Use Table to calculate the following binomial probabilities:a. P(x ≥ 6) when n = 7 and p =...
Use Table to calculate the following binomial probabilities:
a. P(x ≥ 6) when n = 7 and p =.5
b. P(x ≥ 5) when n = 9 and p =.5
c. P(x ≥ 8) when n = 8 and p =.5
d. P(x ≥ 10) when n = 15 and p =.5. Also, use the normal approximation to calculate this probability, and then compare the approximation with the exact value.
e. P(x ≥ 15) when n = 25 and p =.5. Also, use the normal approximation to calculate this probability, and then compare the approximation with the exact value.
Binomial Probabilities
Tabulated values are 
(Computations are rounded at the third decimal place.)
n = 5
a k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .951 | .774 | .590 | .328 | .168 | .078 | .031 | .010 | .002 | .000 | .000 | .000 | .000 |
1 | .999 | .977 | .919 | .737 | .528 | .337 | .188 | .087 | .031 | .007 | .000 | .000 | .000 |
2 | 1.000 | .999 | .991 | .942 | .837 | .683 | .500 | .317 | .163 | .058 | .009 | .001 | .000 |
3 | 1.000 | 1.000 | 1.000 | .993 | .969 | .913 | .812 | .663 | .472 | .263 | .081 | .023 | .001 |
4 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .990 | .969 | .922 | .832 | .672 | .410 | .226 | .049 |
a. n = 5
a k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .951 | .774 | .590 | .328 | .168 | .078 | .031 | .010 | .002 | .000 | .000 | .000 | .000 |
1 | .999 | .977 | .919 | .737 | .528 | .337 | .188 | .087 | .031 | .007 | .000 | .000 | .000 |
2 | 1.000 | .999 | .991 | .942 | .837 | .683 | .500 | .317 | .163 | .058 | .009 | .001 | .000 |
3 | 1.000 | 1.000 | 1.000 | .993 | .969 | .913 | .812 | .663 | .472 | .263 | .081 | .023 | .001 |
4 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .990 | .969 | .922 | .832 | .672 | .410 | .226 | .049 |
b. n = 6
a k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .941 | .735 | .531 | .262 | .118 | .047 | .016 | .004 | .001 | .000 | .000 | .000 | .000 |
1 | .999 | .967 | .886 | .655 | .420 | .233 | .109 | .041 | .011 | .002 | .000 | .000 | .000 |
2 | 1.000 | .998 | .984 | .901 | .744 | .544 | .344 | .179 | .070 | .017 | .001 | .000 | .000 |
3 | 1.000 | 1.000 | .999 | .983 | .930 | .821 | .656 | .456 | .256 | .099 | .016 | .002 | .000 |
4 | 1.000 | 1.000 | 1.000 | .998 | .989 | .959 | .891 | .767 | .580 | .345 | .114 | .033 | .001 |
5 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .996 | .984 | .953 | .882 | .738 | .469 | .265 | .059 |
n = 7
P k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .932 | .698 | .478 | .210 | .082 | .028 | .008 | .002 | .000 | .000 | .000 | .000 | .000 |
1 | .998 | .956 | .850 | .577 | .329 | .159 | .063 | .019 | .004 | .000 | .000 | .000 | .000 |
2 | 1.000 | .996 | .974 | .852 | .647 | .420 | .227 | .096 | .029 | .005 | .000 | .000 | .000 |
3 | 1.000 | 1.000 | .997 | .967 | .874 | .710 | .500 | .290 | .126 | .033 | .003 | .000 | .000 |
4 | 1.000 | 1.000 | 1.000 | .995 | .971 | .904 | .773 | .580 | .353 | .148 | .026 | .004 | .000 |
5 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .981 | .937 | .841 | .671 | .423 | .150 | .044 | .002 |
6 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .992 | .972 | .918 | .790 | .522 | .302 | .068 |
c. = 8
a k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .923 | .663 | .430 | .168 | .058 | .017 | .004 | .001 | .000 | .000 | .000 | .000 | .000 |
1 | .997 | .943 | .813 | .503 | .255 | .106 | .035 | .009 | .001 | .000 | .000 | .000 | .000 |
2 | 1.000 | .994 | .962 | .797 | .552 | .315 | .145 | .050 | .011 | .001 | .000 | .000 | .000 |
3 | 1.000 | 1.000 | .995 | .944 | .806 | .594 | .363 | .174 | .058 | .010 | .000 | .000 | .000 |
4 | 1.000 | 1.000 | 1.000 | .990 | .942 | .826 | .637 | .406 | .194 | .056 | .005 | .000 | .000 |
5 | 1.000 | 1.000 | 1.000 | .999 | .989 | .950 | .855 | .685 | .448 | .203 | .038 | .006 | .000 |
6 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .991 | .965 | .894 | .745 | .497 | .187 | .057 | .003 |
7 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .996 | .983 | .942 | .832 | .570 | .337 | .077 |
d. n = 9
