Introduction to Statistical Analysis_Appendix_B

8/2/2019 Introduction to Statistical Analysis_Appendix_B

1/23328 A p p e n d i x B T a b l e s

Appendix B

TablesTable 1 Random Numbers10 09 73 25 33 76 52 01 35 86 34 67 35 48 76 80 95 90 91 17 39 29 27 49 4537 54 20 48 05 64 89 47 42 96 24 80 52 40 37 20 63 61 04 02 00 82 29 16 65

08 42 26 89 53 19 64 50 93 03 23 20 90 25 60 15 95 33 43 64 35 08 03 36 0699 01 90 25 29 09 37 67 07 15 38 31 13 11 65 88 67 67 43 97 04 43 62 76 59

12 80 79 99 70 80 15 73 61 47 64 03 23 66 53 98 95 11 68 77 12 17 17 68 33

66 06 57 47 17 34 07 27 68 50 36 69 73 61 70 65 81 33 98 85 11 19 92 91 70

31 06 01 08 05 45 57 18 24 06 35 30 34 26 14 86 79 90 74 39 23 40 30 97 32

85 26 97 76 02 02 05 16 56 92 68 66 57 48 18 73 05 38 52 47 18 62 38 85 79

63 57 33 21 35 05 32 54 70 48 90 55 35 75 48 28 46 82 87 09 83 49 12 56 24

73 79 64 57 53 03 52 96 47 78 35 80 83 42 82 60 93 52 03 44 35 27 38 84 3598 52 01 77 67 14 90 56 86 07 22 10 94 05 58 60 97 09 34 33 50 50 07 39 98

11 80 50 54 31 39 80 82 77 32 50 72 56 82 48 29 40 52 42 01 52 77 56 78 51

83 45 29 96 34 06 28 89 80 83 13 74 67 00 78 18 47 54 06 10 68 71 17 78 17

88 68 54 02 00 86 50 75 84 01 36 76 66 79 51 90 36 47 64 93 29 60 91 10 62

99 59 46 73 48 87 51 76 49 69 91 82 60 89 28 93 78 56 13 68 23 47 83 41 1365 48 11 76 74 17 46 85 09 50 58 04 77 69 74 73 03 95 71 86 40 21 81 65 44

80 12 43 56 35 17 72 70 80 15 45 31 82 23 74 21 11 57 82 53 14 38 55 37 63

74 35 09 98 17 77 40 27 72 14 43 23 60 02 10 45 52 16 42 37 96 28 60 26 55

69 91 62 68 03 66 25 22 91 48 36 93 68 72 03 76 62 11 39 90 94 40 05 64 18

09 89 32 05 05 14 22 56 85 14 46 42 75 67 88 96 29 77 88 22 54 38 21 45 98

91 49 91 45 23 68 47 92 76 86 46 16 28 35 54 94 75 08 99 23 37 08 92 00 48

80 33 69 45 98 26 94 03 68 58 70 29 73 41 35 54 14 03 33 40 42 05 08 23 4144 10 48 19 49 85 15 74 79 54 32 97 92 65 75 57 60 04 08 81 22 22 20 64 13

12 55 07 37 42 11 10 00 20 40 12 86 07 46 97 96 64 48 94 39 28 70 72 58 15

63 60 64 93 29 16 50 53 44 84 40 21 95 25 63 43 65 17 70 82 07 20 73 17 90

61 19 69 04 46 26 45 74 77 74 51 92 43 37 29 65 39 45 95 93 42 58 26 05 27

15 47 44 52 66 95 27 07 99 53 59 36 78 38 48 82 39 61 01 18 33 21 15 94 66

94 55 72 85 73 67 89 75 43 87 54 62 24 44 31 91 19 04 25 92 92 92 74 59 7342 48 11 62 13 97 34 40 87 21 16 86 84 87 67 03 07 11 20 59 25 70 14 66 70

23 52 37 83 17 73 20 88 98 37 68 93 59 14 16 26 25 22 96 63 05 52 28 25 62

04 49 35 24 94 75 24 63 38 24 45 86 25 10 25 61 96 27 93 35 65 33 71 24 72

00 54 99 76 54 64 05 18 81 59 96 11 96 38 96 54 69 28 23 91 23 28 72 95 29

35 96 31 53 07 26 89 80 93 54 33 35 13 54 62 77 97 45 00 24 90 10 33 93 33

59 80 80 83 91 45 42 72 68 42 83 60 94 97 00 13 02 12 48 92 78 56 52 01 0646 05 88 52 36 01 39 09 22 86 77 28 14 40 77 93 91 08 36 47 70 61 74 29 41

32 17 90 05 97 87 37 92 52 41 05 56 70 70 07 86 74 31 71 57 85 39 41 18 38

69 23 46 14 06 20 11 74 52 04 15 95 66 00 00 18 74 39 24 23 97 11 89 63 3819 56 54 14 30 01 75 87 53 79 40 41 92 15 85 66 67 43 68 06 84 96 28 52 07

