Tablas de Distribucion Normal Estandar

Tabla 1:

Función de Distribución Normal Estándar 0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-3.0 -3. 0

0.0013 0.0 01350 50 0.0 0.0013 01306 06 0.0 0.0012 01264 64 0.0 0.0012 01223 23 0.0 0.0011 01183 83 0.0 0.0011 01144 44 0.0 0.0011 01107 07 0.0 0.0010 01070 70 0.0 0.0010 01035 35 0.0 0.0010 01001 01

-2.9 -2. 9

0.0018 0.0 01866 66 0.0 0.0018 01807 07 0.0 0.0017 01750 50 0.0 0.0016 01695 95 0.0 0.0016 01641 41 0.0 0.0015 01589 89 0.0 0.0015 01538 38 0.0 0.0014 01489 89 0.0 0.0014 01441 41 0.0 0.0013 01395 95

-2.8 -2. 8

0.0025 0.0 02555 55 0.0 0.0024 02477 77 0.0 0.0024 02401 01 0.0 0.0023 02327 27 0.0 0.0022 02256 56 0.0 0.0021 02186 86 0.0 0.0021 02118 18 0.0 0.0020 02052 52 0.0 0.0019 01988 88 0.0 0.0019 01926 26

-2.7 -2. 7

0.0034 0.0 03467 67 0.0 0.0033 03364 64 0.0 0.0032 03264 64 0.0 0.0031 03167 67 0.0 0.0030 03072 72 0.0 0.0029 02980 80 0.0 0.0028 02890 90 0.0 0.0028 02803 03 0.0 0.0027 02718 18 0.0 0.0026 02635 35

-2.6 -2. 6

0.0046 0.0 04661 61 0.0 0.0045 04527 27 0.0 0.0043 04396 96 0.0 0.0042 04269 69 0.0 0.0041 04145 45 0.0 0.0040 04025 25 0.0 0.0039 03907 07 0.0 0.0037 03793 93 0.0 0.0036 03681 81 0.0 0.0035 03573 73

-2.5 -2. 5

0.0062 0.0 06210 10 0.0 0.0060 06037 37 0.0 0.0058 05868 68 0.0 0.0057 05703 03 0.0 0.0055 05543 43 0.0 0.0053 05386 86 0.0 0.0052 05234 34 0.0 0.0050 05085 85 0.0 0.0049 04940 40 0.0 0.0047 04799 99

-2.4 -2. 4

0.0081 0.0 08198 98 0.0 0.0079 07976 76 0.0 0.0077 07760 60 0.0 0.0075 07549 49 0.0 0.0073 07344 44 0.0 0.0071 07143 43 0.0 0.0069 06947 47 0.0 0.0067 06756 56 0.0 0.0065 06569 69 0.0 0.0063 06387 87

-2.3 -2. 3

0.0107 0.0 10724 24 0.0 0.0104 10444 44 0.0 0.0101 10170 70 0.0 0.0099 09903 03 0.0 0.0096 09642 42 0.0 0.0093 09387 87 0.0 0.0091 09137 37 0.0 0.0088 08894 94 0.0 0.0086 08656 56 0.0 0.0084 08424 24

-2.2 -2. 2

0.0139 0.0 13903 03 0.0 0.0135 13553 53 0.0 0.0132 13209 09 0.0 0.0128 12874 74 0.0 0.0125 12545 45 0.0 0.0122 12224 24 0.0 0.0119 11911 11 0.0 0.0116 11604 04 0.0 0.0113 11304 04 0.0 0.0110 11011 11

-2.1 -2. 1

0.0178 0.0 17864 64 0.0 0.0174 17429 29 0.0 0.0170 17003 03 0.0 0.0165 16586 86 0.0 0.0161 16177 77 0.0 0.0157 15778 78 0.0 0.0153 15386 86 0.0 0.0150 15003 03 0.0 0.0146 14629 29 0.0 0.0142 14262 62

-2.0 -2. 0

0.0227 0.0 22750 50 0.0 0.0222 22216 16 0.0 0.0216 21692 92 0.0 0.0211 21178 78 0.0 0.0206 20675 75 0.0 0.0201 20182 82 0.0 0.0196 19699 99 0.0 0.0192 19226 26 0.0 0.0187 18763 63 0.0 0.0183 18309 09

-1.9 -1. 9

0.0287 0.0 28717 17 0.0 0.0280 28067 67 0.0 0.0274 27429 29 0.0 0.0268 26803 03 0.0 0.0261 26190 90 0.0 0.0255 25588 88 0.0 0.0249 24998 98 0.0 0.0244 24419 19 0.0 0.0238 23852 52 0.0 0.0232 23295 95

-1.8 -1. 8

0.0359 0.0 35930 30 0.0 0.0351 35148 48 0.0 0.0343 34380 80 0.0 0.0336 33625 25 0.0 0.0328 32884 84 0.0 0.0321 32157 57 0.0 0.0314 31443 43 0.0 0.0307 30742 42 0.0 0.0300 30054 54 0.0 0.0293 29379 79

-1.7 -1. 7

0.0445 0.0 44565 65 0.0 0.0436 43633 33 0.0 0.0427 42716 16 0.0 0.0418 41815 15 0.0 0.0409 40930 30 0.0 0.0400 40059 59 0.0 0.0392 39204 04 0.0 0.0383 38364 64 0.0 0.0375 37538 38 0.0 0.0367 36727 27