a k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .914 | .630 | .387 | .134 | .040 | .010 | .002 | .000 | .000 | .000 | .000 | .000 | .000 |
1 | .997 | .929 | .775 | .436 | .196 | .071 | .020 | .004 | .000 | .000 | .000 | .000 | .000 |
2 | 1.000 | .992 | .947 | .738 | .463 | .232 | .090 | .025 | .004 | .000 | .000 | .000 | .000 |
3 | 1.000 | .999 | .992 | .914 | .730 | .483 | .254 | .099 | .025 | .003 | .000 | .000 | .000 |
4 | 1.000 | 1.000 | .999 | .980 | .901 | .733 | .500 | .267 | .099 | .020 | .001 | .000 | .000 |
5 | 1.000 | 1.000 | 1.000 | .997 | .975 | .901 | .746 | .517 | .270 | .086 | .008 | .001 | .000 |
6 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .975 | .910 | .768 | .537 | .262 | .053 | .008 | .000 |
7 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .980 | .929 | .804 | .564 | .225 | .071 | .003 |
8 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .990 | .960 | .866 | .613 | .370 | .086 |
e. n = 10
P k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .904 | .599 | .349 | .107 | .028 | .006 | .001 | .000 | .000 | .000 | .000 | .000 | .000 |
1 | .996 | .914 | .736 | .376 | .149 | .046 | .011 | .002 | .000 | .000 | .000 | .000 | .000 |
2 | 1.000 | .988 | .930 | .678 | .383 | .167 | .055 | .012 | .002 | .000 | .000 | .000 | .000 |
3 | 1.000 | .999 | .987 | .879 | .650 | .382 | .172 | .055 | .011 | .001 | .000 | .000 | .000 |
4 | 1.000 | 1.000 | .998 | .967 | .850 | .633 | .377 | .166 | .047 | .006 | .000 | .000 | .000 |
5 | 1.000 | 1.000 | 1.000 | .994 | .953 | .834 | .623 | .367 | .150 | .033 | .002 | .000 | .000 |
6 | 1.000 | 1.000 | 1.000 | .999 | .989 | .945 | .828 | .618 | .350 | .121 | .013 | .001 | .000 |
7 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .988 | .945 | .833 | .617 | .322 | .070 | .012 | .000 |
8 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .989 | .954 | .851 | .624 | .264 | .086 | .004 |
9 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .994 | .972 | .893 | .651 | .401 | .096 |
f. n = 15
p k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .860 | .463 | .206 | .035 | .005 | .000 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
1 | .990 | .829 | .549 | .167 | .035 | .005 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
2 | 1.000 | .964 | .816 | .398 | .127 | .027 | .004 | .000 | .000 | .000 | .000 | .000 | .000 |
3 | 1.000 | .995 | .944 | .648 | .297 | .091 | .018 | .002 | .000 | .000 | .000 | .000 | .000 |
4 | 1.000 | .999 | .987 | .838 | .515 | .217 | .059 | .009 | .001 | .000 | .000 | .000 | .000 |
5 | 1.000 | 1.000 | .998 | .939 | .722 | .403 | .151 | .034 | .004 | .000 | .000 | .000 | .000 |
6 | 1.000 | 1.000 | 1.000 | .982 | .869 | .610 | .304 | .095 | .015 | .001 | .000 | .000 | .000 |
7 | 1.000 | 1.000 | 1.000 | .996 | .950 | .787 | .500 | .213 | .050 | .004 | .000 | .000 | .000 |
8 | 1.000 | 1.000 | 1.000 | .999 | .985 | .905 | .696 | .390 | .131 | .018 | .000 | .000 | .000 |
9 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .966 | .849 | .597 | .278 | .061 | .002 | .000 | .000 |
10 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .991 | .941 | .783 | .485 | .164 | .013 | .001 | .000 |
11 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .982 | .909 | .703 | .352 | .056 | .005 | .000 |
12 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .973 | .873 | .602 | .184 | .036 | .000 |
13 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .995 | .965 | .833 | .451 | .171 | .010 |
14 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .995 | .965 | .794 | .537 | .140 |
g. n = 20
p k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .818 | .358 | .122 | .012 | .001 | .000 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
1 | .983 | .736 | .392 | .069 | .008 | .001 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
2 | .999 | .925 | .677 | .206 | .035 | .004 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
3 | 1.000 | .984 | .867 | .411 | .107 | .016 | .001 | .000 | .000 | .000 | .000 | .000 | .000 |
4 | 1.000 | .997 | .957 | .630 | .238 | .051 | .006 | .000 | .000 | .000 | .000 | .000 | .000 |