45 15 51 49 38 19 47 60 72 46 43 66 79 45 43 59 04 79 00 33 20 82 66 95 41

94 86 43 19 94 36 16 81 08 51 34 88 88 15 53 01 54 03 54 56 05 01 45 11 7698 08 62 48 26 45 24 02 84 04 44 99 90 88 96 39 09 47 34 07 35 44 13 18 80

33 18 51 62 32 41 94 15 09 49 89 43 54 85 81 88 69 54 19 94 37 54 87 30 43

80 95 10 04 06 96 38 27 07 74 20 15 12 33 87 25 01 62 52 98 94 62 46 11 71

79 75 24 91 40 71 96 12 82 96 69 86 10 25 91 74 85 22 05 39 00 38 75 95 79

18 63 33 25 37 98 14 50 65 71 31 01 02 46 74 05 45 56 14 27 77 93 89 19 36

For specifc details about using this table, see page 14 or go online at 4ltrpress.cengage.com/stat.


2/23 32A p p e n d i x B T a b l e s

Table 1 Random Numbers (continued)

54 17 84 56 11 80 99 33 71 43 05 33 51 29 69 56 12 71 92 55 36 04 09 03 24

11 66 44 98 83 52 07 98 48 27 59 38 17 15 39 09 97 33 34 40 88 46 12 33 56

48 32 47 79 28 31 24 96 47 10 02 29 53 68 70 32 30 75 75 46 15 02 00 99 94

69 07 49 41 38 87 63 79 19 76 35 58 40 44 01 10 51 82 16 15 01 84 87 69 3809 18 82 00 97 32 82 53 95 27 04 22 08 63 04 83 38 98 73 74 64 27 85 80 44

90 04 58 54 97 51 98 15 06 54 94 93 88 19 97 91 87 07 61 50 68 47 66 46 59

73 18 95 02 07 47 67 72 62 69 62 29 06 44 64 27 12 46 70 18 41 36 18 27 6075 76 87 64 90 20 97 18 17 49 90 42 91 22 72 95 37 50 58 71 93 82 34 31 78

54 01 64 40 56 66 28 13 10 03 00 68 22 73 98 20 71 45 32 95 07 70 61 78 13

08 35 86 99 10 78 54 24 27 85 13 66 15 88 73 04 61 89 75 53 31 22 30 84 2028 30 60 32 64 81 33 31 05 91 40 51 00 78 93 32 60 46 04 75 94 11 90 18 40

53 84 08 62 33 81 59 41 36 28 51 21 59 02 90 28 46 66 87 95 77 76 22 07 91

91 75 75 37 41 61 61 36 22 69 50 26 39 02 12 55 78 17 65 14 83 48 34 70 55

89 41 59 26 94 00 39 75 83 91 12 60 71 76 46 48 94 97 23 06 94 54 13 74 08

77 51 30 38 20 86 83 42 99 01 68 41 48 27 74 51 90 81 39 80 72 89 35 55 07

19 50 23 71 74 69 97 92 02 88 55 21 02 97 73 74 28 77 52 51 65 34 46 74 1521 81 85 93 13 93 27 88 17 57 05 68 67 31 56 07 08 28 50 46 31 85 33 84 52

51 47 46 64 99 68 10 72 36 21 94 04 99 13 45 42 83 60 91 91 08 00 74 54 49

99 55 96 83 31 62 53 52 41 70 69 77 71 28 30 74 81 97 81 42 43 86 07 28 34

33 71 34 80 07 93 58 47 28 69 51 92 66 47 21 58 30 32 98 22 93 17 49 39 7285 27 48 68 93 11 30 32 92 70 28 83 43 41 37 73 51 59 04 00 71 14 84 36 43

84 13 38 96 40 44 03 55 21 66 73 85 27 00 91 61 22 26 05 61 62 32 71 84 23

56 73 21 62 34 17 39 59 61 31 10 12 39 16 22 85 49 65 75 60 81 60 41 88 8065 13 85 68 06 87 60 88 52 61 34 31 36 58 61 45 87 52 10 69 85 64 44 72 77

38 00 10 21 76 81 71 91 17 11 71 60 29 29 37 74 21 96 40 49 65 58 44 96 98

37 40 29 63 97 01 30 47 75 86 56 27 11 00 86 47 32 46 26 05 40 03 03 74 38

97 12 54 03 48 87 08 33 14 17 21 81 53 92 50 75 23 76 20 47 15 50 12 95 78

21 82 64 11 34 47 14 33 40 72 64 63 88 59 02 49 13 90 64 41 03 85 65 45 52

73 13 54 27 42 95 71 90 90 35 85 79 47 42 96 08 78 98 81 56 64 69 11 92 0207 63 87 79 29 03 06 11 80 72 96 20 74 41 56 23 82 19 95 38 04 71 36 69 94

