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227
APPENDIX TABLE
A-1 Random numbers
A-2 Binomial probabilities
A-3 Poisson probabilities
A-4a Normal distribution
A-4b Laplace function
A-5 Values of chi-square
A-6 Values of the Student t-distribution
A-7 Values of the F-distribution: 10%, 5%, and 1% levels.
A-8 Critical values for Duncan's multiple range test
A-9a Critical values t (, k - 1, df) to compare control against
each of k - 1 other treatments in one-sided Dunnett's tests.
A-9b Critical values t (, k - 1, df) to compare control against
each of k - 1 other treatments in two-sided Dunnett's tests.
A-10 Orthogonal coefficient for trend comparisons
A-11 Critical values of correlation coefficient for certain levels
(.1, .05, .01 and .001) of significance.
228
'Table A-1 Random numbers
8 6 6 1 8 4 2 5 3 4 2 5 8 1 4 4 8 9 0 1 5 7 8 0
0 9 4 9 2 7 6 9 3 3 4 8 5 5 9 8 6 6 2 9 4 6 7 2
9 4 5 4 4 9 7 1 7 0 5 6 4 3 5 5 9 7 3 7 2 2 8 3
3 5 2 0 6 7 5 1 3 0 2 8 9 9 3 3 5 9 8 8 7 1 4 5
8 4 7 4 7 1 2 7 2 0 2 2 2 8 3 4 3 4 8 6 7 0 0 1
0 5 0 7 8 0 5 6 7 6 8 5 3 9 9 1 4 0 6 4 5 5 7 5
2 7 9 3 0 2 6 6 2 6 0 8 3 6 6 0 9 6 0 3 3 7 8 3
8 1 0 8 0 2 7 2 8 9 7 4 6 9 4 9 0 5 6 4 4 5 3 7
7 7 2 8 6 0 8 7 8 1 4 4 1 7 1 3 5 9 6 1 6 4 4 1
1 9 6 8 1 2 6 8 3 9 3 5 5 2 8 7 0 9 9 1 1 7 2 2
6 1 8 9 1 7 7 9 4 2 4 5 0 5 6 6 1 2 2 8 8 9 1 2
6 6 3 6 9 2 0 8 9 0 1 9 4 6 1 1 7 8 7 3 5 3 8 9
0 4 7 2 3 1 4 1 6 0 1 9 1 7 5 2 5 1 6 3 3 9 4 3
0 4 0 7 6 9 6 1 0 2 3 0 9 0 4 0 1 2 2 7 4 6 9 9
7 3 5 3 3 7 6 0 1 9 3 0 7 6 0 4 5 3 1 0 8 1 3 1
9 7 7 7 0 9 2 2 0 3 8 4 7 3 3 2 0 9 5 4 8 0 5 1
2 0 6 0 3 5 5 7 4 1 4 6 1 3 0 2 3 3 0 5 1 6 8 5
1 0 3 6 2 6 7 8 5 6 6 6 7 2 5 5 7 0 8 4 3 3 4 0
8 9 2 5 4 5 9 2 7 1 6 3 2 4 7 5 1 2 4 6 7 9 2 7
8 7 2 3 8 5 3 4 4 6 0 7 0 6 6 3 4 2 8 5 1 4 8 9
1 5 2 8 6 6 4 6 3 6 9 0 5 6 2 2 9 2 6 0 2 6 8 0
4 4 8 9 3 9 2 5 5 4 5 9 6 7 8 0 9 8 2 7 8 7 2 7
6 4 2 4 8 9 8 6 3 6 7 2 5 2 0 2 7 3 6 7 0 7 2 5
5 1 6 7 5 6 8 8 7 6 0 0 7 5 7 8 9 8 2 4 6 8 2 4
7 4 1 5 7 8 4 0 8 1 4 8 0 8 9 6 8 6 0 8 3 4 2 6
4 3 4 4 2 3 3 9 8 3 0 9 6 5 0 7 2 8 0 6 5 7 2 4
2 4 4 4 1 4 9 3 6 9 2 8 9 4 4 5 1 5 1 9 6 7 5 7
0 9 5 6 3 5 2 1 0 4 5 8 7 4 0 6 8 4 0 7 6 5 0 5
2 8 4 6 8 7 5 3 4 1 9 4 5 1 5 1 8 8 2 1 0 6 8 7
3 6 7 3 1 9 8 5 1 8 6 0 0 5 5 3 2 4 9 6 6 0 1 2
4 4 5 2 6 1 6 9 6 6 1 7 7 9 3 5 3 8 3 3 7 4 9 3
4 8 4 9 1 1 6 2 7 7 7 3 0 1 7 1 1 5 0 0 8 4 4 2
2 1 6 7 1 2 9 6 3 5 0 6 0 7 7 9 6 3 4 7 7 4 7 9
4 3 0 4 0 5 5 9 0 5 6 4 9 1 3 2 0 3 3 6 4 0 6 5
8 1 4 0 0 7 1 5 9 9 9 8 7 5 0 9 2 9 6 4 8 4 3 8
6 6 5 5 9 5 3 9 6 1 4 0 5 6 0 1 1 8 8 5 0 7 9 3
1 1 0 3 0 9 1 5 1 8 6 1 9 1 6 4 3 0 0 9 8 3 7 5
8 9 6 4 7 6 5 7 1 7 4 0 4 5 6 1 7 5 5 2 8 4 5 7
8 9 7 0 0 2 2 4 4 6 4 8 4 0 9 3 3 7 8 7 7 2 3 6
0 4 4 4 4 0 5 3 3 0 7 4 6 1 3 7 6 8 3 9 7 6 5 6
229
'Table A-2 Binomial probabilities [P(r) for values of r]: each value in the table
represents the probability of r occurrences in n trials with single
occurrence probability equal to p.