-1.6 -1. 6

0.0547 0.0 54799 99 0.0 0.0536 53699 99 0.0 0.0526 52616 16 0.0 0.0515 51551 51 0.0 0.0505 50503 03 0.0 0.0494 49471 71 0.0 0.0484 48457 57 0.0 0.0474 47460 60 0.0 0.0464 46479 79 0.0 0.0455 45514 14

-1.5 -1. 5

0.0668 0.0 66807 07 0.0 0.0655 65522 22 0.0 0.0642 64255 55 0.0 0.0630 63008 08 0.0 0.0617 61780 80 0.0 0.0605 60571 71 0.0 0.0593 59380 80 0.0 0.0582 58208 08 0.0 0.0570 57053 53 0.0 0.0559 55917 17

-1.4 -1. 4

0.0807 0.0 80757 57 0.0 0.0792 79270 70 0.0 0.0778 77804 04 0.0 0.0763 76359 59 0.0 0.0749 74934 34 0.0 0.0735 73529 29 0.0 0.0721 72145 45 0.0 0.0707 70781 81 0.0 0.0694 69437 37 0.0 0.0681 68112 12

-1.3 -1. 3

0.0968 0.0 96800 00 0.0 0.0950 95098 98 0.0 0.0934 93418 18 0.0 0.0917 91759 59 0.0 0.0901 90123 23 0.0 0.0885 88508 08 0.0 0.0869 86915 15 0.0 0.0853 85343 43 0.0 0.0837 83793 93 0.0 0.0822 82264 64

-1.2 -1. 2

0.1150 0.1 15070 70 0.1 0.1131 13139 39 0.1 0.1112 11232 32 0.1 0.1093 09349 49 0.1 0.1074 07488 88 0.1 0.1056 05650 50 0.1 0.1038 03835 35 0.1 0.1020 02042 42 0.1 0.1002 00273 73 0.0 0.0985 98525 25

-1.1 -1. 1

0.1356 0.1 35666 66 0.1 0.1335 33500 00 0.1 0.1313 31357 57 0.1 0.1292 29238 38 0.1 0.1271 27143 43 0.1 0.1250 25072 72 0.1 0.1230 23024 24 0.1 0.1210 21000 00 0.1 0.1190 19000 00 0.1 0.1170 17023 23

-1.0 -1. 0

0.1586 0.1 58655 55 0.1 0.1562 56248 48 0.1 0.1538 53864 64 0.1 0.1515 51505 05 0.1 0.1491 49170 70 0.1 0.1468 46859 59 0.1 0.1445 44572 72 0.1 0.1423 42310 10 0.1 0.1400 40071 71 0.1 0.1378 37857 57

-0.9 -0. 9

0.1840 0.1 84060 60 0.1 0.1814 81411 11 0.1 0.1787 78786 86 0.1 0.1761 76186 86 0.1 0.1736 73609 09 0.1 0.1710 71056 56 0.1 0.1685 68528 28 0.1 0.1660 66023 23 0.1 0.1635 63543 43 0.1 0.1610 61087 87

-0.8 -0. 8

0.2118 0.2 11855 55 0.2 0.2089 08970 70 0.2 0.2061 06108 08 0.2 0.2032 03269 69 0.2 0.2004 00454 54 0.1 0.1976 97663 63 0.1 0.1948 94895 95 0.1 0.1921 92150 50 0.1 0.1894 89430 30 0.1 0.1867 86733 33

-0.7 -0. 7

0.2419 0.2 41964 64 0.2 0.2388 38852 52 0.2 0.2357 35762 62 0.2 0.2326 32695 95 0.2 0.2296 29650 50 0.2 0.2266 26627 27 0.2 0.2236 23627 27 0.2 0.2206 20650 50 0.2 0.2176 17695 95 0.2 0.2147 14764 64

-0.6 -0. 6

0.2742 0.2 74253 53 0.2 0.2709 70931 31 0.2 0.2676 67629 29 0.2 0.2643 64347 47 0.2 0.2610 61086 86 0.2 0.2578 57846 46 0.2 0.2546 54627 27 0.2 0.2514 51429 29 0.2 0.2482 48252 52 0.2 0.2450 45097 97

-0.5 -0. 5

0.3085 0.3 08538 38 0.3 0.3050 05026 26 0.3 0.3015 01532 32 0.2 0.2980 98056 56 0.2 0.2945 94599 99 0.2 0.2911 91160 60 0.2 0.2877 87740 40 0.2 0.2843 84339 39 0.2 0.2809 80957 57 0.2 0.2775 77595 95

-0.4 -0. 4

0.3445 0.3 44578 78 0.3 0.3409 40903 03 0.3 0.3372 37243 43 0.3 0.3335 33598 98 0.3 0.3299 29969 69 0.3 0.3263 26355 55 0.3 0.3227 22758 58 0.3 0.3191 19178 78 0.3 0.3156 15614 14 0.3 0.3120 12067 67

-0.3 -0. 3

0.3820 0.3 82089 89 0.3 0.3782 78280 80 0.3 0.3744 74484 84 0.3 0.3707 70700 00 0.3 0.3669 66928 28 0.3 0.3631 63169 69 0.3 0.3594 59424 24 0.3 0.3556 55691 91 0.3 0.3519 51973 73 0.3 0.3482 48268 68

