Kapitel 16 Tabellen

16.1 Normalverteilung

## Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
## Use 'xfun::attr2()' instead.
## See help("Deprecated")
Verteilungsfunktion \(F(z)\) der Standardnormalverteilung \(N(0,1)\)
Beispiel: \(F(z) = P(z \leq 1.96) = 0.9750\)
z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 
# R Example
qnorm(p = 0.975, mean = 0, sd = 1)
## [1] 1.959964

16.2 \(t\)-Verteilung

## Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
## Use 'xfun::attr2()' instead.
## See help("Deprecated")
Quantile \(t_{n;\gamma}\) der \(t_n\)-Verteilung
\(n \setminus \gamma\) 0.995 0.99 0.975 0.95 0.9 0.75 0.5
1 63.6567 31.8205 12.7062 6.3138 3.0777 1.0000 0
2 9.9248 6.9646 4.3027 2.9200 1.8856 0.8165 0
3 5.8409 4.5407 3.1824 2.3534 1.6377 0.7649 0
4 4.6041 3.7469 2.7764 2.1318 1.5332 0.7407 0
5 4.0321 3.3649 2.5706 2.0150 1.4759 0.7267 0
6 3.7074 3.1427 2.4469 1.9432 1.4398 0.7176 0
7 3.4995 2.9980 2.3646 1.8946 1.4149 0.7111 0
8 3.3554 2.8965 2.3060 1.8595 1.3968 0.7064 0
9 3.2498 2.8214 2.2622 1.8331 1.3830 0.7027 0
10 3.1693 2.7638 2.2281 1.8125 1.3722 0.6998 0
11 3.1058 2.7181 2.2010 1.7959 1.3634 0.6974 0
12 3.0545 2.6810 2.1788 1.7823 1.3562 0.6955 0
13 3.0123 2.6503 2.1604 1.7709 1.3502 0.6938 0
14 2.9768 2.6245 2.1448 1.7613 1.3450 0.6924 0
15 2.9467 2.6025 2.1314 1.7531 1.3406 0.6912 0
16 2.9208 2.5835 2.1199 1.7459 1.3368 0.6901 0
17 2.8982 2.5669 2.1098 1.7396 1.3334 0.6892 0
18 2.8784 2.5524 2.1009 1.7341 1.3304 0.6884 0
19 2.8609 2.5395 2.0930 1.7291 1.3277 0.6876 0
20 2.8453 2.5280 2.0860 1.7247 1.3253 0.6870 0
21 2.8314 2.5176 2.0796 1.7207 1.3232 0.6864 0
22 2.8188 2.5083 2.0739 1.7171 1.3212 0.6858 0
23 2.8073 2.4999 2.0687 1.7139 1.3195 0.6853 0
24 2.7969 2.4922 2.0639 1.7109 1.3178 0.6848 0
25 2.7874 2.4851 2.0595 1.7081 1.3163 0.6844 0
26 2.7787 2.4786 2.0555 1.7056 1.3150 0.6840 0
27 2.7707 2.4727 2.0518 1.7033 1.3137 0.6837 0
28 2.7633 2.4671 2.0484 1.7011 1.3125 0.6834 0
29 2.7564 2.4620 2.0452 1.6991 1.3114 0.6830 0
30 2.7500 2.4573 2.0423 1.6973 1.3104 0.6828 0
40 2.7045 2.4233 2.0211 1.6839 1.3031 0.6807 0
50 2.6778 2.4033 2.0086 1.6759 1.2987 0.6794 0
60 2.6603 2.3901 2.0003 1.6706 1.2958 0.6786 0
70 2.6479 2.3808 1.9944 1.6669 1.2938 0.6780 0
80 2.6387 2.3739 1.9901 1.6641 1.2922 0.6776 0
90 2.6316 2.3685 1.9867 1.6620 1.2910 0.6772 0
100 2.6259 2.3642 1.9840 1.6602 1.2901 0.6770 0
200 2.6006 2.3451 1.9719 1.6525 1.2858 0.6757 0
300 2.5923 2.3388 1.9679 1.6499 1.2844 0.6753 0
400 2.5882 2.3357 1.9659 1.6487 1.2837 0.6751 0
500 2.5857 2.3338 1.9647 1.6479 1.2832 0.6750 0
600 2.5840 2.3326 1.9639 1.6474 1.2830 0.6749 0
700 2.5829 2.3317 1.9634 1.6470 1.2828 0.6748 0
800 2.5820 2.3310 1.9629 1.6468 1.2826 0.6748 0
900 2.5813 2.3305 1.9626 1.6465 1.2825 0.6748 0
1000 2.5808 2.3301 1.9623 1.6464 1.2824 0.6747 0