p
n r .05 .10 .15 .20 .25 .30 .35 .40 .45 .50
1 0 .950 .900 .850 .800 .750 .700 .650 .600 .550 .500
1 1 .050 .100 .150 .200 .250 .300 .350 .400 .450 .500
2 0 .903 .810 .723 .640 .563 .490 .423 .360 .303 .250
2 1 .095 .180 .255 .320 .375 .420 .455 .480 .495 .500
2 2 .003 .010 .023 .040 .063 .090 .123 .160 .203 .250
3 0 .857 .729 .614 .512 .422 .343 .275 .216 .166 .125
3 1 .135 .243 .325 .384 .422 .441 .444 .432 .408 .375
3 2 .007 .027 .057 .096 .141 .189 .239 .288 .334 .375
3 3 .000 .001 .003 .008 .016 .027 .043 .064 .091 .125
4 0 .815 .656 .522 .410 .316 .240 .179 .130 .092 .063
4 1 .171 .292 .368 .410 .422 .412 .384 .346 .299 .250
4 2 .014 .049 .098 .154 .211 .265 .311 .346 .368 .375
4 3 .000 .004 .011 .026 .047 .076 .111 .154 .200 .250
4 4 .000 .000 .001 .002 .004 .008 .015 .026 .041 .063
5 0 .774 .590 .444 .328 .237 .168 .116 .078 .050 .031
5 1 .204 .328 .392 .410 .396 .360 .312 .259 .206 .156
5 2 .021 .073 .138 .205 .264 .309 .336 .346 .337 .313
5 3 .001 .008 .024 .051 .088 .132 .181 .230 .276 .313
5 4 .000 .000 .002 .006 .015 .028 .049 .077 .113 .156
5 5 .000 .000 .000 .000 .001 .002 .005 .010 .018 .031
6 0 .735 .531 .377 .262 .178 .118 .075 .047 .028 .016
6 1 .232 .354 .399 .393 .356 .303 .244 .187 .136 .094
6 2 .031 .098 .176 .246 .297 .324 .328 .311 .278 .234
6 3 .002 .015 .041 .082 .132 .185 .235 .276 .303 .313
6 4 .000 .001 .005 .015 .033 .060 .095 .138 .186 .234
6 5 .000 .000 .000 .002 .004 .010 .020 .037 .061 .094
6 6 .000 .000 .000 .000 .000 .001 .002 .004 .008 .016
7 0 .698 .478 .321 .210 .133 .082 .049 .028 .015 .008
7 1 .257 .372 .396 .367 .311 .247 .185 .131 .087 .055
7 2 .041 .124 .210 .275 .311 .318 .298 .261 .214 .164
7 3 .004 .023 .062 .115 .173 .227 .268 .290 .292 .273
230
Table A-2 (continued)
p
n r .05 .10 .15 .20 .25 .30 .35 .40 .45 .50
7 4 .000 .003 .011 .029 .058 .097 .144 .194 .239 .273
7 5 .000 .000 .001 .004 .012 .025 .047 .077 .117 .164
7 6 .000 .000 .000 .000 .001 .004 .008 .017 .032 .055
7 7 .000 .000 .000 .000 .000 .000 .001 .002 .004 .008
8 0 .663 .430 .272 .168 .100 .058 .032 .017 .008 .004
8 1 .279 .383 .385 .336 .267 .198 .137 .090 .055 .031
8 2 .051 .149 .238 .294 .311 .296 .259 .209 .157 .109
8 3 .005 .033 .084 .147 .208 .254 .279 .279 .257 .219
8 4 .000 .005 .018 .046 .087 .136 .188 .232 .263 .273
8 5 .000 .000 .003 .009 .023 .047 .081 .124 .172 .219
8 6 .000 .000 .000 .001 .004 .010 .022 .041 .070 .109
8 7 .000 .000 .000 .000 .000 .001 .003 .008 .016 .031
8 8 .000 .000 .000 .000 .000 .000 .000 .001 .002 .004
9 0 .630 .387 .232 .134 .075 .040 .021 .010 .005 .002
9 1 .299 .387 .368 .302 .225 .156 .100 .060 .034 .018
9 2 .063 .172 .260 .302 .300 .267 .216 .161 .111 .070
9 3 .008 .045 .107 .176 .234 .267 .272 .251 .212 .164
9 4 .001 .007 .028 .066 .117 .172 .219 .251 .260 .246
9 5 .000 .001 .005 .017 .039 .074 .118 .167 .213 .246
9 6 .000 .000 .001 .003 .009 .021 .042 .074 .116 .164
9 7 .000 .000 .000 .000 .001 .004 .010 .021 .041 .070
9 8 .000 .000 .000 .000 .000 .000 .001 .004 .008 .018
9 9 .000 .000 .000 .000 .000 .000 .000 .000 .001 .002
10 0 .599 .349 .197 .107 .056 .028 .013 .006 .003 .001
10 1 .315 .387 .347 .268 .188 .121 .072 .040 .021 .010
10 2 .075 .194 .276 .302 .282 .233 .176 .121 .076 .044
10 3 .010 .057 .130 .201 .250 .267 .252 .215 .166 .117
10 4 .001 .011 .040 .088 .146 .200 .238 .251 .238 .205
10 5 .000 .001 .008 .026 .058 .103 .154 .201 .234 .246
10 6 .000 .000 .001 .006 .016 .037 .069 .111 .160 .205
10 7 .000 .000 .000 .001 .003 .009 .021 .042 .075 .117
10 8 .000 .000 .000 .000 .000 .001 .004 .011 .023 .044
10 9 .000 .000 .000 .000 .000 .000 .001 .002 .004 .010
10 10 .000 .000 .000 .000 .000 .000 .000 .000 .000 .001
231
Table A-2 (continued)
p n r .05 .10 .15 .20 .25 .30 .35 .40 .45 .50
11 0 .569 .314 .167 .086 .042 .020 .009 .004 .001 .000
11 1 .329 .384 .325 .236 .155 .093 .052 .027 .013 .005
11 2 .087 .213 .287 .295 .258 .200 .140 .089 .051 .027
11 3 .014 .071 .152 .221 .258 .257 .225 .177 .126 .081