-0.2 -0. 2

0.4207 0.4 20740 40 0.4 0.4168 16834 34 0.4 0.4129 12936 36 0.4 0.4090 09046 46 0.4 0.4051 05165 65 0.4 0.4012 01294 94 0.3 0.3974 97432 32 0.3 0.3935 93580 80 0.3 0.3897 89739 39 0.3 0.3859 85908 08

-0.1 -0. 1

0.4601 0.4 60172 72 0.4 0.4562 56205 05 0.4 0.4522 52242 42 0.4 0.4482 48283 83 0.4 0.4443 44330 30 0.4 0.4403 40382 82 0.4 0.4364 36441 41 0.4 0.4325 32505 05 0.4 0.4285 28576 76 0.4 0.4246 24655 55

-0.0 -0. 0

0.5000 0.5 00000 00 0.4 0.4960 96011 11 0.4 0.4920 92022 22 0.4 0.4880 88034 34 0.4 0.4840 84047 47 0.4 0.4800 80061 61 0.4 0.4760 76078 78 0.4 0.4720 72097 97 0.4 0.4681 68119 19 0.4 0.4641 64144 44

0.0

0.5000 0.5 00000 00 0.5 0.5039 03989 89 0.5 0.5079 07978 78 0.5 0.5119 11966 66 0.5 0.5159 15953 53 0.5 0.5199 19939 39 0.5 0.5239 23922 22 0.5 0.5279 27903 03 0.5 0.5318 31881 81 0.5 0.5358 35856 56

0.1

0.5398 0.5 39828 28 0.5 0.5437 43795 95 0.5 0.5477 47758 58 0.5 0.5517 51717 17 0.5 0.5556 55670 70 0.5 0.5596 59618 18 0.5 0.5635 63559 59 0.5 0.5674 67495 95 0.5 0.5714 71424 24 0.5 0.5753 75345 45

0.2

0.5792 0.5 79260 60 0.5 0.5831 83166 66 0.5 0.5870 87064 64 0.5 0.5909 90954 54 0.5 0.5948 94835 35 0.5 0.5987 98706 06 0.6 0.6025 02568 68 0.6 0.6064 06420 20 0.6 0.6102 10261 61 0.6 0.6140 14092 92

0.3

0.6179 0.6 17911 11 0.6 0.6217 21720 20 0.6 0.6255 25516 16 0.6 0.6293 29300 00 0.6 0.6330 33072 72 0.6 0.6368 36831 31 0.6 0.6405 40576 76 0.6 0.6443 44309 09 0.6 0.6480 48027 27 0.6 0.6517 51732 32

0.4

0.6554 0.6 55422 22 0.6 0.6590 59097 97 0.6 0.6627 62757 57 0.6 0.6664 66402 02 0.6 0.6700 70031 31 0.6 0.6736 73645 45 0.6 0.6772 77242 42 0.6 0.6808 80822 22 0.6 0.6843 84386 86 0.6 0.6879 87933 33

0.5

0.6914 0.6 91462 62 0.6 0.6949 94974 74 0.6 0.6984 98468 68 0.7 0.7019 01944 44 0.7 0.7054 05401 01 0.7 0.7088 08840 40 0.7 0.7122 12260 60 0.7 0.7156 15661 61 0.7 0.7190 19043 43 0.7 0.7224 22405 05

0.6

0.7257 0.7 25747 47 0.7 0.7290 29069 69 0.7 0.7323 32371 71 0.7 0.7356 35653 53 0.7 0.7389 38914 14 0.7 0.7421 42154 54 0.7 0.7453 45373 73 0.7 0.7485 48571 71 0.7 0.7517 51748 48 0.7 0.7549 54903 03

0.7

0.7580 0.7 58036 36 0.7 0.7611 61148 48 0.7 0.7642 64238 38 0.7 0.7673 67305 05 0.7 0.7703 70350 50 0.7 0.7733 73373 73 0.7 0.7763 76373 73 0.7 0.7793 79350 50 0.7 0.7823 82305 05 0.7 0.7852 85236 36

0.8

0.7881 0.7 88145 45 0.7 0.7910 91030 30 0.7 0.7938 93892 92 0.7 0.7967 96731 31 0.7 0.7995 99546 46 0.8 0.8023 02337 37 0.8 0.8051 05105 05 0.8 0.8078 07850 50 0.8 0.8105 10570 70 0.8 0.8132 13267 67

0.9

0.8159 0.8 15940 40 0.8 0.8185 18589 89 0.8 0.8212 21214 14 0.8 0.8238 23814 14 0.8 0.8263 26391 91 0.8 0.8289 28944 44 0.8 0.8314 31472 72 0.8 0.8339 33977 77 0.8 0.8364 36457 57 0.8 0.8389 38913 13

1.0

0.8413 0.8 41345 45 0.8 0.8437 43752 52 0.8 0.8461 46136 36 0.8 0.8484 48495 95 0.8 0.8508 50830 30 0.8 0.8531 53141 41 0.8 0.8554 55428 28 0.8 0.8576 57690 90 0.8 0.8599 59929 29 0.8 0.8621 62143 43

