0.
• 1.6 0.9280 0.9291 0.9302 0.9312 0.9323 0.9333 0.9344 0.9354 0.9364 0.
• 1.7 0.9383 0.9393 0.9402 0.9412 0.9421 0.9430 0.9439 0.9447 0.9456 0.
• 1.8 0.9473 0.9481 0.9489 0.9498 0.9505 0.9513 0.9521 0.9529 0.9536 0.
• 1.9 0.9551 0.9558 0.9565 0.9572 0.9578 0.9585 0.9592 0.9598 0.9605 0.
  • 2 0.9617 0.9623 0.9629 0.9635 0.9641 0.9647 0.9653 0.9658 0.9664 0.
• 2.1 0.9674 0.9680 0.9685 0.9690 0.9695 0.9700 0.9705 0.9709 0.9714 0.
• 2.2 0.9723 0.9728 0.9732 0.9737 0.9741 0.9745 0.9749 0.9753 0.9757 0.
• 2.3 0.9765 0.9769 0.9773 0.9776 0.9780 0.9783 0.9787 0.9790 0.9794 0.
• 2.4 0.9801 0.9804 0.9807 0.9810 0.9813 0.9816 0.9819 0.9822 0.9825 0.
• 2.5 0.9831 0.9833 0.9836 0.9839 0.9841 0.9844 0.9847 0.9849 0.9852 0.
• 2.6 0.9856 0.9859 0.9861 0.9863 0.9865 0.9868 0.9870 0.9872 0.9874 0.
• 2.7 0.9878 0.9880 0.9882 0.9884 0.9886 0.9888 0.9889 0.9891 0.9893 0.
• 2.8 0.9896 0.9898 0.9900 0.9901 0.9903 0.9905 0.9906 0.9908 0.9909 0.
• 2.9 0.9912 0.9913 0.9915 0.9916 0.9918 0.9919 0.9920 0.9921 0.9923 0.
  • 3 0.9925 0.9926 0.9928 0.9929 0.9930 0.9931 0.9932 0.9933 0.9934 0.
• 3.1 0.9936 0.9937 0.9938 0.9939 0.9940 0.9941 0.9942 0.9943 0.9944 0.
• 3.2 0.9946 0.9947 0.9948 0.9948 0.9949 0.9950 0.9951 0.9952 0.9952 0.
• 3.3 0.9954 0.9955 0.9955 0.9956 0.9957 0.9957 0.9958 0.9959 0.9959 0.
• 3.4 0.9961 0.9961 0.9962 0.9962 0.9963 0.9964 0.9964 0.9965 0.9965 0.
• 3.5 0.9966 0.9967 0.9967 0.9968 0.9968 0.9969 0.9969 0.9970 0.9970 0.
• 3.6 0.9971 0.9972 0.9972 0.9973 0.9973 0.9973 0.9974 0.9974 0.9975 0.
• 3.7 0.9975 0.9976 0.9976 0.9977 0.9977 0.9977 0.9978 0.9978 0.9978 0.
• 3.8 0.9979 0.9979 0.9980 0.9980 0.9980 0.9980 0.9981 0.9981 0.9981 0.
• 3.9 0.9982 0.9982 0.9982 0.9983 0.9983 0.9983 0.9983 0.9984 0.9984 0.
• gl \ F(x) 0.01 0.025 0.05 0.1 0.25 0.5 0.75 0.9 0.925 0.95 0.975 0. 8 Distribuci´o Khi-quadrat - 1 0.000 0.001 0.004 0.016 0.102 0.455 1.323 2.706 3.170 3.841 5.024 6. - 2 0.020 0.051 0.103 0.211 0.575 1.386 2.773 4.605 5.181 5.991 7.378 9. - 3 0.115 0.216 0.352 0.584 1.213 2.366 4.108 6.251 6.905 7.815 9.348 11. - 4 0.297 0.484 0.711 1.064 1.923 3.357 5.385 7.779 8.496 9.488 11.143 13. - 5 0.554 0.831 1.145 1.610 2.675 4.351 6.626 9.236 10.008 11.070 12.833 15. - 6 0.872 1.237 1.635 2.204 3.455 5.348 7.841 10.645 11.466 12.592 14.449 16. - 7 1.239 1.690 2.167 2.833 4.255 6.346 9.037 12.017 12.883 14.067 16.013 18. - 8 1.646 2.180 2.733 3.490 5.071 7.344 10.219 13.362 14.270 15.507 17.535 20. - 9 2.088 2.700 3.325 4.168 5.899 8.343 11.389 14.684 15.631 16.919 19.023 21.
