Consider a 2 by 2 table: with rows and/or columns exchanged so that (1)  and (2)  . The table entries are ordered lexicographically by (ascending). (descending) and (descending). For each triple the table presents critical values for one-sided tests of the hypothesis that the true proportion corresponding to is greater than the true proportion corresponding to . Significance levels of 0.05, 0.025 and 0.01 are considered. For all values where critical values exist are tabulated. For each significance level two columns give: (1) the nominal critical value for (that is, reject the null hypothesis if the observed is less than or equal to the table entry) and (2) the -value corresponding to the critical value (this is less than the nominal significance level in most cases due to the discreteness of the distribution).
 3 3 3 0 .050 -- -- -- -- 4 4 4 0 .014 0 .014 -- -- 4 3 4 0 .029 -- -- -- -- 5 5 5 1 .024 1 .024 0 .004 5 5 4 0 .024 0 .024 -- -- 5 4 5 1 .048 0 .008 0 .008 5 4 4 0 .040 -- -- -- -- 5 3 5 0 .018 0 .018 -- -- 5 2 5 0 .048 -- -- -- -- 6 6 6 2 .030 1 .008 1 .008 6 6 5 1 .040 0 .008 0 .008 6 6 4 0 .030 -- -- -- -- 6 5 6 1 .015 1 .015 0 .002 6 5 5 0 .013 0 .013 -- -- 6 5 4 0 .045 -- -- -- -- 6 4 6 1 .033 0 .005 0 .005 6 4 5 0 .024 0 .024 -- -- 6 3 6 0 .012 0 .012 -- -- 6 3 5 0 .048 -- -- -- -- 6 2 6 0 .036 -- -- -- -- 7 7 7 3 .035 2 .010 1 .002 7 7 6 1 .015 1 .015 0 .002 7 7 5 0 .010 0 .010 -- -- 7 7 4 0 .035 -- -- -- -- 7 6 7 2 .021 2 .021 1 .005 7 6 6 1 .025 0 .004 0 .004 7 6 5 0 .016 0 .016 -- -- 7 6 4 0 .049 -- -- -- -- 7 5 7 2 .045 1 .010 0 .001 7 5 6 1 .045 0 .008 0 .008 7 5 5 0 .027 -- -- -- -- 7 4 7 1 .024 1 .024 0 .003 7 4 6 0 .015 0 .015 -- -- 7 4 5 0 .045 -- -- -- -- 7 3 7 0 .008 0 .008 0 .008 7 3 6 0 .033 -- -- -- -- 7 2 7 0 .028 -- -- -- -- 8 8 8 4 .038 3 .013 2 .003 8 8 7 2 .020 2 .020 1 .005 8 8 6 1 .020 1 .020 0 .003 8 8 5 0 .013 0 .013 -- -- 8 8 4 0 .038 -- -- -- -- 8 7 8 3 .026 2 .007 2 .007 8 7 7 2 .035 1 .009 1 .009 8 7 6 1 .032 0 .006 0 .006 8 7 5 0 .019 0 .019 -- -- 8 6 8 2 .015 2 .015 1 .003 8 6 7 1 .016 1 .016 0 .002 8 6 6 0 .009 0 .009 0 .009 8 6 5 0 .028 -- -- -- -- 8 5 8 2 .035 1 .007 1 .007 8 5 7 1 .032 0 .005 0 .005 8 5 6 0 .016 0 .016 -- -- 8 5 5 0 .044 -- -- -- -- 8 4 8 1 .018 1 .018 0 .002 8 4 7 0 .010 0 .010 -- -- 8 4 6 0 .030 -- -- -- -- 8 3 8 0 .006 0 .006 0 .006 8 3 7 0 .024 0 .024 -- -- 8 2 8 0 .022 0 .022 -- -- 9 9 9 5 .041 4 .015 3 .005 9 9 8 3 .025 3 .025 2 .008 9 9 7 2 .028 1 .008 1 .008 9 9 6 1 .025 1 .025 0 .005 9 9 5 0 .015 0 .015 -- -- 9 9 4 0 .041 -- -- -- -- 9 8 9 4 .029 