2191:. Note that to do this we cannot simply double the one-tailed p-value unless the probability of the event is 1/2. This is because the binomial distribution becomes asymmetric as that probability deviates from 1/2. There are two methods to define the two-tailed p-value. One method is to sum the probability that the total deviation in numbers of events in either direction from the expected value is either more than or less than the expected value. The probability of that occurring in our example is 0.0437. The second method involves computing the probability that the deviation from the expected value is as unlikely or more unlikely than the observed value, i.e. from a comparison of the probability density functions. This can create a subtle difference, but in this example yields the same probability of 0.0437. In both cases, the two-tailed test reveals significance at the 5% level, indicating that the number of 6s observed was significantly different for this die than the expected number at the 5% level.
25:
2783:, use Binom.Dist. The function takes parameters (Number of successes, Trials, Probability of Success, Cumulative). The "Cumulative" parameter takes a boolean True or False, with True giving the Cumulative probability of finding this many successes (a left-tailed test), and False the exact probability of finding this many successes.
2073:(here we are basically testing whether this die is biased towards generating more 6s than expected). In order to calculate the probability of 51 or more 6s in a sample of 235 under the null hypothesis we add up the probabilities of getting exactly 51 6s, exactly 52 6s, and so on up to probability of getting exactly 235 6s:
1841:
times. We have now observed that the number of 6s is higher than what we would expect on average by pure chance had the die been a fair one. But, is the number significantly high enough for us to conclude anything about the fairness of the die? This question can be answered by the binomial test. Our
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Normally, when we are testing for fairness of a die, we are also interested if the die is biased towards generating fewer 6s than expected, and not only more 6s as we considered in the one-tailed test above. In order to consider both the biases, we use a
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1204:), such as a coin toss. Tables are widely available to give the significance observed numbers of observations in the categories for this case. However, as the example below shows, the binomial test is not restricted to this case.
870:-value from the one-tailed test. Recall that we want to consider events that are as, or more, extreme than the one we've seen, so we should consider the probability that we would see an event that is as, or less, likely than
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If we have a significance level of 5%, then this result (0.02654 < 5%) indicates that we have evidence that is significant enough to reject the null hypothesis that the die is fair.
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for this test by considering the probability of seeing an outcome as, or more, extreme. For a one-tailed test, this is straightforward to compute. Suppose that we want to test if
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and attaches special importance to rolling a 6. In a particular game, the die is rolled 235 times, and 6 comes up 51 times. If the die is fair, we would expect 6 to come up
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As we have observed a value greater than the expected value, we could consider the probability of observing 51 6s or higher under the null, which would constitute a
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1146:{\displaystyle p=\sum _{i\in {\mathcal {I}}}\Pr(X=i)=\sum _{i\in {\mathcal {I}}}{\binom {n}{i}}\pi _{0}^{i}(1-\pi _{0})^{n-i}}
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of deviations from a theoretically expected distribution of observations into two categories using sample data.
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711:{\displaystyle p=\sum _{i=0}^{k}\Pr(X=i)=\sum _{i=0}^{k}{\binom {n}{i}}\pi _{0}^{i}(1-\pi _{0})^{n-i}}
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The most usual (and easiest) approximation is through the standard normal distribution, in which a
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in both numerator and denominator, which is a form that may be more familiar to some readers.
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would be that the die is fair (probability of each number coming up on the die is 1/6).
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Binomial tests are available in most software used for statistical purposes. E.g.
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2058:{\displaystyle f(k,n,p)=\Pr(k;n,p)=\Pr(X=k)={\binom {n}{k}}p^{k}(1-p)^{n-k}}
1473:{\displaystyle Z={\frac {k-n\pi \pm {\frac {1}{2}}}{\sqrt {n\pi (1-\pi )}}}}
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When there are more than two categories, and an exact test is required, the
2173:{\displaystyle \sum _{i=51}^{235}{235 \choose i}p^{i}(1-p)^{235-i}=0.02654}
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To find an answer to this question using the binomial test, we use the
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967:{\displaystyle {\mathcal {I}}=\{i\colon \Pr(X=i)\leq \Pr(X=k)\}}
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the above example could be calculated with the following code:
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were correct, then the expected number of successes would be
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1326:{\displaystyle Z={\frac {k-n\pi }{\sqrt {n\pi (1-\pi )}}}}
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One common use of the binomial test is the case where the
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An analogous computation can be done if we're testing if
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is the number of successes observed in a sample of size
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In notation in terms of a measured sample proportion
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2548:DATA=DiceRoll ; TABLES Roll / BINOMIAL (P=
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2585:npar tests /binomial (.5) = node1 node2.