  • 10 2.558 3.247 3.940 4.865 6.737 9.342 12.549 15.987 16.971 18.307 20.483 23.
  • 11 3.053 3.816 4.575 5.578 7.584 10.341 13.701 17.275 18.294 19.675 21.920 24.
  • 12 3.571 4.404 5.226 6.304 8.438 11.340 14.845 18.549 19.602 21.026 23.337 26.
  • 13 4.107 5.009 5.892 7.042 9.299 12.340 15.984 19.812 20.897 22.362 24.736 27.
  • 14 4.660 5.629 6.571 7.790 10.165 13.339 17.117 21.064 22.180 23.685 26.119 29.
  • 15 5.229 6.262 7.261 8.547 11.037 14.339 18.245 22.307 23.452 24.996 27.488 30.
  • 16 5.812 6.908 7.962 9.312 11.912 15.338 19.369 23.542 24.716 26.296 28.845 32.
  • 17 6.408 7.564 8.672 10.085 12.792 16.338 20.489 24.769 25.970 27.587 30.191 33.
  • 18 7.015 8.231 9.390 10.865 13.675 17.338 21.605 25.989 27.218 28.869 31.526 34.
  • 19 7.633 8.907 10.117 11.651 14.562 18.338 22.718 27.204 28.458 30.144 32.852 36.
  • 20 8.260 9.591 10.851 12.443 15.452 19.337 23.828 28.412 29.692 31.410 34.170 37.
  • 21 8.897 10.283 11.591 13.240 16.344 20.337 24.935 29.615 30.920 32.671 35.479 38.
  • 22 9.542 10.982 12.338 14.041 17.240 21.337 26.039 30.813 32.142 33.924 36.781 40.
  • 23 10.196 11.689 13.091 14.848 18.137 22.337 27.141 32.007 33.360 35.172 38.076 41.
  • 24 10.856 12.401 13.848 15.659 19.037 23.337 28.241 33.196 34.572 36.415 39.364 42.
  • 25 11.524 13.120 14.611 16.473 19.939 24.337 29.339 34.382 35.780 37.652 40.646 44.
  • 26 12.198 13.844 15.379 17.292 20.843 25.336 30.435 35.563 36.984 38.885 41.923 45.
  • 27 12.879 14.573 16.151 18.114 21.749 26.336 31.528 36.741 38.184 40.113 43.195 46.
  • 28 13.565 15.308 16.928 18.939 22.657 27.336 32.620 37.916 39.380 41.337 44.461 48.
  • 29 14.256 16.047 17.708 19.768 23.567 28.336 33.711 39.087 40.573 42.557 45.722 49.
  • 30 14.953 16.791 18.493 20.599 24.478 29.336 34.800 40.256 41.762 43.773 46.979 50.
  • 31 15.655 17.539 19.281 21.434 25.390 30.336 35.887 41.422 42.948 44.985 48.232 52.
  • 32 16.362 18.291 20.072 22.271 26.304 31.336 36.973 42.585 44.131 46.194 49.480 53.
  • 33 17.074 19.047 20.867 23.110 27.219 32.336 38.058 43.745 45.311 47.400 50.725 54.
  • 34 17.789 19.806 21.664 23.952 28.136 33.336 39.141 44.903 46.488 48.602 51.966 56.
  • 35 18.509 20.569 22.465 24.797 29.054 34.336 40.223 46.059 47.663 49.802 53.203 57.
  • 36 19.233 21.336 23.269 25.643 29.973 35.336 41.304 47.212 48.835 50.998 54.437 58.
  • 37 19.960 22.106 24.075 26.492 30.893 36.336 42.383 48.363 50.005 52.192 55.668 59.
  • 38 20.691 22.878 24.884 27.343 31.815 37.335 43.462 49.513 51.173 53.384 56.896 61.