3 .009 3 .009 9 8 8 3 .043 2 .013 1 .003 9 8 7 2 .044 1 .012 0 .002 9 8 6 1 .036 0 .007 0 .007 9 8 5 0 .020 0 .020 -- -- 9 7 9 3 .019 3 .019 2 .005 9 7 8 2 .024 2 .024 1 .006 9 7 7 1 .020 1 .020 0 .003 9 7 6 0 .010 0 .010 -- -- 9 7 5 0 .029 -- -- -- -- 9 6 9 3 .044 2 .011 1 .002 9 6 8 2 .047 1 .011 0 .001 9 6 7 1 .035 0 .006 0 .006 9 6 6 0 .017 0 .017 -- -- 9 6 5 0 .042 -- -- -- -- 9 5 9 2 .027 1 .005 1 .005 9 5 8 1 .023 1 .023 0 .003 9 5 7 0 .010 0 .010 -- -- 9 5 6 0 .028 -- -- -- -- 9 4 9 1 .014 1 .014 0 .001 9 4 8 0 .007 0 .007 0 .007 9 4 7 0 .021 0 .021 -- -- 9 4 6 0 .049 -- -- -- -- 9 3 9 1 .045 0 .005 0 .005 9 3 8 0 .018 0 .018 -- -- 9 3 7 0 .045 -- -- -- -- 9 2 9 0 .018 0 .018 -- -- 10 10 10 6 .043 5 .016 4 .005 10 10 9 4 .029 3 .010 3 .010 10 10 8 3 .035 2 .012 1 .003 10 10 7 2 .035 1 .010 1 .010 10 10 6 1 .029 0 .005 0 .005 10 10 5 0 .016 0 .016 -- -- 10 10 4 0 .043 -- -- -- -- 10 9 10 5 .033 4 .011 3 .003 10 9 9 4 .050 3 .017 2 .005 10 9 8 2 .019 2 .019 1 .004 10 9 7 1 .015 1 .015 0 .002 10 9 6 1 .040 0 .008 0 .008 10 9 5 0 .022 0 .022 -- -- 10 8 10 4 .023 4 .023 3 .007 10 8 9 3 .032 2 .009 2 .009 10 8 8 2 .031 1 .008 1 .008 10 8 7 1 .023 1 .023 0 .004 10 8 6 0 .011 0 .011 -- -- 10 8 5 0 .029 -- -- -- -- 10 7 10 3 .015 3 .015 2 .003 10 7 9 2 .018 2 .018 1 .004 10 7 8 1 .013 1 .013 0 .002 10 7 7 1 .036 0 .006 0 .006 10 7 6 0 .017 0 .017 -- -- 10 7 5 0 .041 -- -- -- -- 10 6 10 3 .036 2 .008 2 .008 10 6 9 2 .036 1 .008 1 .008 10 6 8 1 .024 1 .024 0 .003 10 6 7 0 .010 0 .010 -- -- 10 6 6 0 .026 -- -- -- -- 10 5 10 2 .022 2 .022 1 .004 10 5 9 1 .017 1 .017 0 .002 10 5 8 1 .047 0 .007 0 .007 10 5 7 0 .019 0 .019 -- -- 10 5 6 0 .042 -- -- -- -- 10 4 10 1 .011 1 .011 0 .001 10 4 9 1 .041 0 .005 0 .005 10 4 8 0 .015 0 .015 -- -- 10 4 7 0 .035 -- -- -- -- 10 3 10 1 .038 0 .003 0 .003 10 3 9 0 .014 0 .014 -- -- 10 3 8 0 .035 -- -- -- -- 10 2 10 0 .015 0 .015 -- -- 10 2 9 0 .045 -- -- -- -- 11 11 11 7 .045 6 .018 5 .006 11 11 10 5 .032 4 .012 3 .004 11 11 9 4 .040 3 .015 2 .004 11 11 8 3 .043 2 .015 1 .004 11 11 7 2 .040 1 .012 0 .002 11 11 6 1 .032 0 .006 0 .006 11 11 5 0 .018 0 .018 -- -- 11 11 4 0 .045 -- -- -- -- 11 10 11 6 .035 5 .012 4 .004 11 10 10 4 .021 4 .021 3 .007 11 10 9 3 .024 3 .024 2 .007 11 10 8 2 .023 2 .023 1 .006 11 10 7 1 .017 1 .017 0 .003 11 10 6 1 .043 0 .009 0 .009 11 10 5 0 .023 0 .023 -- -- 11 9 11 5 .026 4 .008 4 .008 11 9 10 4 .038 3 .012 2 .003 11 9 9 3 .040 2 .012 1 .003 11 9 8 2 .035 1 .009 1 .009 11 9 7 1 .025 1 .025 0 .004 11 9 6 0 .012 0 .012 -- -- 11 9 5 0 .030 -- -- -- -- 11 8 11 4 .018 