11 4 .001 .016 .054 .111 .172 .220 .243 .236 .206 .161
11 5 .000 .002 .013 .039 .080 .132 .183 .221 .236 .226
11 6 .000 .000 .002 .010 .027 .057 .099 .147 .193 .226
11 7 .000 .000 .000 .002 .006 .017 .038 .070 .113 .161
11 8 .000 .000 .000 .000 .001 .004 .010 .023 .046 .081
11 9 .000 .000 .000 .000 .000 .001 .002 .005 .013 .027
11 10 .000 .000 .000 .000 .000 .000 .000 .001 .002 .005
11 11 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
12 0 .540 .282 .142 .069 .032 .014 .006 .002 .001 .000
12 1 .341 .377 .301 .206 .127 .071 .037 .017 .008 .003
12 2 .099 .230 .292 .283 .232 .168 .109 .064 .034 .016
12 3 .017 .085 .172 .236 .258 .240 .195 .142 .092 .054
12 4 .002 .021 .068 .133 .194 .231 .237 .213 .170 .121
12 5 .000 .004 .019 .053 .103 .158 .204 .227 .222 .193
12 6 .000 .000 .004 .016 .040 .079 .128 .177 .212 .226
12 7 .000 .000 .001 .003 .011 .029 .059 .101 .149 .193
12 8 .000 .000 .000 .001 .002 .008 .020 .042 .076 .121
12 9 .000 .000 .000 .000 .000 .001 .005 .012 .028 .054
12 10 .000 .000 .000 .000 .000 .000 .001 .002 .007 .016
12 11 .000 .000 .000 .000 .000 .000 .000 .000 .001 .003
12 12 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
13 0 .513 .254 .121 .055 .024 .010 .004 .001 .000 .000
13 1 .351 .367 .277 .179 .103 .054 .026 .011 .004 .002
13 2 .111 .245 .294 .268 .206 .139 .084 .045 .022 .010
13 3 .021 .100 .190 .246 .252 .218 .165 .111 .066 .035
13 4 .003 .028 .084 .154 .210 .234 .222 .184 .135 .087
13 5 .000 .006 .027 .069 .126 .180 .215 .221 .199 .157
13 6 .000 .001 .006 .023 .056 .103 .155 .197 .217 .209
13 7 .000 .000 .001 .006 .019 .044 .083 .131 .177 .209
13 8 .000 .000 .000 .001 .005 .014 .034 .066 .109 .157
13 9 .000 .000 .000 .000 .001 .003 .010 .024 .050 .087
13 10 .000 .000 .000 .000 .000 .001 .002 .006 .016 .035
232
Table A-2 (continued)
p n r .05 .10 .15 .20 .25 .30 .35 .40 .45 .50
13 11 .000 .000 .000 .000 .000 .000 .000 .001 .004 .010
13 12 .000 .000 .000 .000 .000 .000 .000 .000 .000 .002
13 13 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
14 0 .488 .229 .103 .044 .018 .007 .002 .001 .000 .000
14 1 .359 .356 .254 .154 .083 .041 .018 .007 .003 .001
14 2 .123 .257 .291 .250 .180 .113 .063 .032 .014 .006
14 3 .026 .114 .206 .250 .240 .194 .137 .085 .046 .022
14 4 .004 .035 .100 .172 .220 .229 .202 .155 .104 .061
14 5 .000 .008 .035 .086 .147 .196 .218 .207 .170 .122
14 6 .000 .001 .009 .032 .073 .126 .176 .207 .209 .183
14 7 .000 .000 .002 .009 .028 .062 .108 .157 .195 .209
14 8 .000 .000 .000 .002 .008 .023 .051 .092 .140 .183
14 9 .000 .000 .000 .000 .002 .007 .018 .041 .076 .122
14 10 .000 .000 .000 .000 .000 .001 .005 .014 .031 .061
14 11 .000 .000 .000 .000 .000 .000 .001 .003 .009 .022
14 12 .000 .000 .000 .000 .000 .000 .000 .001 .002 .006
14 13 .000 .000 .000 .000 .000 .000 .000 .000 .000 .001
14 14 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
15 0 .463 .206 .087 .035 .013 .005 .002 .000 .000 .000
15 1 .366 .343 .231 .132 .067 .031 .013 .005 .002 .000
15 2 .135 .267 .286 .231 .156 .092 .048 .022 .009 .003
15 3 .031 .129 .218 .250 .225 .170 .111 .063 .032 .014
15 4 .005 .043 .116 .188 .225 .219 .179 .127 .078 .042
15 5 .001 .010 .045 .103 .165 .206 .212 .186 .140 .092
15 6 .000 .002 .013 .043 .092 .147 .191 .207 .191 .153
15 7 .000 .000 .003 .014 .039 .081 .132 .177 .201 .196
15 8 .000 .000 .001 .003 .013 .035 .071 .118 .165 .196
15 9 .000 .000 .000 .001 .003 .012 .030 .061 .105 .153
15 10 .000 .000 .000 .000 .001 .003 .010 .024 .051 .092
15 11 .000 .000 .000 .000 .000 .001 .002 .007 .019 .042
15 12 .000 .000 .000 .000 .000 .000 .000 .002 .005 .014
15 13 .000 .000 .000 .000 .000 .000 .000 .000 .001 .003
15 14 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
15 15 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
233
'Table A-3 Poisson Probabilities. Each number in this table represents the
probability of obtaining at least [P(r x) for x = 1, 2, . . . ] for selected
mean values of . This probability is indicated as the shaded area of the
histogram.