1.1

0.8643 0.8 64334 34 0.8 0.8665 66500 00 0.8 0.8686 68643 43 0.8 0.8707 70762 62 0.8 0.8728 72857 57 0.8 0.8749 74928 28 0.8 0.8769 76976 76 0.8 0.8790 79000 00 0.8 0.8810 81000 00 0.8 0.8829 82977 77

1.2

0.8849 0.8 84930 30 0.8 0.8868 86861 61 0.8 0.8887 88768 68 0.8 0.8906 90651 51 0.8 0.8925 92512 12 0.8 0.8943 94350 50 0.8 0.8961 96165 65 0.8 0.8979 97958 58 0.8 0.8997 99727 27 0.9 0.9014 01475 75

1.3

0.9032 0.9 03200 00 0.9 0.9049 04902 02 0.9 0.9065 06582 82 0.9 0.9082 08241 41 0.9 0.9098 09877 77 0.9 0.9114 11492 92 0.9 0.9130 13085 85 0.9 0.9146 14657 57 0.9 0.9162 16207 07 0.9 0.9177 17736 36

1.4

0.9192 0.9 19243 43 0.9 0.9207 20730 30 0.9 0.9221 22196 96 0.9 0.9236 23641 41 0.9 0.9250 25066 66 0.9 0.9264 26471 71 0.9 0.9278 27855 55 0.9 0.9292 29219 19 0.9 0.9305 30563 63 0.9 0.9318 31888 88

1.5

0.9331 0.9 33193 93 0.9 0.9344 34478 78 0.9 0.9357 35745 45 0.9 0.9369 36992 92 0.9 0.9382 38220 20 0.9 0.9394 39429 29 0.9 0.9406 40620 20 0.9 0.9417 41792 92 0.9 0.9429 42947 47 0.9 0.9440 44083 83

1.6

0.9452 0.9 45201 01 0.9 0.9463 46301 01 0.9 0.9473 47384 84 0.9 0.9484 48449 49 0.9 0.9494 49497 97 0.9 0.9505 50529 29 0.9 0.9515 51543 43 0.9 0.9525 52540 40 0.9 0.9535 53521 21 0.9 0.9544 54486 86

1.7

0.9554 0.9 55435 35 0.9 0.9563 56367 67 0.9 0.9572 57284 84 0.9 0.9581 58185 85 0.9 0.9590 59070 70 0.9 0.9599 59941 41 0.9 0.9607 60796 96 0.9 0.9616 61636 36 0.9 0.9624 62462 62 0.9 0.9632 63273 73

1.8

0.9640 0.9 64070 70 0.9 0.9648 64852 52 0.9 0.9656 65620 20 0.9 0.9663 66375 75 0.9 0.9671 67116 16 0.9 0.9678 67843 43 0.9 0.9685 68557 57 0.9 0.9692 69258 58 0.9 0.9699 69946 46 0.9 0.9706 70621 21

i

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1.9

0.971283 0.971933 0.972571 0.973197 0.973810 0.974412 0.975002 0.975581 0.976148 0.976705

2.0

0.977250 0.977784 0.978308 0.978822 0.979325 0.979818 0.980301 0.980774 0.981237 0.981691

2.1

0.982136 0.982571 0.982997 0.983414 0.983823 0.984222 0.984614 0.984997 0.985371 0.985738

2.2

0.986097 0.986447 0.986791 0.987126 0.987455 0.987776 0.988089 0.988396 0.988696 0.988989

2.3

0.989276 0.989556 0.989830 0.990097 0.990358 0.990613 0.990863 0.991106 0.991344 0.991576

2.4

0.991802 0.992024 0.992240 0.992451 0.992656 0.992857 0.993053 0.993244 0.993431 0.993613

2.5

0.993790 0.993963 0.994132 0.994297 0.994457 0.994614 0.994766 0.994915 0.995060 0.995201

2.6

0.995339 0.995473 0.995604 0.995731 0.995855 0.995975 0.996093 0.996207 0.996319 0.996427

2.7

0.996533 0.996636 0.996736 0.996833 0.996928 0.997020 0.997110 0.997197 0.997282 0.997365

2.8

0.997445 0.997523 0.997599 0.997673 0.997744 0.997814 0.997882 0.997948 0.998012 0.998074

2.9

0.998134 0.998193 0.998250 0.998305 0.998359 0.998411 0.998462 0.998511 0.998559 0.998605

3.0

0.998650 0.998694 0.998736 0.998777 0.998817 0.998856 0.998893 0.998930 0.998965 0.998999

Ejemplos: Si X  ∼ N(0,1), entonces Pr(X  ≤ −1.96) = 0.024998 y Pr(X  ≤ 2.00) = 0.977250. Fuente: Tabla construida utilizando la función @cnorm de EViews® 3.1.

ii

Tabla 2:

Valores Críticos de la Distribución t  de Student. FUNCION DE DISTRIBUCION

0.005

0.010

0.025

0.050

0.100

0.900

0.950

0.975

0.990

0.995

1

-63.657

-31.821

-12.706

-6.314

-3.078

3.078

6.314

12.706

31.821

63.657

2

-9.925

-6.965

-4.303

-2.920

-1.886

1.886

2.920

4.303

6.965

9.925

3

-5.841

-4.541

-3.182

-2.353

-1.638

1.638

2.353

3.182

4.541

5.841

4

-4.604

-3.747

-2.776

-2.132

-1.533

1.533

2.132

2.776

3.747

4.604

5

-4.032

-3.365

-2.571

-2.015

-1.476

1.476

2.015

2.571

3.365

4.032

6

-3.707

-3.143

-2.447

-1.943

-1.440

1.440

1.943

2.447

3.143

3.707

7

-3.499

-2.998

-2.365

-1.895

-1.415

1.415

1.895

2.365

2.998

3.499

8

-3.355

-2.896

-2.306

-1.860

-1.397

1.397

1.860

2.306

2.896

3.355

9

-3.250

-2.821

-2.262

-1.833

-1.383

1.383

1.833

2.262

2.821

3.250

10

-3.169

-2.764

-2.228

-1.812

-1.372

1.372

1.812

2.228

2.764

3.169

G

11

-3.106

-2.718

-2.201

-1.796

-1.363

1.363

1.796

2.201

2.718

3.106

R

12

-3.055

-2.681

-2.179

-1.782

-1.356

1.356

1.782

2.179

2.681

3.055

 A

13

-3.012

-2.650

-2.160

-1.771

-1.350

1.350

1.771

2.160

2.650

3.012

D

14

-2.977

-2.624

-2.145

-1.761

-1.345

1.345

1.761

2.145

2.624

2.977

O

15

-2.947

-2.602

-2.131

-1.753

-1.341

1.341

1.753

2.131

2.602

2.947

16

-2.921

-2.583

-2.120

-1.746

-1.337

1.337

1.746

2.120

2.583

2.921

D

17

-2.898

-2.567

-2.110

-1.740

-1.333

1.333

1.740

2.110

2.567

2.898

E

18

-2.878

-2.552

-2.101

-1.734

-1.330

1.330

1.734

2.101

2.552

2.878

19

-2.861

-2.539

-2.093

-1.729

-1.328

1.328

1.729

2.093

2.539

2.861

20

-2.845

-2.528

-2.086

-1.725

-1.325

1.325

1.725

2.086

2.528

2.845

B

21

-2.831

-2.518

-2.080

-1.721

-1.323

1.323

1.721

2.080

2.518

2.831

E

22

-2.819

-2.508

-2.074

-1.717

-1.321

1.321

1.717

2.074

2.508

2.819

R

23

-2.807

-2.500

-2.069

-1.714

-1.319

1.319

1.714

2.069

2.500

2.807

T

24

-2.797

-2.492

-2.064

-1.711

-1.318

1.318

1.711

2.064

2.492

2.797

 A

25

-2.787

-2.485

-2.060

-1.708

-1.316

1.316

1.708

2.060

2.485

2.787

26

-2.779

-2.479

-2.056

-1.706

-1.315

1.315

1.706

2.056

2.479

2.779

27

-2.771

-2.473

-2.052

-1.703

-1.314

1.314

1.703

2.052

2.473

2.771

28

-2.763

-2.467

-2.048

-1.701

-1.313

1.313

1.701

2.048

2.467

2.763

29

-2.756

-2.462

-2.045

-1.699

-1.311

1.311

1.699

2.045

2.462

2.756

30

-2.750

-2.457

-2.042

-1.697

-1.310

1.310

1.697

2.042

2.457

2.750

40

-2.704

-2.423

-2.021

-1.684

-1.303

1.303

1.684

2.021

2.423

2.704

60

-2.660

-2.390

-2.000

-1.671

-1.296

1.296

1.671

2.000

2.390

2.660

90

-2.632

-2.368

-1.987

-1.662

-1.291

1.291

1.662

1.987

2.368

2.632

120

-2.617

-2.358

-1.980

-1.658

-1.289

1.289

1.658

1.980

2.358

2.617

INF

-2.576

-2.327

-1.960

-1.645

-1.282

1.282

1.645

1.960

2.327

2.576

S

L I

D

Ejemplos: Si X  ∼ t (20), entonces Pr(X  ≤ −2.528) = 0.01 y Pr( X  ≤ 1.725) = 0.95; si X  ∼ t (n ) con n  suficientemente grande, entonces Pr(X  ≤ −1.960) ≈ 0.025 y Pr(X  ≤ 2.327) ≈ 0.99. Fuente: Tabla construida utilizando la función @qtdist de EViews® 3.1.

iii

Tabla 3:

Valores Críticos de la Distribución Chi-Cuadrado. FUNCION DE DISTRIBUCION

0.005

0.010

0.025

0.050

0.100

0.900

0.950

0.975

0.990

0.995

1

0.000039 0.000157 0.000982 0.003932 0.0158

2.71

3.84

5.02

6.63

7.88

2

0.0100

0.0201

0.0506

0.10

0.21

4.61

5.99

7.38

9.21

10.60

3

0.0717

0.11

0.22

0.35

0.58

6.25

7.81

9.35

11.34

12.84

4

0.21

0.30

0.48

0.71

1.06

7.78

9.49

11.14

13.28

14.86

5

0.41

0.55

0.83

1.15

1.61

9.24

11.07

12.83

15.09

16.75

6

0.68

0.87

1.24

1.64

2.20

10.64

12.59

14.45

16.81

18.55

7

0.99

1.24

1.69

2.17

2.83

12.02

14.07

16.01

18.48

20.28

G

8

1.34

1.65

2.18

2.73

3.49

13.36

15.51

17.53

20.09

21.95

R

9

1.73

2.09

2.70

3.33

4.17

14.68

16.92

19.02

21.67

23.59

 A

10

2.16

2.56

3.25

3.94

4.87

15.99

18.31

20.48

23.21

25.19

O

11

2.60

3.05

3.82

4.57

5.58

17.28

19.68

21.92

24.72

26.76

S

12

3.07

3.57

4.40

5.23

6.30

18.55

21.03

23.34

26.22

28.30

13

3.57

4.11

5.01

5.89

7.04

19.81

22.36

24.74

27.69

29.82

D

14

4.07

4.66

5.63

6.57

7.79

21.06

23.68

26.12

29.14

31.32

E

15

4.60

5.23

6.26

7.26

8.55

22.31

25.00

27.49

30.58

32.80

16

5.14

5.81

6.91

7.96

9.31

23.54

26.30

28.85

32.00

34.27

L

17

5.70

6.41

7.56

8.67

10.09

24.77

27.59

30.19

33.41

35.72

I

18

6.26

7.01

8.23

9.39

10.86

25.99

28.87

31.53

34.81

37.16

B

19

6.84

7.63

8.91

10.12

11.65

27.20

30.14

32.85

36.19

38.58

E

20

7.43

8.26

9.59

10.85

12.44

28.41

31.41

34.17

37.57

40.00

T

21

8.03

8.90

10.28

11.59

13.24

29.62

32.67

35.48

38.93

41.40

 A

22

8.64

9.54

10.98

12.34

14.04

30.81

33.92

36.78

40.29

42.80

D

23

9.26

10.20

11.69

13.09

14.85

32.01

35.17

38.08

41.64

44.18

24

9.89

10.86

12.40

13.85

15.66

33.20

36.42

39.36

42.98

45.56

25

10.52

11.52

13.12

14.61

16.47

34.38

37.65

40.65

44.31

46.93

26

11.16

12.20

13.84

15.38

17.29

35.56

38.89

41.92

45.64

48.29

27

11.81

12.88

14.57

16.15

18.11

36.74

40.11

43.19

46.96

49.64

28

12.46

13.56

15.31

16.93

18.94

37.92

41.34

44.46

48.28

50.99

29

13.12

14.26

16.05

17.71

19.77

39.09

42.56

45.72

49.59

52.34

30

13.79

14.95

16.79

18.49

20.60

40.26

43.77

46.98

50.89

53.67

D

R

2

Ejemplos: Si X  ∼ χ (20), entonces Pr(X  ≤ 9.59) = 0.025 y Pr(X  ≤ 34.17) = 0.975. Fuente: Tabla construida utilizando la función @qchisq de EViews® 3.1.

iv

Tabla 4-1:

Valores Críticos de la Distribución F  - Función de Distribución = 0.90. GRADOS DE LIBERTAD DEL NUMERADOR