  • 39 21.426 23.654 25.695 28.196 32.737 38.335 44.539 50.660 52.338 54.572 58.120 62.
  • 40 22.164 24.433 26.509 29.051 33.660 39.335 45.616 51.805 53.501 55.758 59.342 63.
    • 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0. UNIVERSITAT DE BARCELON
  • 1 0.6827 0.6851 0.6875 0.6898 0.6922 0.6945 0.6968 0.6991 0.7013 0.
• 1.1 0.7057 0.7079 0.7101 0.7122 0.7143 0.7165 0.7185 0.7206 0.7226 0.
• 1.2 0.7267 0.7287 0.7306 0.7326 0.7345 0.7364 0.7383 0.7402 0.7421 0.
• 1.3 0.7458 0.7476 0.7494 0.7512 0.7530 0.7547 0.7565 0.7582 0.7599 0.
• 1.4 0.7633 0.7649 0.7666 0.7682 0.7699 0.7715 0.7731 0.7747 0.7762 0.
• 1.5 0.7793 0.7809 0.7824 0.7839 0.7854 0.7869 0.7883 0.7898 0.7912 0.
• 1.6 0.7941 0.7955 0.7969 0.7983 0.7997 0.8010 0.8024 0.8037 0.8051 0.
• 1.7 0.8077 0.8090 0.8103 0.8116 0.8129 0.8141 0.8154 0.8166 0.8179 0.
• 1.8 0.8203 0.8215 0.8227 0.8239 0.8250 0.8262 0.8274 0.8285 0.8297 0.
• 1.9 0.8319 0.8330 0.8341 0.8352 0.8363 0.8374 0.8385 0.8396 0.8406 0.
  • 2 0.8427 0.8437 0.8448 0.8458 0.8468 0.8478 0.8488 0.8498 0.8508 0.
• 2.1 0.8527 0.8537 0.8546 0.8556 0.8565 0.8574 0.8584 0.8593 0.8602 0.
• 2.2 0.8620 0.8629 0.8638 0.8646 0.8655 0.8664 0.8672 0.8681 0.8689 0.
• 2.3 0.8706 0.8715 0.8723 0.8731 0.8739 0.8747 0.8755 0.8763 0.8771 0.
• 2.4 0.8787 0.8794 0.8802 0.8810 0.8817 0.8825 0.8832 0.8840 0.8847 0.
• 2.5 0.8862 0.8869 0.8876 0.8883 0.8890 0.8897 0.8904 0.8911 0.8918 0.
• 2.6 0.8931 0.8938 0.8945 0.8951 0.8958 0.8965 0.8971 0.8977 0.8984 0.
• 2.7 0.8997 0.9003 0.9009 0.9015 0.9021 0.9027 0.9034 0.9040 0.9046 0.
• 2.8 0.9057 0.9063 0.9069 0.9075 0.9081 0.9086 0.9092 0.9098 0.9103 0.
• 2.9 0.9114 0.9120 0.9125 0.9131 0.9136 0.9141 0.9147 0.9152 0.9157 0.
  • 3 0.9167 0.9172 0.9178 0.9183 0.9188 0.9193 0.9198 0.9203 0.9207 0.
• 3.1 0.9217 0.9222 0.9227 0.9231 0.9236 0.9241 0.9245 0.9250 0.9255 0.
• 3.2 0.9264 0.9268 0.9273 0.9277 0.9281 0.9286 0.9290 0.9294 0.9299 0.
• 3.3 0.9307 0.9311 0.9316 0.9320 0.9324 0.9328 0.9332 0.9336 0.9340 0.
• 3.4 0.9348 0.9352 0.9356 0.9360 0.9364 0.9367 0.9371 0.9375 0.9379 0.
• 3.5 0.9386 0.9390 0.9394 0.9397 0.9401 0.9405 0.9408 0.9412 0.9415 0.
• 3.6 0.9422 0.9426 0.9429 0.9433 0.9436 0.9439 0.9443 0.9446 0.9449 0.
• 3.7 0.9456 0.9459 0.9462 0.9466 0.9469 0.9472 0.9475 0.9478 0.9481 0.
• 3.8 0.9487 0.9491 0.9494 0.9497 0.9500 0.9503 0.9505 0.9508 0.9511 0.
