DODATEK A. TABLICE STATYSTYCZNE

DODATEK A. TABLICE STATYSTYCZNE

1. Rozkład normalny

2. Rozkład gamma

3. Rozkład Studenta

4. Rozkład P

5. Rozkład F Snedecora

TABELA 1A. Wartości dystrybuanty Φ Φ Φ Φ(u) standaryzowanego rozkładu normalnego N(0,1)

Tabela 1A podaje wartości dystrybuanty Φ(u) rozkładu normalnego N(0,1):

1 1

( ) exp

2 2

u

u x dx

−∞

π

 

Φ =  − 

 

∫ _(A.1)

dla u = 0, 0.01, ..., 3.09.

Przykład 1A1. Odczytać Φ(u) dla u = 0.57.

Rozwiązanie: Odpowiedź znajduje się w tab. 1A na przecięciu wiersza 0.57 i kolumny 0.07:

Φ Φ Φ

Φ(0.57) = 0.7157

Gdy u < 0, korzystamy z równania ( ) u 1 ( u )

Φ = − Φ − (A.2)

objaśnionego na rysunku obok, a które pozwala na wykorzystanie tablicy 1A.

Przykład 1A2. Obliczyć Φ(u) dla u = -0.29.

Rozwiązanie: Φ Φ Φ Φ(-0.29) = 1 - Φ Φ Φ(0.29) =... (tu korzystamy z tablicy 1A) ... = 1 - 0.6141 = Φ 0.3859.

TABELA 1B. Wartości kwantyli u

w standaryzowanym rozkładzie normalnym N(0,1)

Tabela 1B zawiera wartości kwantyli u

w standaryzo- wanym rozkładzie normalnym N(0,1), gdzie Φ jest praw- dopodobieństwem nieprzewyższenia, Φ > 0.5 (zob. rysu- nek obok).

Przykład 1B1. Odczytać u

.

Rozwiązanie: Odpowiedź znajduje się w tab. 1B na przecięciu wiersza 0.97 i kolumny 0.005: u

= 1.960

Gdy Φ < 0.5 należy posługiwać się wzorem

u

= − u

_(A.3)

(patrz rysunek obok) i tablicą 1B.

Przykład 1B2. Obliczyć u

.

x f_Hx_L

0 u Φ Hu_L

x f_Hx_L

x f_Hx_L

0

−u u

ΦHu_L 1_−ΦH−u_L x f_Hx_L

x f_Hx_L

0 u_Φ Φ

x f_Hx_L

x f_Hx_L

0 u₁ u −Φ

Φ

Φ 1_−Φ ^Φ

x f_Hx_L

TABELA 1A. Wartości dystrybuanty Φ Φ Φ Φ (u) standaryzowanego rozkładu normalnego N(0,1) dla u > 0