1

2

3

4

5

6

7

8

9

10

1

39.86

49.50

53.59

55.83

57.24

58.20

58.91

59.44

59.86

60.19

2

8.53

9.00

9.16

9.24

9.29

9.33

9.35

9.37

9.38

9.39

3

5.54

5.46

5.39

5.34

5.31

5.28

5.27

5.25

5.24

5.23

G

4

4.54

4.32

4.19

4.11

4.05

4.01

3.98

3.95

3.94

3.92

R

5

4.06

3.78

3.62

3.52

3.45

3.40

3.37

3.34

3.32

3.30

D

6

3.78

3.46

3.29

3.18

3.11

3.05

3.01

2.98

2.96

2.94

O

7

3.59

3.26

3.07

2.96

2.88

2.83

2.78

2.75

2.72

2.70

S

8

3.46

3.11

2.92

2.81

2.73

2.67

2.62

2.59

2.56

2.54

9

3.36

3.01

2.81

2.69

2.61

2.55

2.51

2.47

2.44

2.42

10

3.29

2.92

2.73

2.61

2.52

2.46

2.41

2.38

2.35

2.32

11

3.23

2.86

2.66

2.54

2.45

2.39

2.34

2.30

2.27

2.25

L

12

3.18

2.81

2.61

2.48

2.39

2.33

2.28

2.24

2.21

2.19

I

13

3.14

2.76

2.56

2.43

2.35

2.28

2.23

2.20

2.16

2.14

B

14

3.10

2.73

2.52

2.39

2.31

2.24

2.19

2.15

2.12

2.10

E

15

3.07

2.70

2.49

2.36

2.27

2.21

2.16

2.12

2.09

2.06

T

16

3.05

2.67

2.46

2.33

2.24

2.18

2.13

2.09

2.06

2.03

 A

17

3.03

2.64

2.44

2.31

2.22

2.15

2.10

2.06

2.03

2.00

D

18

3.01

2.62

2.42

2.29

2.20

2.13

2.08

2.04

2.00

1.98

19

2.99

2.61

2.40

2.27

2.18

2.11

2.06

2.02

1.98

1.96

20

2.97

2.59

2.38

2.25

2.16

2.09

2.04

2.00

1.96

1.94

21

2.96

2.57

2.36

2.23

2.14

2.08

2.02

1.98

1.95

1.92

22

2.95

2.56

2.35

2.22

2.13

2.06

2.01

1.97

1.93

1.90

D

23

2.94

2.55

2.34

2.21

2.11

2.05

1.99

1.95

1.92

1.89

E

24

2.93

2.54

2.33

2.19

2.10

2.04

1.98

1.94

1.91

1.88

N

25

2.92

2.53

2.32

2.18

2.09

2.02

1.97

1.93

1.89

1.87

M

26

2.91

2.52

2.31

2.17

2.08

2.01

1.96

1.92

1.88

1.86

I

27

2.90

2.51

2.30

2.17

2.07

2.00

1.95

1.91

1.87

1.85

N

28

2.89

2.50

2.29

2.16

2.06

2.00

1.94

1.90

1.87

1.84

 A

29

2.89

2.50

2.28

2.15

2.06

1.99

1.93

1.89

1.86

1.83

D

30

2.88

2.49

2.28

2.14

2.05

1.98

1.93

1.88

1.85

1.82

40

2.84

2.44

2.23

2.09

2.00

1.93

1.87

1.83

1.79

1.76

60

2.79

2.39

2.18

2.04

1.95

1.87

1.82

1.77

1.74

1.71

90

2.76

2.36

2.15

2.01

1.91

1.84

1.78

1.74

1.70

1.67

120

2.75

2.35

2.13

1.99

1.90

1.82

1.77

1.72

1.68

1.65

INF

2.71

2.30

2.08

1.95

1.85

1.77

1.72

1.67

1.63

1.60

 A

D E

R

D E L

O

O R

Ejemplos: Si X  ∼ F (2,20), Pr(X  ≤ 2.59) = 0.90; si X  ∼ F (3,n ) con n  suficientemente grande, Pr(X  ≤ 2.08) ≈ 0.90. Fuente: Tabla construida utilizando la función @qfdist de EViews® 3.1.

v

Tabla 4-2:

Valores Críticos de la Distribución F  - Función de Distribución = 0.95. GRADOS DE LIBERTAD DEL NUMERADOR