4 .018 3 .005 11 8 10 3 .024 3 .024 2 .006 11 8 9 2 .022 2 .022 1 .005 11 8 8 1 .015 1 .015 0 .002 11 8 7 1 .037 0 .007 0 .007 11 8 6 0 .017 0 .017 -- -- 11 8 5 0 .040 -- -- -- -- 11 7 11 4 .043 3 .011 2 .002 11 7 10 3 .047 2 .013 1 .002 11 7 9 2 .039 1 .009 1 .009 11 7 8 1 .025 1 .025 0 .004 11 7 7 0 .010 0 .010 -- -- 11 7 6 0 .025 0 .025 -- -- 11 6 11 3 .029 2 .006 2 .006 11 6 10 2 .028 1 .005 1 .005 11 6 9 1 .018 1 .018 0 .002 11 6 8 1 .043 0 .007 0 .007 11 6 7 0 .017 0 .017 -- -- 11 6 6 0 .037 -- -- -- -- 11 5 11 2 .018 2 .018 1 .003 11 5 10 1 .013 1 .013 0 .001 11 5 9 1 .036 0 .005 0 .005 11 5 8 0 .013 0 .013 -- -- 11 5 7 0 .029 -- -- -- -- 11 4 11 1 .009 1 .009 1 .009 11 4 10 1 .033 0 .004 0 .004 11 4 9 0 .011 0 .011 -- -- 11 4 8 0 .026 -- -- -- -- 11 3 11 1 .033 0 .003 0 .003 11 3 10 0 .011 0 .011 -- -- 11 3 9 0 .027 -- -- -- -- 11 2 11 0 .013 0 .013 -- -- 11 2 10 0 .038 -- -- -- -- 12 12 12 8 .047 7 .019 6 .007 12 12 11 6 .034 5 .014 4 .005 12 12 10 5 .045 4 .018 3 .006 12 12 9 4 .050 3 .020 2 .006 12 12 8 3 .050 2 .018 1 .005 12 12 7 2 .045 1 .014 0 .002 12 12 6 1 .034 0 .007 0 .007 12 12 5 0 .019 0 .019 -- -- 12 12 4 0 .047 -- -- -- -- 12 11 12 7 .037 6 .014 5 .005 12 11 11 5 .024 5 .024 4 .008 12 11 10 4 .029 3 .010 2 .003 12 11 9 3 .030 2 .009 2 .009 12 11 8 2 .026 1 .007 1 .007 12 11 7 1 .019 1 .019 0 .003 12 11 6 1 .045 0 .009 0 .009 12 11 5 0 .024 0 .024 -- -- 12 10 12 6 .029 5 .010 5 .010 12 10 11 5 .043 4 .015 3 .005 12 10 10 4 .048 3 .017 2 .005 12 10 9 3 .046 2 .015 1 .004 12 10 8 2 .038 1 .010 0 .002 12 10 7 1 .026 0 .005 0 .005 12 10 6 0 .012 0 .012 -- -- 12 10 5 0 .030 -- -- -- -- 12 9 12 5 .021 5 .021 4 .006 12 9 11 4 .029 3 .009 3 .009 12 9 10 3 .029 2 .008 2 .008 12 9 9 2 .024 2 .024 1 .006 12 9 8 1 .016 1 .016 0 .002 12 9 7 1 .037 0 .007 0 .007 12 9 6 0 .017 0 .017 -- -- 12 9 5 0 .039 -- -- -- -- 12 8 12 5 .049 4 .014 3 .004 12 8 11 3 .018 3 .018 2 .004 12 8 10 2 .015 2 .015 1 .003 12 8 9 2 .040 1 .010 1 .010 12 8 8 1 .025 1 .025 0 .004 12 8 7 0 .010 0 .010 -- -- 12 8 6 0 .024 0 .024 -- -- 12 7 12 4 .036 3 .009 3 .009 12 7 11 3 .038 2 .010 2 .010 12 7 10 2 .029 1 .006 1 .006 12 7 9 1 .017 1 .017 0 .002 12 7 8 1 .040 0 .007 0 .007 12 7 7 0 .016 0 .016 -- -- 12 7 6 0 .034 -- -- -- -- 12 6 12 3 .025 3 .025 2 .005 12 6 11 2 .022 2 .022 1 .004 12 6 10 1 .013 1 .013 0 .002 12 6 9 1 .032 0 .005 0 .005 12 6 8 0 .011 0 .011 -- -- 12 6 7 0 .025 0 .025 -- -- 12 6 6 0 .050 -- -- -- -- 12 5 12 2 .015 2 .015 1 .002 12 5 11 1 .010 1 .010 1 .010 12 5 10 1 .028 0 .003 0 .003 12 5 9 0 .009 0 .009 0 .009 12 5 8 0 .020 0 .020 -- -- 12 5 7 0 .041 -- -- -- -- 12 4 12 2 .050 1 .007 1 .007 12 4 11 1 .027 0 .003 0 .003 12 4 10 0 .008 0 .008 0 .008 12 4 9 0 .019 0 .019 -- -- 12 4 8 0 .038 -- -- -- -- 12 3 12 1 .029 0 .002 0 .002 12 3 11 0 .009 0 .009 0 .009 12 3 10 0 .022 0 .022 -- -- 12 3 9 0 .044 -- -- -- -- 12 2 12 0 .011 0 .011 -- -- 12 2 11 0 .033 -- -- -- -- 13 13 13 9 .048 8 .020 7 .007 13 13 12 7 .037 6 .015 5 .006 13 13 11 6 .048 5 .021 4 .008 13 13 10 4 .024 4 .024 3 .008 13 13 9 3 .024 3 .024 2 .008 13 13 8 2 .021 2 .021 1 .006 13 13 7 2 .048 1 .015 0 .003 13 13 6 1 .037 0 .007 0 .007 13 13 5 0 .020 0 .020 -- -- 13 13 4 0 .048 -- -- -- -- 13 12 13 8 .039 7 .015 6 .005 13 12 12 6 .027 5 .010 5 .010 13 12 11 5 .033 4 .013 3 .004 13 12 10 4 .036 3 .013 2 .004 13 12 9 3 .034 2 .011 1 .003 13 12 8 2 .029 1 .008 1 .008 13 12 7 1 .020 1 .020 0 .004 13 12 6 1 .046 0 .010 0 .010 13 12 5 0 .024 0 .024 -- -- 13 11 13 7 .031 6 .011 5 .003 13 11 12 6 .048 5 .018 4 .006 13 11 11 4 .021 4 .021 3 .007 13 11 10 3 .021 3 .021 2 .006 13 11 9 3 .050 2 .017 1 .004 13 11 8 2 .040 1 .011 0 .002 13 11 7 1 .027 0 .005 0 .005 13 11 6 0 .013 0 .013 -- -- 13 11 5 0 .030 -- -- -- -- 13 10 13 6 .024 6 .024 5 .007 13 10 12 5 .035 4 .012 3 .003 13 10 11 4 .037 3 .012 2 .003 13 10 10 3 .033 2 .010 1 .002 13 10 9 2 .026 1 .006 1 .006 13 10 8 1 .017 1 .017 0 .003 13 10 7 1 .038 0 .007 0 .007 13 10 6 0 .017 0 .017 -- -- 13 10 5 0 .038 -- -- -- -- 13 9 13 5 .017 5 .017 4 .005 13 9 12 4 .023 4 .023 3 .007 13 9 11 3 .022 3 .022 2 .006 13 9 10 2 .017 2 .017 1 .004 13 9 9 2 .040 1 .010 0 .001 13 9 8 1 .025 1 .025 0 .004 13 9 7 0 .010 0 .010 -- -- 13 9 6 0 .023 0 .023 -- -- 13 9 5 0 .049 -- -- -- -- 13 8 13 5 .042 4 .012 3 .003 13 8 12 4 .047 3 .014 2 .003 13 8 11 3 .041 2 .011 1 .002 13 8 10 2 .029 1 .007 1 .007 13 8 9 1 .017 1 .017 0 .002 13 8 8 1 .037 0 .006 0 .006 13 8 7 0 .015 0 .015 -- -- 13 8 6 0 .032 -- -- -- -- 13 7 13 4 .031 3 .007 3 .007 13 7 12 3 .031 2 .007 2 .007 13 7 11 2 .022 2 .022 1 .004 13 7 10 1 .012 1 .012 0 .002 13 7 9 1 .029 0 .004 0 .004 13 7 8 0 .010 0 .010 -- -- 13 7 7 0 .022 0 .022 -- -- 13 7 6 0 .044 -- -- -- -- 13 6 13 3 .021 3 .021 2 .004 13 6 12 2 .017 2 .017 1 .003 13 6 11 2 .046 1 .010 1 .010 13 6 10 1 .024 1 .024 0 .003 13 6 9 1 .050 0 .008 0 .008 13 6 8 0 .017 0 .017 -- -- 13 6 7 0 .034 -- -- -- -- 13 5 13 2 .012 2 .012 1 .002 13 5 12 2 .044 1 .008 1 .008 13 5 11 1 .022 1 .022 0 .002 13 5 10 1 .047 0 .007 0 .007 13 5 9 0 .015 0 .015 -- -- 13 5 8 0 .029 -- -- -- -- 13 4 13 2 .044 1 .006 1 .006 13 4 12 1 .022 1 .022 0 .002 13 4 11 0 .006 0 .006 0 .006 13 4 10 0 .015 0 .015 -- -- 13 4 9 0 .029 -- -- -- -- 13 3 13 1 .025 1 .025 0 .002 13 3 12 0 .007 0 .007 0 .007 13 3 11 0 .018 0 .018 -- -- 13 3 10 0 .036 -- -- -- -- 13 2 13 0 .010 0 .010 0 .010 13 2 12 0 .029 -- -- -- -- 14 14 14 10 .049 9 .020 8 .008 14 14 13 8 .038 7 .016 6 .006 14 14 12 6 .023 6 .023 5 .009 14 14 11 5 .027 4 .011 3 .004 14 14 10 4 .028 3 .011 2 .003 14 14 9 3 .027 2 .009 2 .009 14 14 8 2 .023 2 .023 1 .006 14 14 7 1 .016 1 .016 0 .003 14 14 6 1 .038 0 .008 0 .008 14 14 5 0 .020 0 .020 -- -- 14 14 4 0 .049 -- -- -- -- 14 13 14 9 .041 8 .016 7 .006 14 13 13 7 .029 6 .011 5 .004 14 13 12 6 .037 5 .015 4 .005 14 13 11 5 .041 4 .017 3 .006 14 13 10 4 .041 3 .016 2 .005 14 13 9 3 .038 2 .013 1 .003 14 13 8 2 .031 1 .009 1 .009 14 13 7 1 .021 1 .021 0 .004 14 13 6 1 .048 0 .010 -- -- 14 13 5 0 .025 0 .025 -- -- 14 12 14 8 .033 7 .012 6 .004 14 12 13 6 .021 6 .021 5 .007 14 12 12 5 .025 4 .009 4 .009 14 12 11 4 .026 3 .009 3 .009 14 12 10 3 .024 3 .024 2 .007 14 12 9 2 .019 2 .019 1 .005 14 12 8 2 .042 1 .012 0 .002 14 12 7 1 .028 0 .005 0 .005 14 12 6 0 .013 0 .013 -- -- 14 12 5 0 .030 -- -- -- -- 14 11 14 7 .026 6 .009 6 .009 14 11 13 6 .039 5 .014 4 .004 14 11 12 5 .043 4 .016 3 .005 14 11 11 4 .042 3 .015 2 .004 14 11 10 3 .036 2 .011 1 .003 14 11 9 2 .027 1 .007 1 .007 14 11 8 1 .017 1 .017 0 .003 14 11 7 1 .038 0 .007 0 .007 14 11 6 0 .017 0 .017 -- -- 14 11 5 0 .038 -- -- -- -- 14 10 14 6 .020 6 .020 5 .006 14 10 13 5 .028 4 .009 4 .009 14 10 12 4 .028 3 .009 3 .009 14 10 11 3 .024 3 .024 2 .007 14 10 10 2 .018 2 .018 1 .004 14 10 9 2 .040 1 .011 0 .002 14 10 8 1 .024 1 .024 0 .004 14 10 7 0 .010 0 .010 0 .010 14 10 6 0 .022 0 .022 -- -- 14 10 5 0 .047 -- -- -- -- 14 9 14 6 .047 5 .014 4 .004 14 9 13 4 .018 4 .018 3 .005 14 9 12 3 .017 3 .017 2 .004 14 9 11 3 .042 2 .012 1 .002 14 9 10 2 .029 1 .007 1 .007 14 9 9 1 .017 1 .017 0 .002 14 9 8 1 .036 0 .006 0 .006 14 9 7 0 .014 0 .014 -- -- 14 9 6 0 .030 -- -- -- -- 14 8 14 5 .036 4 .010 4 .010 14 8 13 4 .039 3 .011 2 .002 14 8 12 3 .032 2 .008 2 .008 14 8 11 2 .022 2 .022 1 .005 14 8 10 2 .048 1 .012 0 .002 14 8 9 1 .026 0 .004 0 .004 14 8 8 0 .009 0 .009 0 .009 14 8 7 0 .020 0 .020 -- -- 14 8 6 0 .040 -- -- -- -- 14 7 14 4 .026 3 .006 3 .006 14 7 13 3 .025 2 .006 2 .006 14 7 12 2 .017 2 .017 1 .003 14 7 11 2 .041 1 .009 1 .009 14 7 10 1 .021 1 .021 0 .003 14 7 9 1 .043 0 .007 0 .007 14 7 8 0 .015 0 .015 -- -- 14 7 7 0 .030 -- -- -- -- 14 6 14 3 .018 3 .018 2 .003 14 6 13 2 .014 2 .014 1 .002 14 6 12 2 .037 1 .007 1 .007 14 6 11 1 .018 