850:. This means that we can't just double the
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2825:(6. ed.). Belmont, Calif.: Thomson.
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109:Learn how and when to remove this message
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754:using the summation of the range from
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1197:{\displaystyle H_{0}\colon \pi =0.5}
47:adding citations to reliable sources
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2823:Statistical methods for psychology
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2195:In statistical software packages
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843:{\displaystyle \pi _{0}\neq 0.5}
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530:{\displaystyle \pi <\pi _{0}}
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16:Test of statistical significance
2864:Binomial Probability Calculator
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1233:Pearson's chi-squared test
2821:Howell, David C. (2007).
994:-value is calculated as,
474:{\displaystyle n\pi _{0}}
309:{\displaystyle n\pi _{0}}
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1213:multinomial distribution
236:{\displaystyle \pi _{0}}
130:statistical significance
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246:If in a sample of size
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43:improve this article
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2295:"greater"
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2577:Nonparametric test
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2879:Statistical tests
2720:(two-tailed test)
2661:(one-tailed test)
2655:'greater'
2531:(two-tailed test)
2475:(one-tailed test)
2419:(one-tailed test)
2348:(two-tailed test)
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767:{\displaystyle k}
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495:{\displaystyle p}
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2423:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2197:
2189:two-tailed test
2148:
2126:
2105:
2078:
2077:
2071:one-tailed test
2039:
2017:
1996:
1915:
1914:
1857:
1856:
1844:null hypothesis
1800:
1799:
1785:
1761:
1760:
1759:by dividing by
1725:
1706:
1705:
1692:
1676:
1663:
1662:
1632:
1627:
1626:
1584:
1583:
1564:
1563:
1542:
1537:
1536:
1508:
1507:
1485:
1484:
1483:For very large
1418:
1405:
1404:
1378:
1377:
1358:
1357:
1338:
1337:
1284:
1271:
1270:
1248:
1247:
1221:
1211:, based on the
1172:
1167:
1166:
1159:
1127:
1117:
1072:
999:
998:
976:
975:
898:
897:
872:
871:
852:
851:
824:
819:
818:
799:
798:
776:
775:
756:
755:
734:
723:
722:
692:
682:
637:
562:
561:
539:
538:
517:
506:
505:
484:
483:
461:
453:
452:
431:
426:
425:
395:
373:
352:
325:
324:
296:
288:
287:
268:
267:
248:
247:
223:
218:
217:
193:
174:
169:
168:
146:
145:
142:test hypotheses
138:
115:
104:
98:
95:
58:"Binomial test"
52:
50:
40:
28:
17:
12:
11:
5:
2892:
2890:
2882:
2881:
2871:
2870:
2867:
2866:
2859:
2858:External links
2856:
2855:
2854:
2839:
2838:
2832:978-0495012870
2831:
2812:
2811:
2809:
2806:
2805:
2804:
2799:
2789:
2786:
2785:
2784:
2777:
2770:
2723:
2722:
2721:
2662:
2588:
2584:
2565:
2544:
2535:
2534:
2533:
2532:
2476:
2420:
2361:Apache Commons
2352:
2351:
2350:
2349:
2302:
2255:
2196:
2193:
2181:
2180:
2169:
2166:
2161:
2158:
2155:
2151:
2147:
2144:
2141:
2138:
2133:
2129:
2122:
2117:
2114:
2109:
2101:
2096:
2093:
2090:
2086:
2067:
2066:
2052:
2049:
2046:
2042:
2038:
2035:
2032:
2029:
2024:
2020:
2013:
2008:
2005:
2000:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
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1940:
1937:
1934:
1931:
1928:
1925:
1922:
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1892:
1888:
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1882:
1879:
1876:
1873:
1870:
1867:
1864:
1839:
1838:
1827:
1824:
1821:
1817:
1813:
1810:
1807:
1784:
1781:
1768:
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1756:
1741:
1737:
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1728:
1724:
1721:
1718:
1713:
1709:
1699:
1695:
1691:
1685:
1682:
1673:
1670:
1647:
1644:
1639:
1635:
1614:
1610:
1606:
1603:
1597:
1594:
1571:
1549:
1545:
1521:
1518:
1492:
1481:
1480:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1441:
1438:
1433:
1430:
1427:
1424:
1421:
1415:
1412:
1385:
1365:
1345:
1334:
1333:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1296:
1293:
1290:
1287:
1281:
1278:
1255:
1220:
1217:
1193:
1190:
1187:
1184:
1179:
1175:
1158:
1155:
1154:
1153:
1140:
1137:
1134:
1130:
1124:
1120:
1116:
1113:
1110:
1105:
1100:
1096:
1089:
1084:
1081:
1076:
1066:
1061:
1058:
1054:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1025:
1020:
1017:
1013:
1009:
1006:
983:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
907:
885:
882:
879:
859:
839:
836:
831:
827:
806:
797:Calculating a
783:
763:
741:
737:
733:
730:
719:
718:
705:
702:
699:
695:
689:
685:
681:
678:
675:
670:
665:
661:
654:
649:
646:
641:
633:
628:
625:
622:
618:
614:
611:
608:
605:
602:
599:
596:
591:
586:
583:
580:
576:
572:
569:
546:
524:
520:
516:
513:
491:
481:. We find our
468:
464:
460:
438:
434:
422:
421:
408:
405:
402:
398:
394:
391:
388:
385:
380:
376:
369:
364:
361:
356:
350:
347:
344:
341:
338:
335:
332:
303:
299:
295:
275:
255:
230:
226:
214:
213:
200:
196:
192:
189:
186:
181:
177:
164:) of success:
153:
137:
134:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2891:
2880:
2877:
2876:
2874:
2865:
2862:
2861:
2857:
2851:
2847:
2843:
2842:
2834:
2828:
2824:
2817:
2814:
2807:
2803:
2800:
2798:
2796:
2792:
2791:
2787:
2782:
2778:
2776:, use bitest.
2775:
2771:
2732:
2728:
2724:
2663:
2604:
2603:
2601:
2597:
2593:
2589:
2582:
2578:
2574:
2570:
2566:
2541:
2537:
2536:
2477:
2421:
2365:
2364:
2362:
2358:
2354:
2353:
2303:
2256:
2209:
2208:
2206:
2202:
2201:
2200:
2194:
2192:
2190:
2184:
2167:
2164:
2159:
2156:
2153:
2145:
2142:
2139:
2131:
2127:
2115:
2112:
2099:
2094:
2091:
2088:
2084:
2076:
2075:
2074:
2072:
2050:
2047:
2044:
2036:
2033:
2030:
2022:
2018:
2006:
2003:
1992:
1986:
1983:
1980:
1971:
1965:
1962:
1959:
1956:
1953:
1944:
1938:
1935:
1932:
1929:
1926:
1920:
1913:
1894:
1890:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1862:
1855:
1854:
1853:
1852:
1847:
1845:
1825:
1822:
1819:
1815:
1811:
1808:
1805:
1798:
1797:
1796:
1794:
1790:
1782:
1780:
1766:
1739:
1730:
1726:
1722:
1719:
1711:
1707:
1697:
1693:
1689:
1680:
1671:
1668:
1661:
1660:
1659:
1645:
1642:
1637:
1633:
1612:
1608:
1604:
1601:
1592:
1569:
1547:
1543:
1516:
1504:
1490:
1461:
1458:
1455:
1449:
1446:
1439:
1436:
1431:
1428:
1425:
1422:
1419:
1413:
1410:
1403:
1402:
1401:
1399:
1383:
1363:
1343:
1314:
1311:
1308:
1302:
1299:
1294:
1291:
1288:
1285:
1279:
1276:
1269:
1268:
1267:
1253:
1245:
1240:
1238:
1234:
1230:
1226:
1219:Large samples
1218:
1216:
1214:
1210:
1205:
1191:
1188:
1185:
1182:
1177:
1173:
1164:
1156:
1138:
1135:
1132:
1122:
1118:
1114:
1111:
1103:
1098:
1094:
1082:
1079:
1059:
1056:
1052:
1048:
1042:
1039:
1036:
1018:
1015:
1011:
1007:
1004:
997:
996:
995:
981:
955:
952:
949:
940:
934:
931:
928:
919:
916:
910:
883:
880:
877:
857:
837:
834:
829:
825:
804:
795:
781:
761:
739:
735:
731:
728:
703:
700:
697:
687:
683:
679:
676:
668:
663:
659:
647:
644:
631:
626:
623:
620:
616:
612:
606:
603:
600:
589:
584:
581:
578:
574:
570:
567:
560:
559:
558:
544:
522:
518:
514:
511:
503:
489:
466:
462:
458:
436:
432:
406:
403:
400:
392:
389:
386:
378:
374:
362:
359:
348:
342:
339:
336:
323:
322:
321:
319:
301:
297:
293:
273:
253:
244:
228:
224:
198:
194:
190:
187:
184:
179:
175:
167:
166:
165:
151:
143:
135:
133:
131:
127:
123:
122:Binomial test
113:
110:
102:
99:November 2016
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
2849:
2822:
2816:
2794:
2580:
2576:
2572:
2489:binomialTest
2483:BinomialTest
2469:GREATER_THAN
2433:binomialTest
2427:BinomialTest
2377:binomialTest
2371:BinomialTest
2198:
2185:
2182:
2068:
1848:
1840:
1786:
1758:
1505:
1482:
1335:
1241:
1222:
1206:
1160:
796:
720:
423:
245:
215:
139:
121:
120:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
2739:myBinomTest
2731:myBinomTest
2708:alternative
2649:alternative
2336:alternative
2289:alternative
2242:alternative
1266:, given by
537:. Then our
2808:References
2546:PROC FREQ
2359:using the
2306:binom.test
2259:binom.test
2212:binom.test
1789:board game
1157:Common use
266:there are
126:exact test
69:newspapers
2678:binomtest
2619:binomtest
2600:binomtest
2525:TWO_SIDED
2413:LESS_THAN
2363:library:
2157:−
2143:−
2085:∑
2048:−
2034:−
1809:×
1723:−
1690:−
1684:^
1646:π
1596:^
1520:^
1462:π
1459:−
1450:π
1432:±
1429:π
1423:−
1384:π
1315:π
1312:−
1303:π
1295:π
1289:−
1186:π
1183::
1136:−
1119:π
1115:−
1095:π
1060:∈
1053:∑
1019:∈
1012:∑
941:≤
920::
835:≠
826:π
794:instead.
736:π
729:π
701:−
684:π
680:−
660:π
617:∑
575:∑
519:π
512:π
463:π
404:−
390:−
298:π
225:π
195:π
188:π
185::
152:π
2873:Category
2788:See also
2581:Binomial
2552:) ALPHA=
2550:0.166667
1582:, where
1235:and the
2573:Analyze
2168:0.02654
1783:Example
128:of the
83:scholar
2829:
2797:-value
2729:, use
2727:MATLAB
2594:, use
2592:Python
1336:where
1244:z-test
1237:G-test
896:. Let
502:-value
216:where
124:is an
85:
78:
71:
64:
56:
2774:Stata
2672:stats
2666:scipy
2613:stats
2607:scipy
2596:SciPy
2579:>
2575:>
1910:with
1826:39.17
136:Usage
90:JSTOR
76:books
2827:ISBN
2569:SPSS
2561:RUN;
2554:0.05
2357:Java
1625:and
1376:and
732:>
515:<
62:news
2779:In
2772:In
2751:235
2725:In
2696:1.0
2690:235
2637:1.0
2631:235
2598:'s
2590:In
2567:In
2540:SAS
2538:In
2507:1.0
2495:235
2486:().
2480:new
2451:1.0
2439:235
2430:().
2424:new
2395:1.0
2383:235
2374:().