u 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 u

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

u 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 u

TABELA 1B. Kwantyle u

w standaryzowanym rozkładzie normalnym N(0,1) dla Φ

Φ Φ

Φ > 0.500 Φ

Φ Φ

Φ

Φ Φ Φ Φ

0.50 0.000 0.002 0.005 0.007 0.010 0.012 0.015 0.017 0.020 0.022 0.50 0.51 0.025 0.027 0.030 0.033 0.035 0.038 0.040 0.043 0.045 0.048 0.51 0.52 0.050 0.053 0.055 0.058 0.060 0.063 0.065 0.068 0.070 0.073 0.52 0.53 0.075 0.078 0.080 0.083 0.085 0.088 0.090 0.093 0.095 0.098 0.53 0.54 0.100 0.103 0.105 0.108 0.110 0.113 0.115 0.118 0.120 0.123 0.54 0.55 0.125 0.128 0.130 0.133 0.135 0.138 0.141 0.143 0.146 0.148 0.55 0.56 0.151 0.153 0.156 0.158 0.161 0.163 0.166 0.168 0.171 0.173 0.56 0.57 0.176 0.179 0.181 0.184 0.186 0.189 0.191 0.194 0.196 0.199 0.57 0.58 0.202 0.204 0.207 0.209 0.212 0.214 0.217 0.219 0.222 0.225 0.58 0.59 0.227 0.230 0.232 0.235 0.237 0.240 0.243 0.245 0.248 0.250 0.59 0.60 0.253 0.256 0.258 0.261 0.263 0.266 0.268 0.271 0.274 0.276 0.60 0.61 0.279 0.281 0.284 0.287 0.289 0.292 0.295 0.297 0.300 0.302 0.61 0.62 0.305 0.308 0.310 0.313 0.316 0.318 0.321 0.323 0.326 0.329 0.62 0.63 0.331 0.334 0.337 0.339 0.342 0.345 0.347 0.350 0.353 0.355 0.63 0.64 0.358 0.361 0.363 0.366 0.369 0.371 0.374 0.377 0.379 0.382 0.64 0.65 0.385 0.388 0.390 0.393 0.396 0.398 0.401 0.404 0.407 0.409 0.65 0.66 0.412 0.415 0.417 0.420 0.423 0.426 0.428 0.431 0.434 0.437 0.66 0.67 0.439 0.442 0.445 0.448 0.451 0.453 0.456 0.459 0.462 0.464 0.67 0.68 0.467 0.470 0.473 0.476 0.478 0.481 0.484 0.487 0.490 0.493 0.68 0.69 0.495 0.498 0.501 0.504 0.507 0.510 0.513 0.515 0.518 0.521 0.69 0.70 0.524 0.527 0.530 0.533 0.536 0.538 0.541 0.544 0.547 0.550 0.70 0.71 0.553 0.556 0.559 0.562 0.565 0.568 0.571 0.574 0.577 0.580 0.71 0.72 0.582 0.585 0.588 0.591 0.594 0.597 0.600 0.603 0.606 0.609 0.72 0.73 0.612 0.615 0.619 0.622 0.625 0.628 0.631 0.634 0.637 0.640 0.73 0.74 0.643 0.646 0.649 0.652 0.655 0.659 0.662 0.665 0.668 0.671 0.74 0.75 0.674 0.677 0.681 0.684 0.687 0.690 0.693 0.696 0.700 0.703 0.75 0.76 0.706 0.709 0.712 0.716 0.719 0.722 0.725 0.729 0.732 0.735 0.76 0.77 0.739 0.742 0.745 0.749 0.752 0.755 0.759 0.762 0.765 0.769 0.77 0.78 0.772 0.775 0.779 0.782 0.786 0.789 0.792 0.796 0.799 0.803 0.78 0.79 0.806 0.810 0.813 0.817 0.820 0.824 0.827 0.831 0.834 0.838 0.79 0.80 0.841 0.845 0.849 0.852 0.856 0.859 0.863 0.867 0.870 0.874 0.80 0.81 0.878 0.881 0.885 0.889 0.893 0.896 0.900 0.904 0.908 0.911 0.81 0.82 0.915 0.919 0.923 0.927 0.931 0.935 0.938 0.942 0.946 0.950 0.82 0.83 0.954 0.958 0.962 0.966 0.970 0.974 0.978 0.982 0.986 0.990 0.83 0.84 0.994 0.999 1.003 1.007 1.011 1.015 1.019 1.024 1.028 1.032 0.84 0.85 1.036 1.041 1.045 1.049 1.054 1.058 1.063 1.067 1.071 1.076 0.85 0.86 1.080 1.085 1.089 1.094 1.099 1.103 1.108 1.112 1.117 1.122 0.86 0.87 1.126 1.131 1.136 1.141 1.146 1.150 1.155 1.160 1.165 1.170 0.87 0.88 1.175 1.180 1.185 1.190 1.195 1.200 1.206 1.211 1.216 1.221 0.88 0.89 1.227 1.232 1.237 1.243 1.248 1.254 1.259 1.265 1.270 1.276 0.89 0.90 1.282 1.287 1.293 1.299 1.305 1.311 1.317 1.323 1.329 1.335 0.90 0.91 1.341 1.347 1.353 1.360 1.366 1.372 1.379 1.385 1.392 1.399 0.91 0.92 1.405 1.412 1.419 1.426 1.433 1.440 1.447 1.454 1.461 1.469 0.92 0.93 1.476 1.484 1.491 1.499 1.507 1.514 1.522 1.530 1.539 1.547 0.93 0.94 1.555 1.564 1.572 1.581 1.590 1.599 1.608 1.617 1.626 1.636 0.94 0.95 1.645 1.655 1.665 1.675 1.685 1.696 1.706 1.717 1.728 1.740 0.95 0.96 1.751 1.763 1.775 1.787 1.800 1.812 1.825 1.839 1.853 1.867 0.96 0.97 1.881 1.896 1.911 1.927 1.944 1.960 1.978 1.996 2.015 2.034 0.97 0.98 2.054 2.075 2.097 2.121 2.145 2.171 2.198 2.227 2.258 2.291 0.98 0.99 2.327 2.366 2.409 2.458 2.513 2.576 2.652 2.748 2.879 3.091 0.99