mu 1 2 3 4 5 6 7 8 9 10 11 12
0.1 .095 .005
0.2 .181 .018 .001
0.3 .259 .037 .004
0.4 .330 .062 .008
0.5 .393 .090 .014 .002
0.6 .451 .122 .023 .003
0.7 .503 .156 .034 .006
0.8 .551 .191 .047 .009 .001
0.9 .593 .228 .063 .013 .002
1.0 .632 .264 .080 .019 .004
1.1 .667 .301 .100 .026 .005
1.2 .699 .337 .121 .034 .008 .002
1.3 .727 .373 .143 .043 .011 .002
1.4 .753 .408 .167 .054 .014 .003
1.5 .777 .442 .191 .066 .019 .004
1.6 .798 .475 .217 .079 .024 .006 .001
1.7 .817 .507 .243 .093 .030 .008 .002
1.8 .835 .537 .269 .109 .036 .010 .003
1.9 .850 .566 .296 .125 .044 .013 .003
2.0 .865 .594 .323 .143 .053 .017 .005 .001
2.5 .918 .713 .456 .242 .109 .042 .014 .004 .001
3.0 .950 .801 .577 .353 .185 .084 .034 .012 .004 .001
3.5 .970 .864 .679 .463 .275 .142 .065 .027 .010 .003 .001
4.0 .982 .908 .762 .567 .371 .215 .111 .051 .021 .008 .003
4.5 .989 .939 .826 .658 .468 .297 .169 .087 .040 .017 .007 .002
5.0 .993 .960 .875 .735 .560 .384 .238 .133 .068 .032 .014 .005
5.5 .996 .973 .912 .798 .642 .471 .314 .191 .106 .054 .025 .011
6.0 .998 .983 .938 .849 .715 .554 .394 .256 .153 .084 .043 .020
234
'Table A-4a. Normal Distribution. This table contains the cumulative probabilities
exceeding the z-values under the right-hand side of a normal
distribution curve. The z-values are shown in the first column up to
one digit after the decimal point, and in the top row for the second
digit after the decimal point.
x .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.0 .500 .496 .492 .488 .484 .480 .476 .472 .468 .464
.1 .460 .456 .452 .448 .444 .440 .436 .433 .429 .425
.2 .421 .417 .413 .409 .405 .401 .397 .394 .390 .386
.3 .382 .378 .374 .371 .367 .363 .359 .356 .352 .348
.4 .345 .341 .337 .334 .330 .326 .323 .319 .316 .312
.5 .309 .305 .302 .298 .295 .291 .288 .284 .281 .278
.6 .274 .271 .268 .264 .261 .258 .255 .251 .248 .245
.7 .242 .239 .236 .233 .230 .227 .224 .221 .218 .215
.8 .212 .209 .206 .203 .200 .198 .195 .192 .189 .187
.9 .184 .181 .179 .176 .174 .171 .169 .166 .164 .161
1.0 .159 .156 .154 .152 .149 .147 .145 .142 .140 .138
1.1 .136 .133 .131 .129 .127 .125 .123 .121 .119 .117
1.2 .115 .113 .111 .109 .107 .106 .104 .102 .100 .099
1.3 .097 .095 .093 .092 .090 .089 .087 .085 .084 .082
1.4 .081 .079 .078 .076 .075 .074 .072 .071 .069 .068
1.5 .067 .066 .064 .063 .062 .061 .059 .058 .057 .056
1.6 .055 .054 .053 .052 .051 .049 .048 .047 .046 .046
1.7 .045 .044 .043 .042 .041 .040 .039 .038 .038 .037
1.8 .036 .035 .034 .034 .033 .032 .031 .031 .030 .029
1.9 .029 .028 .027 .027 .026 .026 .025 .024 .024 .023
2.0 .023 .022 .022 .021 .021 .020 .020 .019 .019 .018
2.1 .018 .017 .017 .017 .016 .016 .015 .015 .015 .014
2.2 .014 .014 .013 .013 .013 .012 .012 .012 .011 .011
2.3 .011 .010 .010 .010 .010 .009 .009 .009 .009 .008
2.4 .008 .008 .008 .008 .007 .007 .007 .007 .007 .006
2.5 .006 .006 .006 .006 .006 .005 .005 .005 .005 .005
2.6 .005 .005 .004 .004 .004 .004 .004 .004 .004 .004
2.7 .003 .003 .003 .003 .003 .003 .003 .003 .003 .003
2.8 .003 .002 .002 .002 .002 .002 .002 .002 .002 .002
2.9 .002 .002 .002 .002 .002 .002 .002 .001 .001 .001
3.0 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001
3.1 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001
3.2 .001 .001 .001 .001 .001 .001 .001 .001 .001 .001
3.3 .000 .000 .000 .000 .000 .000 .000 .000 .000 .000
235
'Table A-4b. AREA (Z) UNDER THE STANDARD NORMAL CURVE
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09
.0 .500 .504 .508 .512 .516 .520 .524 .528 .532 .536
.1 .540 .544 .548 .552 .556 .560 .564 .567 .571 .575
.2 .579 .583 .587 .591 .595 .599 .603 .606 .610 .614
.3 .618 .622 .626 .629 .633 .637 .641 .644 .648 .652
.4 .655 .659 .663 .666 .670 .674 .677 .681 .684 .688
.5 .691 .695 .698 .702 .705 .709 .712 .716 .719 .722
.6 .726 .729 .732 .736 .739 .742 .745 .749 .752 .755
.7 .758 .761 .764 .767 .770 .773 .776 .779 .782 .785