• 3.9 0.9517 0.9520 0.9523 0.9526 0.9528 0.9531 0.9534 0.9537 0.9540 0.
  • 4 0.9545 0.9548 0.9550 0.9553 0.9556 0.9558 0.9561 0.9563 0.9566 0.
• 4.1 0.9571 0.9574 0.9576 0.9579 0.9581 0.9584 0.9586 0.9589 0.9591 0.
• 4.2 0.9596 0.9598 0.9600 0.9603 0.9605 0.9607 0.9610 0.9612 0.9614 0.
• 4.3 0.9619 0.9621 0.9623 0.9626 0.9628 0.9630 0.9632 0.9634 0.9636 0.
• 4.4 0.9641 0.9643 0.9645 0.9647 0.9649 0.9651 0.9653 0.9655 0.9657 0.
• 4.5 0.9661 0.9663 0.9665 0.9667 0.9669 0.9671 0.9673 0.9675 0.9677 0.
• 4.6 0.9680 0.9682 0.9684 0.9686 0.9688 0.9689 0.9691 0.9693 0.9695 0.
• 4.7 0.9698 0.9700 0.9702 0.9704 0.9705 0.9707 0.9709 0.9710 0.9712 0.
• 4.8 0.9715 0.9717 0.9719 0.9720 0.9722 0.9724 0.9725 0.9727 0.9728 0.
• 4.9 0.9731 0.9733 0.9735 0.9736 0.9738 0.9739 0.9741 0.9742 0.9744 0.
  • 5 0.9747 0.9748 0.9749 0.9751 0.9752 0.9754 0.9755 0.9757 0.9758 0.
• 5.1 0.9761 0.9762 0.9763 0.9765 0.9766 0.9768 0.9769 0.9770 0.9772 0.
• 5.2 0.9774 0.9775 0.9777 0.9778 0.9779 0.9781 0.9782 0.9783 0.9784 0.
• 5.3 0.9787 0.9788 0.9789 0.9790 0.9792 0.9793 0.9794 0.9795 0.9796 0.
• 5.4 0.9799 0.9800 0.9801 0.9802 0.9803 0.9804 0.9805 0.9807 0.9808 0.
• 5.5 0.9810 0.9811 0.9812 0.9813 0.9814 0.9815 0.9816 0.9817 0.9818 0.
• 5.6 0.9820 0.9821 0.9822 0.9823 0.9824 0.9825 0.9826 0.9827 0.9828 0.
• 5.7 0.9830 0.9831 0.9832 0.9833 0.9834 0.9835 0.9836 0.9837 0.9838 0.
• 5.8 0.9840 0.9841 0.9842 0.9842 0.9843 0.9844 0.9845 0.9846 0.9847 0.
• 5.9 0.9849 0.9849 0.9850 0.9851 0.9852 0.9853 0.9854 0.9854 0.9855 0.
  • 6 0.9857 0.9858 0.9859 0.9859 0.9860 0.9861 0.9862 0.9863 0.9863 0.
• 6.1 0.9865 0.9866 0.9866 0.9867 0.9868 0.9869 0.9869 0.9870 0.9871 0.
• 6.2 0.9872 0.9873 0.9874 0.9874 0.9875 0.9876 0.9877 0.9877 0.9878 0.
• 6.3 0.9879 0.9880 0.9881 0.9881 0.9882 0.9883 0.9883 0.9884 0.9885 0.
• 6.4 0.9886 0.9887 0.9887 0.9888 0.9888 0.9889 0.9890 0.9890 0.9891 0.
• 6.5 0.9892 0.9893 0.9893 0.9894 0.9895 0.9895 0.9896 0.9896 0.9897 0.
• 6.6 0.9898 0.9899 0.9899 0.9900 0.9900 0.9901 0.9901 0.9902 0.9902 0.
• 6.7 0.9904 0.9904 0.9905 0.9905 0.9906 0.9906 0.9907 0.9907 0.9908 0.
• 6.8 0.9909 0.9909 0.9910 0.9910 0.9911 0.9911 0.9912 0.9912 0.9913 0.
• 6.9 0.9914 0.9914 0.9915 0.9915 0.9916 0.9916 0.9917 0.9917 0.9918 0.