Φ

Φ Φ Φ

TABELA 2. Kwantyle t

( λ λ λ λ ) w rozkładzie gamma w zależności od wartości para- metru λ λ λ λ i prawdopodobieństwa nieprzewyższenia F

1 0

( , ) 1

( )

t

F t λ u

e du

λ

− −

Γ

=

∫ Γ _(A.4)

Przykład odczytu: dla λ = 4.0 i F = 20% dostajemy t

( λ )

= 2.297.

Uwaga. Pożyteczny związek: x

( α , λ ) = t

( λ )/ α

F, % λ

λ λ

λ ^{1% 2% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 98% 99%}

0.5 0.000 0.000 0.002 0.008 0.032 0.074 0.137 0.227 0.354 0.537 0.821 1.353 1.921 2.706 3.317 1.0 0.010 0.020 0.051 0.105 0.223 0.357 0.511 0.693 0.916 1.204 1.609 2.303 2.996 3.912 4.605 1.5 0.057 0.092 0.176 0.292 0.503 0.712 0.935 1.183 1.473 1.832 2.321 3.126 3.907 4.919 5.672 2.0 0.149 0.215 0.355 0.532 0.824 1.097 1.376 1.678 2.022 2.439 2.994 3.890 4.744 5.834 6.638 2.5 0.277 0.376 0.573 0.805 1.171 1.500 1.828 2.176 2.566 3.032 3.645 4.618 5.535 6.694 7.543 3.0 0.436 0.567 0.818 1.102 1.535 1.914 2.285 2.674 3.105 3.616 4.279 5.322 6.296 7.517 8.406 3.5 0.620 0.782 1.084 1.417 1.911 2.336 2.747 3.173 3.642 4.192 4.902 6.009 7.034 8.311 9.238 4.0 0.823 1.016 1.366 1.745 2.297 2.764 3.211 3.672 4.175 4.762 5.515 6.681 7.754 9.084 10.05 4.5 1.044 1.266 1.663 2.084 2.690 3.197 3.679 4.171 4.707 5.328 6.121 7.342 8.459 9.839 10.83 5.0 1.279 1.530 1.970 2.433 3.090 3.634 4.148 4.671 5.237 5.890 6.721 7.994 9.154 10.58 11.60 5.5 1.527 1.804 2.287 2.789 3.494 4.074 4.619 5.171 5.765 6.449 7.316 8.638 9.838 11.31 12.36 6.0 1.785 2.089 2.613 3.152 3.904 4.517 5.091 5.670 6.292 7.006 7.906 9.275 10.51 12.03 13.11 6.5 2.053 2.383 2.946 3.521 4.317 4.963 5.565 6.170 6.818 7.559 8.492 9.906 11.18 12.74 13.84 7.0 2.330 2.684 3.285 3.895 4.734 5.411 6.039 6.670 7.343 8.111 9.075 10.53 11.84 13.44 14.57 7.5 2.615 2.992 3.630 4.273 5.153 5.861 6.515 7.169 7.867 8.661 9.655 11.15 12.50 14.13 15.29 8.0 2.906 3.307 3.981 4.656 5.576 6.312 6.991 7.669 8.390 9.209 10.23 11.77 13.15 14.82 16.00 8.5 3.204 3.628 4.336 5.043 6.001 6.765 7.469 8.169 8.912 9.756 10.81 12.38 13.79 15.50 16.70 9.0 3.507 3.953 4.695 5.432 6.428 7.220 7.947 8.669 9.434 10.30 11.38 12.99 14.43 