2368:new
2355:In
2318:235
2271:235
2224:235
2203:In
2154:235
2113:235
2100:235
1912:pmf
1875:235
1806:235
1793:die
1192:0.5
838:0.5
774:to
45:by
2875::
2848:.
2745:51
2684:51
2625:51
2602::
2501:51
2445:51
2389:51
2312:51
2265:51
2218:51
2095:51
1975:Pr
1948:Pr
1400::
1031:Pr
944:Pr
923:Pr
595:Pr
331:Pr
2852:.
2835:.
2795:p
2766:)
2763:6
2760:/
2757:1
2754:,
2748:,
2742:(
2736:=
2717:)
2711:=
2705:,
2702:6
2699:/
2693:,
2687:,
2681:(
2675:.
2669:.
2658:)
2652:=
2646:,
2643:6
2640:/
2634:,
2628:,
2622:(
2616:.
2610:.
2558:;
2528:)
2522:.
2516:,
2513:6
2510:/
2504:,
2498:,
2492:(
2472:)
2466:.
2460:,
2457:6
2454:/
2448:,
2442:,
2436:(
2416:)
2410:.
2404:,
2401:6
2398:/
2392:,
2386:,
2380:(
2345:)
2339:=
2333:,
2330:6
2327:/
2324:1
2321:,
2315:,
2309:(
2298:)
2292:=
2286:,
2283:6
2280:/
2277:1
2274:,
2268:,
2262:(
2251:)
2245:=
2239:,
2236:6
2233:/
2230:1
2227:,
2221:,
2215:(
2205:R
2165:=
2160:i
2150:)
2146:p
2140:1
2137:(
2132:i
2128:p
2121:)
2116:i
2108:(
2092:=
2089:i
2065:.
2051:k
2045:n
2041:)
2037:p
2031:1
2028:(
2023:k
2019:p
2012:)
2007:k
2004:n
1999:(
1993:=
1990:)
1987:k
1984:=
1981:X
1978:(
1972:=
1969:)
1966:p
1963:,
1960:n
1957:;
1954:k
1951:(
1945:=
1942:)
1939:p
1936:,
1933:n
1930:,
1927:k
1924:(
1921:f
1898:)
1895:6
1891:/
1887:1
1884:=
1881:p
1878:,
1872:=
1869:N
1866:(
1863:B
1823:=
1820:6
1816:/
1812:1
1767:n
1740:n
1736:)
1731:0
1727:p
1720:1
1717:(
1712:0
1708:p
1698:0
1694:p
1681:p
1672:=
1669:Z
1643:=
1638:0
1634:p
1613:n
1609:/
1605:k
1602:=
1593:p
1570:n
1548:0
1544:p
1517:p
1491:n
1465:)
1456:1
1453:(
1447:n
1440:2
1437:1
1426:n
1420:k
1414:=
1411:Z
1364:n
1344:k
1318:)
1309:1
1306:(
1300:n
1292:n
1286:k
1280:=
1277:Z
1254:Z
1189:=
1178:0
1174:H
1139:i
1133:n
1129:)
1123:0
1112:1
1109:(
1104:i
1099:0
1088:)
1083:i
1080:n
1075:(
1065:I
1057:i
1049:=
1046:)
1043:i
1040:=
1037:X
1034:(
1024:I
1016:i
1008:=
1005:p
982:p
962:}
959:)
956:k
953:=
950:X
947:(
938:)
935:i
932:=
929:X
926:(
917:i
914:{
911:=
906:I
884:k
881:=
878:X
858:p
830:0
805:p
782:n
762:k
740:0
704:i
698:n
694:)
688:0
677:1
674:(
669:i
664:0
653:)
648:i
645:n
640:(
632:k
627:0
624:=
621:i
613:=
610:)
607:i
604:=
601:X
598:(
590:k
585:0
582:=
579:i
571:=
568:p
545:p
523:0
490:p
467:0
459:n
437:0
433:H
407:k
401:n
397:)
393:p
387:1
384:(
379:k
375:p
368:)
363:k
360:n
355:(
349:=
346:)
343:k
340:=
337:X
334:(
302:0
294:n
274:k
254:n
229:0
199:0
191:=
180:0
176:H
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