5 | 1.000 | 1.000 | .989 | .804 | .416 | .126 | .021 | .002 | .000 | .000 | .000 | .000 | .000 |
6 | 1.000 | 1.000 | .998 | .913 | .608 | .250 | .058 | .006 | .000 | .000 | .000 | .000 | .000 |
7 | 1.000 | 1.000 | 1.000 | .968 | .772 | .416 | .132 | .021 | .001 | .000 | .000 | .000 | .000 |
8 | 1.000 | 1.000 | 1.000 | .990 | .887 | .596 | .252 | .057 | .005 | .000 | .000 | .000 | .000 |
9 | 1.000 | 1.000 | 1.000 | .997 | .952 | .755 | .412 | .128 | .017 | .001 | .000 | .000 | .000 |
10 | 1.000 | 1.000 | 1.000 | .999 | .983 | .872 | .588 | .245 | .048 | .003 | .000 | .000 | .000 |
11 | 1.000 | 1.000 | 1.000 | 1.000 | .995 | .943 | .748 | .404 | .113 | .010 | .000 | .000 | .000 |
12 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .979 | .868 | .584 | .228 | .032 | .000 | .000 | .000 |
13 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .994 | .942 | .750 | .392 | .087 | .002 | .000 | .000 |
14 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .979 | .874 | .584 | .196 | .011 | .000 | .000 |
15 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .994 | .949 | .762 | .370 | .043 | .003 | .000 |
16 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .984 | .893 | .589 | .133 | .016 | .000 |
17 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .965 | .794 | .323 | .075 | .001 |
18 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .992 | .931 | .608 | .264 | .017 |
19 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .988 | .878 | .642 | .182 |
h. n = 25
p k | .01 | .05 | .10 | .20 | .30 | .40 | .50 | .60 | .70 | .80 | .90 | .95 | .99 |
0 | .778 | .277 | .072 | .004 | .000 | .000 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
1 | .974 | .642 | .271 | .027 | .002 | .000 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
2 | .998 | .873 | .537 | .098 | .009 | .000 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
3 | 1.000 | .966 | .764 | .234 | .033 | .002 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
4 | 1.000 | .993 | .902 | .421 | .090 | .009 | .000 | .000 | .000 | .000 | .000 | .000 | .000 |
5 | 1.000 | .999 | .967 | .617 | .193 | .029 | .002 | .000 | .000 | .000 | .000 | .000 | .000 |
6 | 1.000 | 1.000 | .991 | .780 | .341 | .074 | .007 | .000 | .000 | .000 | .000 | .000 | .000 |
7 | 1.000 | 1.000 | .998 | .891 | .512 | .154 | .022 | .001 | .000 | .000 | .000 | .000 | .000 |
8 | 1.000 | 1.000 | 1.000 | .953 | .677 | .274 | .054 | .004 | .000 | .000 | .000 | .000 | .000 |
9 | 1.000 | 1.000 | 1.000 | .983 | .811 | .425 | .115 | .013 | .000 | .000 | .000 | .000 | .000 |
10 | 1.000 | 1.000 | 1.000 | .994 | .902 | .586 | .212 | .034 | .002 | .000 | .000 | .000 | .000 |
11 | 1.000 | 1.000 | 1.000 | .998 | .956 | .732 | .345 | .078 | .006 | .000 | .000 | .000 | .000 |
12 | 1.000 | 1.000 | 1.000 | 1.000 | .983 | .846 | .500 | .154 | .017 | .000 | .000 | .000 | .000 |
13 | 1.000 | 1.000 | 1.000 | 1.000 | .994 | .922 | .655 | .268 | .044 | .002 | .000 | .000 | .000 |
14 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .966 | .788 | .414 | .098 | .006 | .000 | .000 | .000 |
15 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .987 | .885 | .575 | .189 | .017 | .000 | .000 | .000 |
16 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .946 | .726 | .323 | .047 | .000 | .000 | .000 |
17 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .999 | .978 | .846 | .488 | .109 | .002 | .000 | .000 |
18 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .993 | .926 | .659 | .220 | .009 | .000 | .000 |
19 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .971 | .807 | .383 | .033 | .001 | .000 |
20 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .991 | .910 | .579 | .098 | .007 | .000 |
21 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .967 | .766 | .236 | .034 | .000 |
22 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .991 | .902 | .463 | .127 | .002 |
23 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .998 | .973 | .729 | .358 | .026 |
24 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | .996 | .928 | .723 | .222 |
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