16.17 17.40 9.5 3.816 4.284 5.059 5.825 6.858 7.676 8.425 9.169 9.955 10.84 11.95 13.60 15.07 16.84 18.10 10.0 4.130 4.618 5.425 6.221 7.289 8.133 8.904 9.669 10.48 11.39 12.52 14.21 15.71 17.51 18.78 11.0 4.771 5.300 6.169 7.021 8.157 9.050 9.864 10.67 11.52 12.47 13.65 15.41 16.96 18.83 20.14 12.0 5.428 5.996 6.924 7.829 9.031 9.972 10.83 11.67 12.55 13.55 14.78 16.60 18.21 20.14 21.49 13.0 6.099 6.704 7.690 8.646 9.910 10.90 11.79 12.67 13.59 14.62 15.90 17.78 19.44 21.43 22.82 14.0 6.782 7.424 8.464 9.470 10.79 11.82 12.75 13.67 14.62 15.70 17.01 18.96 20.67 22.71 24.14 15.0 7.477 8.153 9.246 10.30 11.68 12.75 13.72 14.67 15.66 16.77 18.13 20.13 21.89 23.98 25.45 16.0 8.181 8.891 10.04 11.14 12.57 13.69 14.69 15.67 16.69 17.83 19.23 21.29 23.10 25.24 26.74 17.0 8.895 9.638 10.83 11.98 13.47 14.62 15.66 16.67 17.72 18.90 20.34 22.45 24.30 26.50 28.03 18.0 9.616 10.39 11.63 12.82 14.37 15.56 16.63 17.67 18.75 19.96 21.44 23.61 25.50 27.74 29.31 19.0 10.35 11.15 12.44 13.67 15.27 16.50 17.60 18.67 19.78 21.02 22.54 24.76 26.69 28.98 30.58 20.0 11.08 11.92 13.25 14.53 16.17 17.44 18.57 19.67 20.81 22.08 23.63 25.90 27.88 30.22 31.85 21.0 11.83 12.69 14.07 15.38 17.08 18.38 19.54 20.67 21.84 23.14 24.73 27.05 29.06 31.45 33.10 22.0 12.57 13.47 14.89 16.24 17.99 19.32 20.51 21.67 22.87 24.20 25.82 28.18 30.24 32.67 34.35 23.0 13.33 14.25 15.72 17.11 18.90 20.26 21.48 22.67 23.89 25.25 26.91 29.32 31.41 33.89 35.60 24.0 14.09 15.04 16.55 17.97 19.81 21.21 22.46 23.67 24.92 26.31 28.00 30.45 32.59 35.10 36.84 25.0 14.85 15.83 17.38 18.84 20.72 22.16 23.43 24.67 25.95 27.36 29.08 31.58 33.75 36.31 38.08 26.0 15.62 16.63 18.22 19.72 21.64 23.10 24.41 25.67 26.97 28.41 30.17 32.71 34.92 37.51 39.31 27.0 16.40 17.43 19.06 20.59 22.56 24.05 25.38 26.67 28.00 29.46 31.25 33.84 36.08 38.71 40.53 28.0 17.17 18.23 19.90 21.47 23.48 25.00 26.36 27.67 29.02 30.52 32.33 34.96 37.23 39.91 41.76 29.0 17.96 19.04 20.75 22.35 24.40 25.95 27.33 28.67 30.04 31.56 33.41 36.08 38.39 41.10 42.98 30.0 18.74 19.85 21.59 23.23 25.32 26.90 28.31 29.67 31.07 32.61 34.49 37.20 39.54 42.29 44.19