16.3 \(\mathcal{X}^{2}_{n;\,\gamma}\)-Verteilung

## Warning in attr(x, "align"): 'xfun::attr()' is deprecated.
## Use 'xfun::attr2()' instead.
## See help("Deprecated")
Quantile \(\mathcal{X}^2_{n;\,\gamma}\) der \(\mathcal{X}^2_n\)-Verteilung
Beispiel: \(P(\mathcal{X}^2_{10} \leq 20.4832) = 0.975\)
\(n \setminus \gamma\) 0.995 0.99 0.975 0.95 0.9 0.75 0.5 0.25 0.1 0.05 0.01 0.005
1 7.8794 6.6349 5.0239 3.8415 2.7055 1.3233 0.4549 0.1015 0.0158 0.0039 0.0002 0.0000
2 10.5966 9.2103 7.3778 5.9915 4.6052 2.7726 1.3863 0.5754 0.2107 0.1026 0.0201 0.0100
3 12.8382 11.3449 9.3484 7.8147 6.2514 4.1083 2.3660 1.2125 0.5844 0.3518 0.1148 0.0717
4 14.8603 13.2767 11.1433 9.4877 7.7794 5.3853 3.3567 1.9226 1.0636 0.7107 0.2971 0.2070
5 16.7496 15.0863 12.8325 11.0705 9.2364 6.6257 4.3515 2.6746 1.6103 1.1455 0.5543 0.4117
6 18.5476 16.8119 14.4494 12.5916 10.6446 7.8408 5.3481 3.4546 2.2041 1.6354 0.8721 0.6757
7 20.2777 18.4753 16.0128 14.0671 12.0170 9.0371 6.3458 4.2549 2.8331 2.1673 1.2390 0.9893
8 21.9550 20.0902 17.5345 15.5073 13.3616 10.2189 7.3441 5.0706 3.4895 2.7326 1.6465 1.3444
9 23.5894 21.6660 19.0228 16.9190 14.6837 11.3888 8.3428 5.8988 4.1682 3.3251 2.0879 1.7349
10 25.1882 23.2093 20.4832 18.3070 15.9872 12.5489 9.3418 6.7372 4.8652 3.9403 2.5582 2.1559
11 26.7568 24.7250 21.9200 19.6751 17.2750 13.7007 10.3410 7.5841 5.5778 4.5748 3.0535 2.6032
12 28.2995 26.2170 23.3367 21.0261 18.5493 14.8454 11.3403 8.4384 6.3038 5.2260 3.5706 3.0738
13 29.8195 27.6882 24.7356 22.3620 19.8119 15.9839 12.3398 9.2991 7.0415 5.8919 4.1069 3.5650
14 31.3193 29.1412 26.1189 23.6848 21.0641 17.1169 13.3393 10.1653 7.7895 6.5706 4.6604 4.0747
15 32.8013 30.5779 27.4884 24.9958 22.3071 18.2451 14.3389 11.0365 8.5468 7.2609 5.2293 4.6009
16 34.2672 31.9999 28.8454 26.2962 23.5418 19.3689 15.3385 11.9122 9.3122 7.9616 5.8122 5.1422
17 35.7185 33.4087 30.1910 27.5871 24.7690 20.4887 16.3382 12.7919 10.0852 8.6718 6.4078 5.6972
18 37.1565 34.8053 31.5264 28.8693 25.9894 21.6049 17.3379 13.6753 10.8649 9.3905 7.0149 6.2648
19 38.5823 36.1909 32.8523 30.1435 27.2036 22.7178 18.3377 14.5620 11.6509 10.1170 7.6327 6.8440
20 39.9968 37.5662 34.1696 31.4104 28.4120 23.8277 19.3374 15.4518 12.4426 10.8508 8.2604 7.4338
21 41.4011 38.9322 35.4789 32.6706 29.6151 24.9348 20.3372 16.3444 13.2396 11.5913 8.8972 8.0337
22 42.7957 40.2894 36.7807 33.9244 30.8133 26.0393 21.3370 17.2396 14.0415 12.3380 9.5425 8.6427