1 .018 0 .002 14 6 10 1 .038 0 .005 0 .005 14 6 9 0 .012 0 .012 -- -- 14 6 8 0 .024 0 .024 -- -- 14 6 7 0 .044 -- -- -- -- 14 5 14 2 .010 2 .010 1 .001 14 5 13 2 .037 1 .006 1 .006 14 5 12 1 .017 1 .017 0 .002 14 5 11 1 .038 0 .005 0 .005 14 5 10 0 .011 0 .011 -- -- 14 5 9 0 .022 0 .022 -- -- 14 5 8 0 .040 -- -- -- -- 14 4 14 2 .039 1 .005 1 .005 14 4 13 1 .019 1 .019 0 .002 14 4 12 1 .044 0 .005 0 .005 14 4 11 0 .011 0 .011 -- -- 14 4 10 0 .023 0 .023 -- -- 14 4 9 0 .041 -- -- -- -- 14 3 14 1 .022 1 .022 0 .001 14 3 13 0 .006 0 .006 0 .006 14 3 12 0 .015 0 .015 -- -- 14 3 11 0 .029 -- -- -- -- 14 2 14 0 .008 0 .008 0 .008 14 2 13 0 .025 0 .025 -- -- 14 2 12 0 .050 -- -- -- -- 15 15 15 11 .050 10 .021 9 .008 15 15 14 9 .040 8 .018 7 .007 15 15 13 7 .025 6 .010 5 .004 15 15 12 6 .030 5 .013 4 .005 15 15 11 5 .033 4 .013 3 .005 15 15 10 4 .033 3 .013 2 .004 15 15 9 3 .030 2 .010 1 .003 15 15 8 2 .025 1 .007 1 .007 15 15 7 1 .018 1 .018 0 .003 15 15 6 1 .040 0 .008 0 .008 15 15 5 0 .021 0 .021 -- -- 15 15 4 0 .050 -- -- -- -- 15 14 15 10 .042 9 .017 8 .006 15 14 14 8 .031 7 .013 6 .005 15 14 13 7 .041 6 .017 5 .007 15 14 12 6 .046 5 .020 4 .007 15 14 11 5 .048 4 .020 3 .007 15 14 10 4 .046 3 .018 2 .006 15 14 9 3 .041 2 .014 1 .004 15 14 8 2 .033 1 .009 1 .009 15 14 7 1 .022 1 .022 0 .004 15 14 6 1 .049 0 .011 -- -- 15 14 5 0 .025 -- -- -- -- 15 13 15 9 .035 8 .013 7 .005 15 13 14 7 .023 7 .023 6 .009 15 13 13 6 .029 5 .011 4 .004 15 13 12 5 .031 4 .012 3 .004 15 13 11 4 .030 3 .011 2 .003 15 13 10 3 .026 2 .008 2 .008 15 13 9 2 .020 2 .020 1 .005 15 13 8 2 .043 1 .013 0 .002 15 13 7 1 .029 0 .005 0 .005 15 13 6 0 .013 0 .013 -- -- 15 13 5 0 .031 -- -- -- -- 15 12 15 8 .028 7 .010 7 .010 15 12 14 7 .043 6 .016 5 .006 15 12 13 6 .049 5 .019 4 .007 15 12 12 5 .049 4 .019 3 .006 15 12 11 4 .045 3 .017 2 .005 15 12 10 3 .038 2 .012 1 .003 15 12 9 2 .028 1 .007 1 .007 15 12 8 1 .018 1 .018 0 .003 15 12 7 1 .038 0 .007 0 .007 15 12 6 0 .017 0 .017 -- -- 15 12 5 0 .037 -- -- -- -- 15 11 15 7 .022 7 .022 6 .007 15 11 14 6 .032 5 .011 4 .003 15 11 13 5 .034 4 .012 3 .003 15 11 12 4 .032 3 .010 2 .003 15 11 11 3 .026 2 .008 2 .008 15 11 10 2 .019 2 .019 1 .004 15 11 9 2 .040 1 .011 0 .002 15 11 8 1 .024 1 .024 0 .004 15 11 7 1 .049 0 .010 0 .010 15 11 6 0 .022 0 .022 -- -- 15 11 5 0 .046 -- -- -- -- 15 10 15 6 .017 6 .017 5 .005 15 10 14 5 .023 5 .023 4 .007 15 10 13 4 .022 4 .022 3 .007 15 10 12 3 .018 3 .018 2 .005 15 10 11 3 .042 2 .013 1 .003 15 10 10 2 .029 1 .007 1 .007 15 10 9 1 .016 1 .016 0 .002 15 10 8 1 .034 0 .006 0 .006 