λ λ λ

λ 1% 2% 5% 10% 20% 30% 40% 50% 60% 70% 80% 90% 95% 98% 99%

t f_GHtL

0 t=tFHl_L F

‡0 t

f_GHt,l_L„t=F

TABELA 3. Kwantyle t

( ν νν ν ) w rozkładzie t Studenta z ν νν ν stopniami swobody

( )

( )

p

t t

p f t dt

ν

∞

= ∫ _(A.5)

ν – liczba stopni swobody, p – prawdopodobieństwo prze- wyższenia. W ostatnim wierszu tabeli 3 zamieszczono odpowiednie kwantyle rozkładu N(0,1), które można stosować jako dobre przybliżenie rozkładu Studenta dla ν > 30.

p

ν

ν

1 0.325 0.727 1.376 3.078 6.314 12.71 31.82 63.66 318.3 1

2 0.289 0.617 1.061 1.886 2.920 4.303 6.965 9.925 22.33 2

3 0.277 0.584 0.978 1.638 2.353 3.182 4.541 5.841 10.21 3

4 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.604 7.173 4

5 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.032 5.893 5

6 0.265 0.553 0.906 1.440 1.943 2.447 3.143 3.707 5.208 6

7 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.499 4.785 7

8 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.355 4.501 8

9 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.250 4.297 9

10 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.169 4.144 10

11 0.260 0.540 0.876 1.363 1.796 2.201 2.718 3.106 4.025 11

12 0.259 0.539 0.873 1.356 1.782 2.179 2.681 3.055 3.930 12

13 0.259 0.538 0.870 1.350 1.771 2.160 2.650 3.012 3.852 13

14 0.258 0.537 0.868 1.345 1.761 2.145 2.624 2.977 3.787 14

15 0.258 0.536 0.866 1.341 1.753 2.131 2.602 2.947 3.733 15

16 0.258 0.535 0.865 1.337 1.746 2.120 2.583 2.921 3.686 16

17 0.257 0.534 0.863 1.333 1.740 2.110 2.567 2.898 3.646 17

18 0.257 0.534 0.862 1.330 1.734 2.101 2.552 2.878 3.610 18

19 0.257 0.533 0.861 1.328 1.729 2.093 2.539 2.861 3.579 19

20 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.845 3.552 20

21 0.257 0.532 0.859 1.323 1.721 2.080 2.518 2.831 3.527 21

22 0.256 0.532 0.858 1.321 1.717 2.074 2.508 2.819 3.505 22

23 0.256 0.532 0.858 1.319 1.714 2.069 2.500 2.807 3.485 23

24 0.256 0.531 0.857 1.318 1.711 2.064 2.492 2.797 3.467 24

25 0.256 0.531 0.856 1.316 1.708 2.060 2.485 2.787 3.450 25

26 0.256 0.531 0.856 1.315 1.706 2.056 2.479 2.779 3.435 26

27 0.256 0.531 0.855 1.314 1.703 2.052 2.473 2.771 3.421 27

28 0.256 0.530 0.855 1.313 1.701 2.048 2.467 2.763 3.408 28

29 0.256 0.530 0.854 1.311 1.699 2.045 2.462 2.756 3.396 29

30 0.256 0.530 0.854 1.310 1.697 2.042 2.457 2.750 3.385 30

N(0,1) 0.253 0.524 0.842 1.282 1.645 1.960 2.326 2.576 3.090 N(0,1) t f_Ht_L

0 t= tpHn_L p

‡t

¶

f_Ht_L„t = p

TABELA 4. Kwantyle χ χ χ χ

(F, < << <) rozkładu P P P P

(chi-kwadrat),

< - liczba stopni swobody, F = prawdopodobieństwo nieprzewyższenia

2

2

( , )

0

( )

F

F f u du

χ ν

= ∫

(A.6)

Uwaga. Tablice wartości dystrybuanty rozkładu P

kończą się zwykle na < = 30, gdyż przyjmuje się, że od tej liczby rozkład zmiennej (2P

)

można zastąpić granicznym rozkładem normalnym N[(2<-1)

, 1].