23 44.1813 41.6384 38.0756 35.1725 32.0069 27.1413 22.3369 18.1373 14.8480 13.0905 10.1957 9.2604
24 45.5585 42.9798 39.3641 36.4150 33.1962 28.2412 23.3367 19.0373 15.6587 13.8484 10.8564 9.8862
25 46.9279 44.3141 40.6465 37.6525 34.3816 29.3389 24.3366 19.9393 16.4734 14.6114 11.5240 10.5197
26 48.2899 45.6417 41.9232 38.8851 35.5632 30.4346 25.3365 20.8434 17.2919 15.3792 12.1981 11.1602
27 49.6449 46.9629 43.1945 40.1133 36.7412 31.5284 26.3363 21.7494 18.1139 16.1514 12.8785 11.8076
28 50.9934 48.2782 44.4608 41.3371 37.9159 32.6205 27.3362 22.6572 18.9392 16.9279 13.5647 12.4613
29 52.3356 49.5879 45.7223 42.5570 39.0875 33.7109 28.3361 23.5666 19.7677 17.7084 14.2565 13.1211
30 53.6720 50.8922 46.9792 43.7730 40.2560 34.7997 29.3360 24.4776 20.5992 18.4927 14.9535 13.7867
40 66.7660 63.6907 59.3417 55.7585 51.8051 45.6160 39.3353 33.6603 29.0505 26.5093 22.1643 20.7065
50 79.4900 76.1539 71.4202 67.5048 63.1671 56.3336 49.3349 42.9421 37.6886 34.7643 29.7067 27.9907
60 91.9517 88.3794 83.2977 79.0819 74.3970 66.9815 59.3347 52.2938 46.4589 43.1880 37.4849 35.5345
70 104.2149 100.4252 95.0232 90.5312 85.5270 77.5767 69.3345 61.6983 55.3289 51.7393 45.4417 43.2752
80 116.3211 112.3288 106.6286 101.8795 96.5782 88.1303 79.3343 71.1445 64.2778 60.3915 53.5401 51.1719
90 128.2989 124.1163 118.1359 113.1453 107.5650 98.6499 89.3342 80.6247 73.2911 69.1260 61.7541 59.1963
100 140.1695 135.8067 129.5612 124.3421 118.4980 109.1412 99.3341 90.1332 82.3581 77.9295 70.0649 67.3276
150 198.3602 193.2077 185.8004 179.5806 172.5812 161.2912 149.3339 137.9829 128.2751 122.6918 112.6676 109.1422
200 255.2642 249.4451 241.0579 233.9943 226.0210 213.1022 199.3337 186.1717 174.8353 168.2786 156.4320 152.2410
250 311.3462 304.9396 295.6886 287.8815 279.0504 264.6970 249.3337 234.5768 221.8059 214.3916 200.9386 196.1606
300 366.8444 359.9064 349.8745 341.3951 331.7885 316.1384 299.3336 283.1353 269.0679 260.8781 245.9725 240.6634
400 476.6064 468.7245 457.3055 447.6325 436.6490 418.6969 399.3335 380.5767 364.2074 354.6410 337.1553 330.9028
600 692.9816 683.5156 669.7692 658.0936 644.8004 622.9876 599.3335 576.2859 556.0560 544.1801 522.3651 514.5289
800 906.7862 895.9843 880.2753 866.9114 851.6712 826.6040 799.3334 772.6694 749.1852 735.3623 709.8969 700.7250
1000 1118.9481 1106.9690 1089.5309 1074.6794 1057.7239 1029.7898 999.3334 969.4836 943.1326 927.5944 898.9124 888.5635