15 10 7 0 .013 0 .013 -- -- 15 10 6 0 .028 -- -- -- -- 15 9 15 6 .042 5 .012 4 .003 15 9 14 5 .047 4 .015 3 .004 15 9 13 4 .042 3 .013 2 .003 15 9 12 3 .032 2 .009 2 .009 15 9 11 2 .021 2 .021 1 .005 15 9 10 2 .045 1 .011 0 .002 15 9 9 1 .024 1 .024 0 .004 15 9 8 1 .048 0 .009 0 .009 15 9 7 0 .019 0 .019 -- -- 15 9 6 0 .037 -- -- -- -- 15 8 15 5 .032 4 .008 4 .008 15 8 14 4 .033 3 .009 3 .009 15 8 13 3 .026 2 .006 2 .006 15 8 12 2 .017 2 .017 1 .003 15 8 11 2 .037 1 .008 1 .008 15 8 10 1 .019 1 .019 0 .003 15 8 9 1 .038 0 .006 0 .006 15 8 8 0 .013 0 .013 -- -- 15 8 7 0 .026 -- -- -- -- 15 8 6 0 .050 -- -- -- -- 15 7 15 4 .023 4 .023 3 .005 15 7 14 3 .021 3 .021 2 .004 15 7 13 2 .014 2 .014 1 .002 15 7 12 2 .032 1 .007 1 .007 15 7 11 1 .015 1 .015 0 .002 15 7 10 1 .032 0 .005 0 .005 15 7 9 0 .010 0 .010 -- -- 15 7 8 0 .020 0 .020 -- -- 15 7 7 0 .038 -- -- -- -- 15 6 15 3 .015 3 .015 2 .003 15 6 14 2 .011 2 .011 1 .002 15 6 13 2 .031 1 .006 1 .006 15 6 12 1 .014 1 .014 0 .002 15 6 11 1 .029 0 .004 0 .004 15 6 10 0 .009 0 .009 0 .009 15 6 9 0 .017 0 .017 -- -- 15 6 8 0 .032 -- -- -- -- 15 5 15 2 .009 2 .009 2 .009 15 5 14 2 .032 1 .005 1 .005 15 5 13 1 .014 1 .014 0 .001 15 5 12 1 .031 0 .004 0 .004 15 5 11 0 .008 0 .008 0 .008 15 5 10 0 .016 0 .016 -- -- 15 5 9 0 .030 -- -- -- -- 15 4 15 2 .035 1 .004 1 .004 15 4 14 1 .016 1 .016 0 .001 15 4 13 1 .037 0 .004 0 .004 15 4 12 0 .009 0 .009 0 .009 15 4 11 0 .018 0 .018 -- -- 15 4 10 0 .033 -- -- -- -- 15 3 15 1 .020 1 .020 0 .001 15 3 14 0 .005 0 .005 0 .005 15 3 13 0 .012 0 .012 -- -- 15 3 12 0 .025 0 .025 -- -- 15 3 11 0 .043 -- -- -- -- 15 2 15 0 .007 0 .007 0 .007 15 2 14 0 .022 0 .022 -- -- 15 2 13 0 .044 -- -- -- -- 23 10 21 5 .016 5 .016 4 .004 32 13 32 10 .020 10 .020 9 .005

Let and be the true proportions in two populations. The sample size, , for two equally sized groups is tabulated for one-sided significance level and probability of not rejecting the null hypothesis. Each rectangular portion of the table contains sample sizes for two pairs of and values, one above the diagonal and one below it. The arcsine approximation was used to estimate .
 and .001 .01 .05 .10 .15 .20 .25 .30 .40 .50 .60 .70 .80 .90 .001 -- 2305 288 129 81 58 45 37 26 20 15 12 10 8 .01 1679 -- 689 221 123 82 61 48 32 24 18 14 11 9 .05 210 502 -- 1169 366 191 122 87 52 35 25 19 14 11 .10 94 161 852 -- 1877 538 266 163 83 51 34 25 18 13 .15 59 90 266 1368 -- 2489 683 327 132 73 46 31 22 15 .20 43 60 140 392 1814 -- 3012 805 222 105 61 39 27 18 .25 33 44 89 194 498 2194 -- 3447 417 158 83 50 32 21 .30 27 35 63 119 239 587 2511 -- 981 256 116 64 39 25 .40 19 24 38 60 96 162 304 715 -- 1068 267 116 61 34 .50 14 17 26 37 53 77 116 187 778 -- 1068 256 105 51 .60 11 13 19 25 34 45 61 84 195 778 -- 981 222 83 .70 9 10 14 18 23 29 37 47 84 187 715 -- 805 163 .80 7 8 11 13 16 20 24 29 45 77 162 587 -- 538 .90 6 6 8 10 11 13 15 18 25 37 60 119 392 -- and (or and and (or and .001 .01 .05 .10 .15 .20 .25 .30 .40 .50 .60 .70 .80 .90 .001 -- 1384 173 78 49 35 27 22 16 12 9 8 6 5 .01 1119 -- 414 133 74 50 37 29 20 14 11 9 7 5 .05 140 335 -- 702 220 115 74 52 31 21 15 12 9 7 .10 63 108 568 -- 1127 323 160 98 50 31 21 15 11 8 .15 40 60 178 911 -- 1494 410 197 79 44 28 19 13 9 .20 29 40 93 261 1208 -- 1808 483 133 63 37 24 16 11 .25 22 30 60 129 332 1462 -- 2069 251 95 50 30 20 13 .30 18 23 42 79 159 391 1673 -- 589 154 70 39 24 15 .40 13 16 25 40 64 108 203 476 -- 641 161 70 37 21 .50 10 12 17 25 35 51 77 125 519 -- 641 154 63 31 .60 8 9 13 17 23 30 40 56 130 519 -- 589 133 50 .70 6 7 9 12 15 19 25 32 56 125 476 -- 483 98 .80 5 6 7 9 11 13 16 19 30 51 108 391 -- 323 .90 4 4 6 7 8 9 10 12 17 25 40 79 261 -- and (or and and .001 .01 .05 .10 .15 .20 .25 .30 .40 .50 .60 .70 .80 .90 .001 -- 1152 144 65 41 29 23 19 13 10 8 6 5 4 .01 912 -- 345 111 62 41 31 24 16 12 9 7 6 5 .05 114 273 -- 585 183 96 61 44 26 18 13 10 7 6 .10 51 88 463 -- 939 269 133 82 42 26 17 13 9 7 .15 32 49 145 743 -- 1245 342 164 66 36 23 16 11 8 .20 23 33 76 213 985 -- 1506 403 111 53 31 20 14 9 .25 18 24 49 106 271 1192 -- 1723 209 79 42 25 16 11 .30 15 19 35 65 130 319 1364 -- 491 128 58 32 20 13 .40 11 13 21 33 52 88 165 388 -- 534 134 58 31 17 .50 8 10 14 20 29 42 63 102 423 -- 534 128 53 26 .60 6 7 10 14 18 24 33 46 106 423 -- 491 111 42 .70 5 6 8 10 13 16 20 26 46 102 388 -- 403 82 .80 4 5 6 7 9 11 13 16 24 42 88 319 -- 269 .90 3 4 5 5 6 7 9 10 14 20 33 65 213 -- and (or and and .001 .01 .05 .10 .15 .20 .25 .30 .40 .50 .60 .70 .80 .90 .001 -- 700 88 40 25 18 14 11 8 6 5 4 3 3 .01 480 -- 210 67 38 25 19 15 10 7 6 5 4 3 .05 60 144 -- 355 111 58 37 27 16 11 8 6 5 4 .10 27 46 244 -- 570 164 81 50 25 16 11 8 6 4 .15 17 26 77 391 -- 756 208 100 40 22 14 10 7 5 .20 13 18 40 112 519 -- 914 245 68 32 19 12 8 6 .25 10 13 26 56 143 628 -- 1046 127 48 25 16 10 7 .30 8 10 18 34 69 168 718 -- 298 78 35 20 12 8 .40 6 7 11 18 28 47 87 205 -- 325 82 35 19 11 .50 4 5 8 11 15 22 33 54 223 -- 325 78 32 16 .60 4 4 6 8 10 13 18 25 56 223 -- 298 68 25 .70 3 3 4 6 7 9 11 14 25 54 205 -- 245 50 .80 2 3 3 4 5 6 7 9 13 22 47 168 -- 164 .90 2 2 3 3 4 4 5 6 8 11 18 34 112 -- and (or and