prawdopodobieństwo nieprzewyższenia,

<

<

1 0.000 0.001 0.004 0.016 0.064 0.148 0.275 0.455 0.708 1.074 1.642 2.706 3.841 5.024 6.635 1 2 0.020 0.051 0.103 0.211 0.446 0.713 1.022 1.386 1.833 2.408 3.219 4.605 5.991 7.378 9.210 2 3 0.115 0.216 0.352 0.584 1.005 1.424 1.869 2.366 2.946 3.665 4.642 6.251 7.815 9.348 11.34 3 4 0.297 0.484 0.711 1.064 1.649 2.195 2.753 3.357 4.045 4.878 5.989 7.779 9.488 11.14 13.28 4 5 0.554 0.831 1.145 1.610 2.343 3.000 3.656 4.351 5.132 6.064 7.289 9.236 11.07 12.83 15.09 5 6 0.872 1.237 1.635 2.204 3.070 3.828 4.570 5.348 6.211 7.231 8.558 10.64 12.59 14.45 16.81 6 7 1.239 1.690 2.167 2.833 3.822 4.671 5.493 6.346 7.283 8.383 9.803 12.02 14.07 16.01 18.48 7 8 1.647 2.180 2.733 3.490 4.594 5.527 6.423 7.344 8.351 9.524 11.03 13.36 15.51 17.53 20.09 8 9 2.088 2.700 3.325 4.168 5.380 6.393 7.357 8.343 9.414 10.66 12.24 14.68 16.92 19.02 21.67 9 10 2.558 3.247 3.940 4.865 6.179 7.267 8.295 9.342 10.47 11.78 13.44 15.99 18.31 20.48 23.21 10 11 3.053 3.816 4.575 5.578 6.989 8.148 9.237 10.34 11.53 12.90 14.63 17.28 19.68 21.92 24.73 11 12 3.571 4.404 5.226 6.304 7.807 9.034 10.18 11.34 12.58 14.01 15.81 18.55 21.03 23.34 26.22 12 13 4.107 5.009 5.892 7.041 8.634 9.926 11.13 12.34 13.64 15.12 16.98 19.81 22.36 24.74 27.69 13 14 4.660 5.629 6.571 7.790 9.467 10.82 12.08 13.34 14.69 16.22 18.15 21.06 23.68 26.12 29.14 14 15 5.229 6.262 7.261 8.547 10.31 11.72 13.03 14.34 15.73 17.32 19.31 22.31 25.00 27.49 30.58 15 16 5.812 6.908 7.962 9.312 11.15 12.62 13.98 15.34 16.78 18.42 20.47 23.54 26.30 28.85 32.00 16 17 6.408 7.564 8.672 10.09 12.00 13.53 14.94 16.34 17.82 19.51 21.61 24.77 27.59 30.19 33.41 17 18 7.015 8.231 9.390 10.86 12.86 14.44 15.89 17.34 18.87 20.60 22.76 25.99 28.87 31.53 34.81 18 19 7.633 8.907 10.12 11.65 13.72 15.35 16.85 18.34 19.91 21.69 23.90 27.20 30.14 32.85 36.19 19 20 8.260 9.591 10.85 12.44 14.58 16.27 17.81 19.34 20.95 22.77 25.04 28.41 31.41 34.17 37.57 20 21 8.897 10.28 11.59 13.24 15.44 17.18 18.77 20.34 21.99 23.86 26.17 29.62 32.67 35.48 38.93 21 22 9.542 10.98 12.34 14.04 16.31 18.10 19.73 21.34 23.03 24.94 27.30 30.81 33.92 36.78 40.29 22 23 10.20 11.69 13.09 14.85 17.19 19.02 20.69 22.34 24.07 26.02 28.43 32.01 35.17 38.08 41.64 23 24 10.86 12.40 13.85 15.66 18.06 19.94 21.65 23.34 25.11 27.10 29.55 33.20 36.42 39.36 42.98 24 25 11.52 13.12 14.61 16.47 18.94 20.87 22.62 24.34 26.14 28.17 30.68 34.38 37.65 40.65 44.31 25 26 12.20 13.84 15.38 17.29 19.82 21.79 23.58 25.34 27.18 29.25 31.79 35.56 38.89 41.92 45.64 26 27 12.88 14.57 16.15 18.11 20.70 22.72 24.54 26.34 28.21 30.32 32.91 36.74 40.11 43.19 46.96 27 28 13.56 15.31 16.93 18.94 21.59 23.65 25.51 27.34 29.25 31.39 34.03 37.92 41.34 44.46 48.28 28 29 14.26 16.05 17.71 19.77 22.48 24.58 26.48 28.34 30.28 32.46 35.14 39.09 42.56 45.72 49.59 29 30 14.95 16.79 18.49 20.60 23.36 25.51 27.44 29.34 31.32 33.53 36.25 40.26 43.77 46.98 50.89 30

u f_c²_Hu_L

0 u= c²_HF,n_L F

‡0

c²

f_c²_Hu,n_L„u = F

TABELA 5. Kwantyle F( α α α α ,n,m) rozkładu F Snedecora dla prawdopodobieństwa przewyższenia α = 0.05 (górny wiersz) i 0.01 (dolny wiersz) oraz liczby stopni swo- body (n,m).

Przydatny wzór:

( , , ) 1

(1 , , ) F n m

F m n

α ⁼ − α (A.7)

Przykład odczytu: F( α = 5%, n = 6, m = 4) = 6.16.

n

m 2 3 4 5 6 8 10 15 15 20 25 30 35 40 50 60 2 ^19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.4 19.5 19.5 19.5 19.5 19.5 19.5