.8 .788 .791 .794 .797 .800 .802 .805 .808 .811 .813
.9 .816 .819 .821 .824 .826 .829 .831 .834 .836 .839
1.0 .841 .844 .846 .848 .851 .853 .855 .858 .860 .862
1.1 .864 .867 .869 .871 .873 .875 .877 .879 .881 .883
1.2 .885 .887 .889 .891 .893 .894 .896 .898 .900 .901
1.3 .903 .905 .907 .908 .910 .911 .913 .915 .916 .918
1.4 .919 .921 .922 .924 .925 .926 .928 .929 .931 .932
1.5 .933 .934 .936 .937 .938 .939 .941 .942 .943 .944
1.6 .945 .946 .947 .948 .949 .951 .952 .953 .954 .954
1.7 .955 .956 .957 .958 .959 .960 .961 .962 .962 .963
1.8 .964 .965 .966 .966 .967 .968 .969 .969 .970 .971
1.9 .971 .972 .973 .973 .974 .974 .975 .976 .976 .977
2.0 .977 .978 .978 .979 .979 .980 .980 .981 .981 .982
2.1 .982 .983 .983 .983 .984 .984 .985 .985 .985 .986
2.2 .986 .986 .987 .987 .987 .988 .988 .988 .989 .989
2.3 .989 .990 .990 .990 .990 .991 .991 .991 .991 .992
2.4 .992 .992 .992 .992 .993 .993 .993 .993 .993 .994
2.5 .994 .994 .994 .994 .994 .995 .995 .995 .995 .995
2.6 .995 .995 .996 .996 .996 .996 .996 .996 .996 .996
2.7 .997 .997 .997 .997 .997 .997 .997 .997 .997 .997
2.8 .997 .998 .998 .998 .998 .998 .998 .998 .998 .998
2.9 .998 .998 .998 .998 .998 .998 .998 .999 .999 .999
3.0 .999 .999 .999 .999 .999 .999 .999 .999 .999 .999
3.1 .999 .999 .999 .999 .999 .999 .999 .999 .999 .999
3.2 .999 .999 .999 .999 .999 .999 .999 .999 .999 .999
3.3 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
236
'Table A-5 Values of Chi-squares. This table is used to find values of
chi-square corresponding to selected probabilities. Each
number in this table is chi-square value for an indicated
degree of freedom (left column). Each number (i.e., .99,
.95, ...) in the top row of the table represents a probability of
obtaining a value as large of larger than the corresponding
chi-square value in the table.
df Probability of obtaining a value as large or larger
0.99 0.95 0.90 0.50 0.10 0.05 0.01 0.001
1 0.000 0.004 0.016 0.455 2.706 3.841 6.635 10.828
2 0.020 0.103 0.211 1.386 4.605 5.991 9.210 13.816
3 0.115 0.352 0.584 2.366 6.251 7.815 11.345 16.266
4 0.297 0.711 1.064 3.357 7.779 9.488 13.277 18.467
5 0.554 1.145 1.610 4.351 9.236 11.070 15.086 20.515
6 0.872 1.635 2.204 5.348 10.645 12.592 16.812 22.458
7 1.239 2.167 2.833 6.346 12.017 14.067 18.475 24.322
8 1.646 2.733 3.490 7.344 13.362 15.507 20.090 26.124
9 2.088 3.325 4.168 8.343 14.684 16.919 21.666 27.877
10 2.558 3.940 4.865 9.342 15.987 18.307 23.209 29.588
11 3.053 4.575 5.578 10.341 17.275 19.675 24.725 31.264
12 3.571 5.226 6.304 11.340 18.549 21.026 26.217 32.909
13 4.107 5.892 7.042 12.340 19.812 22.362 27.688 34.528
14 4.660 6.571 7.790 13.339 21.064 23.685 29.141 36.123
15 5.229 7.261 8.547 14.339 22.307 24.996 30.578 37.697
16 5.812 7.962 9.312 15.338 23.542 26.296 32.000 39.252
17 6.408 8.672 10.085 16.338 24.769 27.587 33.409 40.790
18 7.015 9.390 10.865 17.338 25.989 28.869 34.805 42.312
19 7.633 10.117 11.651 18.338 27.204 30.144 36.191 43.820
20 8.260 10.851 12.443 19.337 28.412 31.410 37.566 45.315
21 8.897 11.591 13.240 20.337 29.615 32.671 38.932 46.797
22 9.542 12.338 14.041 21.337 30.813 33.924 40.289 48.268
23 10.196 13.091 14.848 22.337 32.007 35.172 41.638 49.728
24 10.856 13.848 15.659 23.337 33.196 36.415 42.980 51.179
25 11.524 14.611 16.473 24.337 34.382 37.652 44.314 52.620
26 12.198 15.379 17.292 25.336 35.563 38.885 45.642 54.052
27 12.879 16.151 18.114 26.336 36.741 40.113 46.963 55.476
28 13.565 16.928 18.939 27.336 37.916 41.337 48.278 56.892
29 14.256 17.708 19.768 28.336 39.087 42.557 49.588 58.301
30 14.953 18.493 20.599 29.336 40.256 43.773 50.892 59.703
237
'Table A-6 Values of the Student t-distribution. The values in this table
are Student t-values which correspond to the indicated
cumulative probabilities (on the right-hand side under a t-
distribution curve) and to the indicated degrees of freedom.
