15  Referenztabellen

15.1 Verteilungsfunktion und Quantile der Standardnormalverteilung

Bei der z-Transformation benötigt man die Werte dieser Tabelle beispielsweise für Fragen wie

Wie hoch ist der prozentuale Anteil für Werte, die höchstens X sind?

Tabelle 15.1: Verteilungsfunktion und Quantile der Standardnormalverteilung
\(\phi\) z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359
0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714 .5753
0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103 .6141
0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517
0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879
0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224
0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549
0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852
0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133
0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389
1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599 .8621
1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .8830
1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015
1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177
1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319
1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441
1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545
1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633
1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706
1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767
2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817
2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854 .9857
2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890
2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916
2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934 .9936
2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952
2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 .9964
2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974
2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981
2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986
3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990
p = 0.5000 0.7500 0.8000 0.9000 0.9500 0.9750 0.9900 0.9950 0.9975 0.9990
\(z_p\) = 0.0000 0.6745 0.8416 1.2816 1.6449 1.9600 2.3263 2.5758 2.8070 3.0902

15.2 Überschreitungswahrscheinlichkeiten \(P\) für Abszissenwerte \(z\) der Standardnormalverteilung

Im Gegensatz zu Tabelle 15.1 zeigt Tabelle 15.2 die Überschreitungswahrscheinlichkeiten \(P\) für Abszissenwerte \(z\) der Standardnormalverteilung (\(\mu=0\) und \(\sigma=1\)).

Bei der z-Transformation benötigt man die Werte dieser Tabelle beispielsweise für Fragen wie

Wie hoch ist der prozentuale Anteil für Werte, die mindestens X sind?

Tabelle 15.2: Überschreitungswahrscheinlichkeiten \(P\) für Abszissenwerte \(z\) der Standardnormalverteilung (\(\mu=0\) und \(\sigma=1\))
\(\phi\) z .00 .01 .02 .03 .04 .05 .06 .07 .08 .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 .2887 .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 .1464 .1423 .1401 .1379
1.1 .1357 .1335 .1314 .1291 .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 .0643 .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
p = 0.5000 0.7500 0.8000 0.9000 0.9500 0.9750 0.9900 0.9950 0.9975 0.9990
\(z_p\) = 0.0000 0.6745 0.8416 1.2816 1.6449 1.9600 2.3263 2.5758 2.8070 3.0902

15.3 Kritische Werte \(t^*\) der \(t\)-Verteilung

Tabelle 15.3: Kritische Werte \(t^*\) der \(t\)-Verteilung
\(\alpha\)/2 0.10 0.05 0.025 0.01 0.005 0.0005
\(\alpha\) 0.20 0.10 0.050 0.020 0.010 0.0010
1 3.078 6.314 12.706 31.821 63.657 318.309
2 1.886 2.920 4.303 6.965 9.925 22.327
3 1.638 2.353 3.182 4.541 5.841 10.215
4 1.533 2.132 2.776 3.747 4.604 7.173
5 1.476 2.015 2.571 3.365 4.032 5.893
6 1.440 1.943 2.447 3.143 3.707 5.208
7 1.415 1.895 2.365 2.998 3.499 4.785
8 1.397 1.860 2.306 2.896 3.355 4.501
9 1.383 1.833 2.262 2.821 3.250 4.297
10 1.372 1.812 2.228 2.764 3.169 4.144
11 1.363 1.796 2.201 2.718 3.106 4.025
12 1.356 1.782 2.179 2.681 3.055 3.930
13 1.350 1.771 2.160 2.650 3.012 3.852
14 1.345 1.761 2.145 2.624 2.977 3.787
15 1.341 1.753 2.131 2.602 2.947 3.733
16 1.337 1.746 2.120 2.583 2.921 3.686
17 1.333 1.740 2.110 2.567 2.898 3.646
18 1.330 1.734 2.101 2.552 2.878 3.611
19 1.328 1.729 2.093 2.539 2.861 3.579
20 1.325 1.725 2.086 2.528 2.845 3.552
21 1.323 1.721 2.080 2.518 2.831 3.527
22 1.321 1.717 2.074 2.508 2.819 3.505
23 1.319 1.714 2.069 2.500 2.807 3.485
24 1.318 1.711 2.064 2.492 2.797 3.467
25 1.316 1.708 2.060 2.485 2.787 3.450
26 1.315 1.706 2.056 2.479 2.779 3.435
27 1.314 1.703 2.052 2.473 2.771 3.421
28 1.313 1.701 2.048 2.467 2.763 3.408
29 1.311 1.699 2.045 2.462 2.756 3.396
30 1.310 1.697 2.042 2.457 2.750 3.385
40 1.303 1.684 2.021 2.423 2.704 3.307
50 1.299 1.676 2.009 2.403 2.678 3.261
60 1.296 1.671 2.000 2.390 2.660 3.232
70 1.294 1.667 1.994 2.381 2.648 3.211
80 1.292 1.664 1.990 2.374 2.639 3.195
90 1.291 1.662 1.987 2.368 2.632 3.183
100 1.290 1.660 1.984 2.364 2.626 3.174
200 1.286 1.653 1.972 2.345 2.601 3.131
300 1.284 1.650 1.968 2.339 2.592 3.118
400 1.284 1.649 1.966 2.336 2.588 3.111
500 1.283 1.648 1.965 2.334 2.586 3.107
\(\infty\) 1.282 1.645 1.960 2.326 2.576 3.090