99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.4 99.4 99.5 99.4 99.4 99.5 99.5 99.4 3 9.55 9.28 9.12 9.01 8.94 8.85 8.79 8.70 8.70 8.66 8.63 8.62 8.60 8.59 8.58 8.57 30.8 29.5 28.7 28.2 27.9 27.5 27.2 26.9 26.9 26.7 26.6 26.5 26.5 26.4 26.4 26.3 4 6.94 6.59 6.39 6.26 6.16 6.04 5.96 5.86 5.86 5.80 5.77 5.75 5.73 5.72 5.70 5.69 18.0 16.7 16.0 15.5 15.2 14.8 14.5 14.2 14.2 14.0 13.9 13.8 13.8 13.7 13.7 13.7 5 5.79 5.41 5.19 5.05 4.95 4.82 4.74 4.62 4.62 4.56 4.52 4.50 4.48 4.46 4.44 4.43 13.3 12.1 11.4 11.0 10.7 10.3 10.1 9.72 9.72 9.55 9.45 9.38 9.33 9.29 9.24 9.20 6 5.14 4.76 4.53 4.39 4.28 4.15 4.06 3.94 3.94 3.87 3.83 3.81 3.79 3.77 3.75 3.74 10.9 9.78 9.15 8.75 8.47 8.10 7.87 7.56 7.56 7.40 7.30 7.23 7.18 7.14 7.09 7.06 8 4.46 4.07 3.84 3.69 3.58 3.44 3.35 3.22 3.22 3.15 3.11 3.08 3.06 3.04 3.02 3.01 8.65 7.59 7.01 6.63 6.37 6.03 5.81 5.52 5.52 5.36 5.26 5.20 5.15 5.12 5.07 5.03 10 4.10 3.71 3.48 3.33 3.22 3.07 2.98 2.85 2.85 2.77 2.73 2.70 2.68 2.66 2.64 2.62 7.56 6.55 5.99 5.64 5.39 5.06 4.85 4.56 4.56 4.41 4.31 4.25 4.20 4.17 4.12 4.08 15 3.68 3.29 3.06 2.90 2.79 2.64 2.54 2.40 2.40 2.33 2.28 2.25 2.22 2.20 2.18 2.16 6.36 5.42 4.89 4.56 4.32 4.00 3.80 3.52 3.52 3.37 3.28 3.21 3.17 3.13 3.08 3.05 20 3.49 3.10 2.87 2.71 2.60 2.45 2.35 2.20 2.20 2.12 2.07 2.04 2.01 1.99 1.97 1.95 5.85 4.94 4.43 4.10 3.87 3.56 3.37 3.09 3.09 2.94 2.84 2.78 2.73 2.69 2.64 2.61 25 3.39 2.99 2.76 2.60 2.49 2.34 2.24 2.09 2.09 2.01 1.96 1.92 1.89 1.87 1.84 1.82 5.57 4.68 4.18 3.85 3.63 3.32 3.13 2.85 2.85 2.70 2.60 2.54 2.49 2.45 2.40 2.36 30 3.32 2.92 2.69 2.53 2.42 2.27 2.16 2.01 2.01 1.93 1.88 1.84 1.81 1.79 1.76 1.74 5.39 4.51 4.02 3.70 3.47 3.17 2.98 2.70 2.70 2.55 2.45 2.39 2.34 2.30 2.25 2.21 35 3.27 2.87 2.64 2.49 2.37 2.22 2.11 1.96 1.96 1.88 1.82 1.79 1.76 1.74 1.70 1.68 5.27 4.40 3.91 3.59 3.37 3.07 2.88 2.60 2.60 2.44 2.35 2.28 2.23 2.19 2.14 2.10 40 3.23 2.84 2.61 2.45 2.34 2.18 2.08 1.92 1.92 1.84 1.78 1.74 1.72 1.69 1.66 1.64 ^5.18 4.31 3.83 3.51 3.29 2.99 2.80 2.52 2.52 2.37 2.27 2.20 2.15 2.11 2.06 2.02 50 3.18 2.79 2.56 2.40 2.29 2.13 2.03 1.87 1.87 1.78 1.73 1.69 1.66 1.63 1.60 1.58 5.06 4.20 3.72 3.41 3.19 2.89 2.70 2.42 2.42 2.27 2.17 2.10 2.05 2.01 1.95 1.91 60 3.15 2.76 2.53 2.37 2.25 2.10 1.99 1.84 1.84 1.75 1.69 1.65 1.62 1.59 1.56 1.53

m 2 3 4 5 6 8 10 15 15 20 25 30 35 40 50 60

u fFHu,n,m_L

0 u= F_Ha,n,m_L 1- a

‡0 F

fFHu,n,m_L„u = 1 - a