1

1

2

3

4

5

6

7

8

9

10

161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88

2

18.51

19.00

19.16

19.25

19.30

19.33

19.35

19.37

19.38

19.40

3

10.13

9.55

9.28

9.12

9.01

8.94

8.89

8.85

8.81

8.79

G

4

7.71

6.94

6.59

6.39

6.26

6.16

6.09

6.04

6.00

5.96

R

5

6.61

5.79

5.41

5.19

5.05

4.95

4.88

4.82

4.77

4.74

D

6

5.99

5.14

4.76

4.53

4.39

4.28

4.21

4.15

4.10

4.06

O

7

5.59

4.74

4.35

4.12

3.97

3.87

3.79

3.73

3.68

3.64

S

8

5.32

4.46

4.07

3.84

3.69

3.58

3.50

3.44

3.39

3.35

9

5.12

4.26

3.86

3.63

3.48

3.37

3.29

3.23

3.18

3.14

10

4.96

4.10

3.71

3.48

3.33

3.22

3.14

3.07

3.02

2.98

11

4.84

3.98

3.59

3.36

3.20

3.09

3.01

2.95

2.90

2.85

L

12

4.75

3.89

3.49

3.26

3.11

3.00

2.91

2.85

2.80

2.75

I

13

4.67

3.81

3.41

3.18

3.03

2.92

2.83

2.77

2.71

2.67

B

14

4.60

3.74

3.34

3.11

2.96

2.85

2.76

2.70

2.65

2.60

E

15

4.54

3.68

3.29

3.06

2.90

2.79

2.71

2.64

2.59

2.54

T

16

4.49

3.63

3.24

3.01

2.85

2.74

2.66

2.59

2.54

2.49

 A

17

4.45

3.59

3.20

2.96

2.81

2.70

2.61

2.55

2.49

2.45

D

18

4.41

3.55

3.16

2.93

2.77

2.66

2.58

2.51

2.46

2.41

19

4.38

3.52

3.13

2.90

2.74

2.63

2.54

2.48

2.42

2.38

20

4.35

3.49

3.10

2.87

2.71

2.60

2.51

2.45

2.39

2.35

21

4.32

3.47

3.07

2.84

2.68

2.57

2.49

2.42

2.37

2.32

22

4.30

3.44

3.05

2.82

2.66

2.55

2.46

2.40

2.34

2.30

D

23

4.28

3.42

3.03

2.80

2.64

2.53

2.44

2.37

2.32

2.27

E

24

4.26

3.40

3.01

2.78

2.62

2.51

2.42

2.36

2.30

2.25

N

25

4.24

3.39

2.99

2.76

2.60

2.49

2.40

2.34

2.28

2.24

M

26

4.23

3.37

2.98

2.74

2.59

2.47

2.39

2.32

2.27

2.22

I

27

4.21

3.35

2.96

2.73

2.57

2.46

2.37

2.31

2.25

2.20

N

28

4.20

3.34

2.95

2.71

2.56

2.45

2.36

2.29

2.24

2.19

 A

29

4.18

3.33

2.93

2.70

2.55

2.43

2.35

2.28

2.22

2.18

D

30

4.17

3.32

2.92

2.69

2.53

2.42

2.33

2.27

2.21

2.16

40

4.08

3.23

2.84

2.61

2.45

2.34

2.25

2.18

2.12

2.08

60

4.00

3.15

2.76

2.53

2.37

2.25

2.17

2.10

2.04

1.99

90

3.95

3.10

2.71

2.47

2.32

2.20

2.11

2.04

1.99

1.94

120

3.92

3.07

2.68

2.45

2.29

2.18

2.09

2.02

1.96

1.91

INF

3.84

3.00

2.61

2.37

2.21

2.10

2.01

1.94

1.88

1.83

 A

D E

R

D E L

O

O R

Ejemplos: Si X  ∼ F (2,20), Pr(X  ≤ 3.49) = 0.95; si X  ∼ F (3,n ) con n  suficientemente grande, Pr(X  ≤ 2.61) ≈ 0.95. Fuente: Tabla construida utilizando la función @qfdist de EViews® 3.1.

vi

Tabla 4-3:

Valores Críticos de la Distribución F  - Función de Distribución = 0.99. GRADOS DE LIBERTAD DEL NUMERADOR

1

2

3

4

5

6

7

8

9

10

1

4052

5000

5403

5625

5764

5859

5928

5981

6022

6056

2

98.50

99.00

99.17

99.25

99.30

99.33

99.36

99.37

99.39

99.40

3

34.12

30.82

29.46

28.71

28.24

27.91

27.67

27.49

27.35

27.23

G

4

21.20

18.00

16.69

15.98

15.52

15.21

14.98

14.80

14.66

14.55

R

5

16.26

13.27

12.06

11.39

10.97

10.67

10.46

10.29

10.16

10.05

D

6

13.75

10.92

9.78

9.15

8.75

8.47

8.26

8.10

7.98

7.87

O

7

12.25

9.55

8.45

7.85

7.46

7.19

6.99

6.84

6.72

6.62

S

8

11.26

8.65

7.59

7.01

6.63

6.37

6.18

6.03

5.91

5.81

9

10.56

8.02

6.99

6.42

6.06

5.80

5.61

5.47

5.35

5.26

10

10.04

7.56

6.55

5.99

5.64

5.39

5.20

5.06

4.94

4.85

11

9.65

7.21

6.22

5.67

5.32

5.07

4.89

4.74

4.63

4.54

L

12

9.33

6.93

5.95

5.41

5.06

4.82

4.64

4.50

4.39

4.30

I

13

9.07

6.70

5.74

5.21

4.86

4.62

4.44

4.30

4.19

4.10

B

14

8.86

6.51

5.56

5.04

4.69

4.46

4.28

4.14

4.03

3.94

E

15

8.68

6.36

5.42

4.89

4.56

4.32

4.14

4.00

3.89

3.80

T

16

8.53

6.23

5.29

4.77

4.44

4.20

4.03

3.89

3.78

3.69

 A

17

8.40

6.11

5.18

4.67

4.34

4.10

3.93

3.79

3.68

3.59

D

18

8.29

6.01

5.09

4.58

4.25

4.01

3.84

3.71

3.60

3.51

19

8.18

5.93

5.01

4.50

4.17

3.94

3.77

3.63

3.52

3.43

20

8.10

5.85

4.94

4.43

4.10

3.87

3.70

3.56

3.46

3.37

21

8.02

5.78

4.87

4.37

4.04

3.81

3.64

3.51

3.40

3.31

22

7.95

5.72

4.82

4.31

3.99

3.76

3.59

3.45

3.35

3.26

D

23

7.88

5.66

4.76

4.26

3.94

3.71

3.54

3.41

3.30

3.21

E

24

7.82

5.61

4.72

4.22

3.90

3.67

3.50

3.36

3.26

3.17

N

25

7.77

5.57

4.68

4.18

3.85

3.63

3.46

3.32

3.22

3.13

M

26

7.72

5.53

4.64

4.14

3.82

3.59

3.42

3.29

3.18

3.09

I

27

7.68

5.49

4.60

4.11

3.78

3.56

3.39

3.26

3.15

3.06

N

28

7.64

5.45

4.57

4.07

3.75

3.53

3.36

3.23

3.12

3.03

 A

29

7.60

5.42

4.54

4.04

3.73

3.50

3.33

3.20

3.09

3.00

D

30

7.56

5.39

4.51

4.02

3.70

3.47

3.30

3.17

3.07

2.98

40

7.31

5.18

4.31

3.83

3.51

3.29

3.12

2.99

2.89

2.80

60

7.08

4.98

4.13

3.65

3.34

3.12

2.95

2.82

2.72

2.63

90

6.93

4.85

4.01

3.53

3.23

3.01

2.84

2.72

2.61

2.52

120

6.85

4.79

3.95

3.48

3.17

2.96

2.79

2.66

2.56

2.47

INF

6.64

4.61

3.78

3.32

3.02

2.80

2.64

2.51

2.41

2.32

 A

D E

R

D E L

O

O R

Ejemplos: Si X  ∼ F (2,20), Pr(X  ≤ 5.85) = 0.99; si X  ∼ F (3,n ) con n  suficientemente grande, Pr(X  ≤ 3.78) ≈ 0.99. Fuente: Tabla construida utilizando la función @qfdist de EViews® 3.1.

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