df Probability of obtaining a larger value of t
0.25 0.20 0.15 0.10 0.05 0.025 0.01 0.005 0.001
1 1.000 1.376 1.963 3.078 6.314 12.706 31.821 63.657 636.62
2 0.816 1.061 1.386 1.886 2.920 4.303 6.965 9.925 31.599
3 0.765 0.978 1.250 1.638 2.353 3.182 4.541 5.841 12.924
4 0.741 0.941 1.190 1.533 2.132 2.776 3.747 4.604 8.610
5 0.727 0.920 1.156 1.476 2.015 2.571 3.365 4.032 6.869
6 0.718 0.906 1.134 1.440 1.943 2.447 3.143 3.707 5.959
7 0.711 0.896 1.119 1.415 1.895 2.365 2.998 3.499 5.408
8 0.706 0.889 1.108 1.397 1.860 2.306 2.896 3.355 5.041
9 0.703 0.883 1.100 1.383 1.833 2.262 2.821 3.250 4.781
10 0.700 0.879 1.093 1.372 1.812 2.228 2.764 3.169 4.587
11 0.697 0.876 1.088 1.363 1.796 2.201 2.718 3.106 4.437
12 0.695 0.873 1.083 1.356 1.782 2.179 2.681 3.055 4.318
13 0.694 0.870 1.079 1.350 1.771 2.160 2.650 3.012 4.221
14 0.692 0.868 1.076 1.345 1.761 2.145 2.624 2.977 4.140
15 0.691 0.866 1.074 1.341 1.753 2.131 2.602 2.947 4.073
16 0.690 0.865 1.071 1.337 1.746 2.120 2.583 2.921 4.015
17 0.689 0.863 1.069 1.333 1.740 2.110 2.567 2.898 3.965
18 0.688 0.862 1.067 1.330 1.734 2.101 2.552 2.878 3.922
19 0.688 0.861 1.066 1.328 1.729 2.093 2.539 2.861 3.883
20 0.687 0.860 1.064 1.325 1.725 2.086 2.528 2.845 3.850
21 0.686 0.859 1.063 1.323 1.721 2.080 2.518 2.831 3.819
22 0.686 0.858 1.061 1.321 1.717 2.074 2.508 2.819 3.792
23 0.685 0.858 1.060 1.319 1.714 2.069 2.500 2.807 3.768
24 0.685 0.857 1.059 1.318 1.711 2.064 2.492 2.797 3.745
25 0.684 0.856 1.058 1.316 1.708 2.060 2.485 2.787 3.725
26 0.684 0.856 1.058 1.315 1.706 2.056 2.479 2.779 3.707
27 0.684 0.855 1.057 1.314 1.703 2.052 2.473 2.771 3.690
28 0.683 0.855 1.056 1.313 1.701 2.048 2.467 2.763 3.674
29 0.683 0.854 1.055 1.311 1.699 2.045 2.462 2.756 3.659
30 0.683 0.854 1.055 1.310 1.697 2.042 2.457 2.750 3.646
40 0.681 0.851 1.050 1.303 1.684 2.021 2.423 2.704 3.551
0.675 0.842 1.036 1.282 1.645 1.960 2.327 2.576 3.291
238
'Table A-7 Values of the F-Distribution - 10%, 5% and 1% level. This
table is used to find F-values at three probability levels (.10,
.05, .01) in the right tail of the F-distribution.
df for nume
degrees of freedom for denominator
1 1 1 2 2 2 3 3 3
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 39.86 161 4,052 8.53 18.51 98.50 5.54 10.13 34.12
2 49.50 199 4,999 9.00 19.00 99.00 5.46 9.55 30.82
3 53.59 216 5,403 9.16 19.16 99.17 5.39 9.28 29.46
4 55.83 225 5,625 9.24 19.25 99.25 5.34 9.12 28.71
5 57.24 230 5,764 9.29 19.30 99.30 5.31 9.01 28.24
6 58.20 234 5,859 9.33 19.33 99.33 5.28 8.94 27.91
7 58.91 237 5,928 9.35 19.35 99.36 5.27 8.89 27.67
8 59.44 239 5,981 9.37 19.37 99.37 5.25 8.85 27.49
9 59.86 241 6,022 9.38 19.38 99.39 5.24 8.81 27.35
10 60.19 242 6,056 9.39 19.40 99.40 5.23 8.79 27.23
12 60.71 244 6,106 9.41 19.41 99.42 5.22 8.74 27.05
14 61.07 245 6,143 9.42 19.42 99.43 5.20 8.71 26.92
16 61.35 246 6,170 9.43 19.43 99.44 5.20 8.69 26.83
18 61.57 247 6,192 9.44 19.44 99.44 5.19 8.67 26.75
.
df for nume
degrees of freedom for denominator
4 4 4 5 5 5 6 6 6
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 4.54 7.71 21.20 4.06 6.61 16.26 3.78 5.99 13.75
2 4.32 6.94 18.00 3.78 5.79 13.27 3.46 5.14 10.92
3 4.19 6.59 16.69 3.62 5.41 12.06 3.29 4.76 9.78
4 4.11 6.39 15.98 3.52 5.19 11.39 3.18 4.53 9.15
5 4.05 6.26 15.52 3.45 5.05 10.97 3.11 4.39 8.75
6 4.01 6.16 15.21 3.40 4.95 10.67 3.05 4.28 8.47
7 3.98 6.09 14.98 3.37 4.88 10.46 3.01 4.21 8.26
8 3.95 6.04 14.80 3.34 4.82 10.29 2.98 4.15 8.10
9 3.94 6.00 14.66 3.32 4.77 10.16 2.96 4.10 7.98
10 3.92 5.96 14.55 3.30 4.74 10.05 2.94 4.06 7.87
12 3.90 5.91 14.37 3.27 4.68 9.89 2.90 4.00 7.72
14 3.88 5.87 14.25 3.25 4.64 9.77 2.88 3.96 7.60
16 3.86 5.84 14.15 3.23 4.60 9.68 2.86 3.92 7.52
18 3.85 5.82 14.08 3.22 4.58 9.61 2.85 3.90 7.45
239
Table A-7 (continued)
df for nume
degrees of freedom for denominator
7 7 7 8 8 8 9 9 9
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 3.59 5.59 12.25 3.46 5.32 11.26 3.36 5.12 10.56
2 3.26 4.74 9.55 3.11 4.46 8.65 3.01 4.26 8.02
3 3.07 4.35 8.45 2.92 4.07 7.59 2.81 3.86 6.99
4 2.96 4.12 7.85 2.81 3.84 7.01 2.69 3.63 6.42
5 2.88 3.97 7.46 2.73 3.69 6.63 2.61 3.48 6.06
6 2.83 3.87 7.19 2.67 3.58 6.37 2.55 3.37 5.80
7 2.78 3.79 6.99 2.62 3.50 6.18 2.51 3.29 5.61
8 2.75 3.73 6.84 2.59 3.44 6.03 2.47 3.23 5.47
9 2.72 3.68 6.72 2.56 3.39 5.91 2.44 3.18 5.35
10 2.70 3.64 6.62 2.54 3.35 5.81 2.42 3.14 5.26
12 2.67 3.57 6.47 2.50 3.28 5.67 2.38 3.07 5.11
14 2.64 3.53 6.36 2.48 3.24 5.56 2.35 3.03 5.01
16 2.62 3.49 6.28 2.45 3.20 5.48 2.33 2.99 4.92
18 2.61 3.47 6.21 2.44 3.17 5.41 2.31 2.96 4.86
.
df for nume
degrees of freedom for denominator
10 10 10 11 11 11 12 12 12
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 3.29 4.96 10.04 3.23 4.84 9.65 3.18 4.75 9.33
2 2.92 4.10 7.56 2.86 3.98 7.21 2.81 3.89 6.93
3 2.73 3.71 6.55 2.66 3.59 6.22 2.61 3.49 5.95
4 2.61 3.48 5.99 2.54 3.36 5.67 2.48 3.26 5.41
5 2.52 3.33 5.64 2.45 3.20 5.32 2.39 3.11 5.06
6 2.46 3.22 5.39 2.39 3.09 5.07 2.33 3.00 4.82
7 2.41 3.14 5.20 2.34 3.01 4.89 2.28 2.91 4.64
8 2.38 3.07 5.06 2.30 2.95 4.74 2.24 2.85 4.50
9 2.35 3.02 4.94 2.27 2.90 4.63 2.21 2.80 4.39
10 2.32 2.98 4.85 2.25 2.85 4.54 2.19 2.75 4.30
12 2.28 2.91 4.71 2.21 2.79 4.40 2.15 2.69 4.16
14 2.26 2.86 4.60 2.18 2.74 4.29 2.12 2.64 4.05
16 2.23 2.83 4.52 2.16 2.70 4.21 2.09 2.60 3.97
18 2.22 2.80 4.46 2.14 2.67 4.15 2.08 2.57 3.91
240
Table A-7 (continued)
df for nume
degrees of freedom for denominator
13 13 13 14 14 14 15 15 15
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 3.14 4.67 9.07 3.10 4.60 8.86 3.07 4.54 8.68
2 2.76 3.81 6.70 2.73 3.74 6.51 2.70 3.68 6.36
3 2.56 3.41 5.74 2.52 3.34 5.56 2.49 3.29 5.42
4 2.43 3.18 5.21 2.39 3.11 5.04 2.36 3.06 4.89
5 2.35 3.03 4.86 2.31 2.96 4.69 2.27 2.90 4.56
6 2.28 2.92 4.62 2.24 2.85 4.46 2.21 2.79 4.32
7 2.23 2.83 4.44 2.19 2.76 4.28 2.16 2.71 4.14
8 2.20 2.77 4.30 2.15 2.70 4.14 2.12 2.64 4.00
9 2.16 2.71 4.19 2.12 2.65 4.03 2.09 2.59 3.89
10 2.14 2.67 4.10 2.10 2.60 3.94 2.06 2.54 3.80
12 2.10 2.60 3.96 2.05 2.53 3.80 2.02 2.48 3.67
14 2.07 2.55 3.86 2.02 2.48 3.70 1.99 2.42 3.56
16 2.04 2.51 3.78 2.00 2.44 3.62 1.96 2.38 3.49
18 2.02 2.48 3.72 1.98 2.41 3.56 1.94 2.35 3.42
.
df for nume
degrees of freedom for denominator
16 16 16 17 17 17 18 18 18
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 3.05 4.49 8.53 3.03 4.45 8.40 3.01 4.41 8.29
2 2.67 3.63 6.23 2.64 3.59 6.11 2.62 3.55 6.01
3 2.46 3.24 5.29 2.44 3.20 5.18 2.42 3.16 5.09
4 2.33 3.01 4.77 2.31 2.96 4.67 2.29 2.93 4.58
5 2.24 2.85 4.44 2.22 2.81 4.34 2.20 2.77 4.25
6 2.18 2.74 4.20 2.15 2.70 4.10 2.13 2.66 4.01
7 2.13 2.66 4.03 2.10 2.61 3.93 2.08 2.58 3.84
8 2.09 2.59 3.89 2.06 2.55 3.79 2.04 2.51 3.71
9 2.06 2.54 3.78 2.03 2.49 3.68 2.00 2.46 3.60
10 2.03 2.49 3.69 2.00 2.45 3.59 1.98 2.41 3.51
12 1.99 2.42 3.55 1.96 2.38 3.46 1.93 2.34 3.37
14 1.95 2.37 3.45 1.93 2.33 3.35 1.90 2.29 3.27
16 1.93 2.33 3.37 1.90 2.29 3.27 1.87 2.25 3.19
18 1.91 2.30 3.31 1.88 2.26 3.21 1.85 2.22 3.13
241
Table A-7 (continued)
df for nume
degrees of freedom for denominator
20 20 20 22 22 22 24 24 24
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 2.97 4.35 8.10 2.95 4.30 7.95 2.93 4.26 7.82
2 2.59 3.49 5.85 2.56 3.44 5.72 2.54 3.40 5.61
3 2.38 3.10 4.94 2.35 3.05 4.82 2.33 3.01 4.72
4 2.25 2.87 4.43 2.22 2.82 4.31 2.19 2.78 4.22
5 2.16 2.71 4.10 2.13 2.66 3.99 2.10 2.62 3.90
6 2.09 2.60 3.87 2.06 2.55 3.76 2.04 2.51 3.67
7 2.04 2.51 3.70 2.01 2.46 3.59 1.98 2.42 3.50
8 2.00 2.45 3.56 1.97 2.40 3.45 1.94 2.36 3.36
9 1.96 2.39 3.46 1.93 2.34 3.35 1.91 2.30 3.26
10 1.94 2.35 3.37 1.90 2.30 3.26 1.88 2.25 3.17
12 1.89 2.28 3.23 1.86 2.23 3.12 1.83 2.18 3.03
14 1.86 2.22 3.13 1.83 2.17 3.02 1.80 2.13 2.93
16 1.83 2.18 3.05 1.80 2.13 2.94 1.77 2.09 2.85
18 1.81 2.15 2.99 1.78 2.10 2.88 1.75 2.05 2.79
df for nume
degrees of freedom for denominator
26 26 26 28 28 28 30 30 30
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 2.91 4.23 7.72 2.89 4.20 7.64 2.88 4.17 7.56
2 2.52 3.37 5.53 2.50 3.34 5.45 2.49 3.32 5.39
3 2.31 2.98 4.64 2.29 2.95 4.57 2.28 2.92 4.51
4 2.17 2.74 4.14 2.16 2.71 4.07 2.14 2.69 4.02
5 2.08 2.59 3.82 2.06 2.56 3.75 2.05 2.53 3.70
6 2.01 2.47 3.59 2.00 2.45 3.53 1.98 2.42 3.47
7 1.96 2.39 3.42 1.94 2.36 3.36 1.93 2.33 3.30
8 1.92 2.32 3.29 1.90 2.29 3.23 1.88 2.27 3.17
9 1.88 2.27 3.18 1.87 2.24 3.12 1.85 2.21 3.07
10 1.86 2.22 3.09 1.84 2.19 3.03 1.82 2.16 2.98
12 1.81 2.15 2.96 1.79 2.12 2.90 1.77 2.09 2.84
14 1.77 2.09 2.86 1.75 2.06 2.79 1.74 2.04 2.74
16 1.75 2.05 2.78 1.73 2.02 2.72 1.71 1.99 2.66
18 1.72 2.02 2.72 1.70 1.99 2.65 1.69 1.96 2.60
242
Table A-7 (continued)
df for nume
degrees of freedom for denominator
35 35 35 40 40 40 45 45 45
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 2.85 4.12 7.42 2.84 4.08 7.31 2.82 4.06 7.23
2 2.46 3.27 5.27 2.44 3.23 5.18 2.42 3.20 5.11
3 2.25 2.87 4.40 2.23 2.84 4.31 2.21 2.81 4.25
4 2.11 2.64 3.91 2.09 2.61 3.83 2.07 2.58 3.77
5 2.02 2.49 3.59 2.00 2.45 3.51 1.98 2.42 3.45
6 1.95 2.37 3.37 1.93 2.34 3.29 1.91 2.31 3.23
7 1.90 2.29 3.20 1.87 2.25 3.12 1.85 2.22 3.07
8 1.85 2.22 3.07 1.83 2.18 2.99 1.81 2.15 2.94
9 1.82 2.16 2.96 1.79 2.12 2.89 1.77 2.10 2.83
10 1.79 2.11 2.88 1.76 2.08 2.80 1.74 2.05 2.74
12 1.74 2.04 2.74 1.71 2.00 2.66 1.70 1.97 2.61
14 1.70 1.99 2.64 1.68 1.95 2.56 1.66 1.92 2.51
16 1.67 1.94 2.56 1.65 1.90 2.48 1.63 1.87 2.43
18 1.65 1.91 2.50 1.62 1.87 2.42 1.60 1.84 2.36
df for nume
degrees of freedom for denominator
50 50 50 60 60 60
10% 5% 1% 10% 5% 1% 10% 5% 1%
1 2.81 4.03 7.17 2.79 4.00 7.08 2.71 3.84 6.64
2 2.41 3.18 5.06 2.39 3.15 4.98 2.30 3.00 4.61
3 2.20 2.79 4.20 2.18 2.76 4.13 2.08 2.61 3.78
4 2.06 2.56 3.72 2.04 2.53 3.65 1.95 2.37 3.32
5 1.97 2.40 3.41 1.95 2.37 3.34 1.85 2.21 3.02
6 1.90 2.29 3.19 1.87 2.25 3.12 1.77 2.10 2.80
7 1.84 2.20 3.02 1.82 2.17 2.95 1.72 2.01 2.64
8 1.80 2.13 2.89 1.77 2.10 2.82 1.67 1.94 2.51
9 1.76 2.07 2.78 1.74 2.04 2.72 1.63 1.88 2.41
10 1.73 2.03 2.70 1.71 1.99 2.63 1.60 1.83 2.32
12 1.68 1.95 2.56 1.66 1.92 2.50 1.55 1.75 2.19
14 1.64 1.89 2.46 1.62 1.86 2.39 1.51 1.69 2.08
16 1.61 1.85 2.38 1.59 1.82 2.31 1.47 1.64 2.00
18 1.59 1.81 2.32 1.56 1.78 2.25 1.44 1.60 1.94