60 52 88 34 41 07 95 41 98 14 59 17 52 06 95 05 53 35 21 39 61 21 20 64 55

83 59 63 56 55 06 95 89 29 83 05 12 80 97 19 77 43 35 37 83 92 30 15 04 98

10 85 06 27 46 99 59 91 05 07 13 49 90 63 19 53 07 57 18 39 06 41 01 93 62

39 82 09 89 52 43 62 26 31 47 64 42 18 08 14 43 80 00 93 51 31 02 47 31 67

59 58 00 64 78 75 56 97 88 00 88 83 55 44 86 23 76 80 61 56 04 11 10 84 0838 50 80 73 41 23 79 34 87 63 90 82 29 70 22 17 71 90 42 07 95 95 44 99 53

30 69 27 06 68 94 68 81 61 27 56 19 68 00 91 82 06 76 34 00 05 46 26 92 00

65 44 39 56 59 18 28 82 74 37 49 63 22 40 41 08 33 76 56 76 96 29 99 08 36

27 26 75 02 64 13 19 27 22 94 07 47 74 46 06 17 98 54 89 11 97 34 13 03 58

91 30 70 69 91 19 07 22 42 10 36 69 95 37 28 28 82 53 57 93 28 97 66 62 52

68 43 49 46 88 84 47 31 36 22 62 12 69 84 08 12 84 38 25 90 09 81 59 31 46

48 90 81 58 77 54 74 52 45 91 35 70 00 47 54 83 82 45 26 92 54 13 05 51 6006 91 34 51 97 42 67 27 86 01 11 88 30 95 28 63 01 19 89 01 14 97 44 03 44

10 45 51 60 19 14 21 03 37 12 91 34 23 78 21 88 32 58 08 51 43 66 77 08 83

12 88 39 73 43 65 02 76 11 84 04 28 50 13 92 17 97 41 50 77 90 71 22 67 69

21 77 83 09 76 38 80 73 69 61 31 64 94 20 96 63 28 10 20 23 08 81 64 74 49

19 52 35 95 15 65 12 25 96 59 86 28 36 82 58 69 57 21 37 98 16 43 59 15 2967 24 55 26 70 35 58 31 65 63 79 24 68 66 86 76 46 33 42 22 26 65 59 08 0260 58 44 73 77 07 50 03 79 92 45 13 42 65 29 26 76 08 36 37 41 32 64 43 44

53 85 34 13 77 36 06 69 48 50 58 83 87 38 59 49 36 47 33 31 96 24 04 36 42

24 63 73 97 36 74 38 48 93 42 52 62 30 79 92 12 36 91 86 01 03 74 28 38 73

83 08 01 24 51 38 99 22 28 15 07 75 95 17 77 97 37 72 75 85 51 97 23 78 67

16 44 42 43 34 36 15 19 90 73 27 49 37 09 39 85 13 03 25 52 54 84 65 47 59

60 79 01 81 57 57 17 86 57 62 11 16 17 85 76 45 81 95 29 79 65 13 00 48 60

From tables o the RAND Corporation. Reprinted rom Wilred J. Dixon and Frank J. Massey, Jr., Introduction to Statistical Analysis. 3rd ed. (New York: McGraw-Hill, 1969), pp. 446447. Re printed by permiss

o the RAND Corporation.



Table 2 Binomial Probabilities [(nx) px qn-x]P

n x 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 x

2 0 .980 .902 .810 .640 .490 .360 .250 .160 .090 .040 .010 .002 0+ 0

1 .020 .095 .180 .320 .420 .480 .500 .480 .420 .320 .180 .095 .020 1

2 0+ .002 .010 .040 .090 .160 .250 .360 .490 .640 .810 .902 .980 23 0 .970 .857 .729 .512 .343 .216 .125 .064 .027 .008 .001 0+ 0+ 0

1 .029 .135 .243 .384 .441 .432 .375 .288 .189 .096 .027 .007 0+ 1

2 0+ .007 .027 .096 .189 .288 .375 .432 .441 .384 .243 .135 .029 2

3 0+ 0+ .001 .008 .027 .064 .125 .216 .343 .512 .729 .857 .970 3

4 0 .961 .815 .656 .410 .240 .130 .062 .026 .008 .002 0+ 0+ 0+ 0

1 .039 .171 .292 .410 .412 .346 .250 .154 .076 .026 .004 0+ 0+ 1

2 .001 .014 .049 .154 .265 .346 .375 .346 .265 .154 .049 .014 .001 2

3 0+ 0+ .004 .026 .076 .154 .250 .346 .412 .410 .292 .171 .039 3

4 0+ 0+ 0+ .002 .008 .026 .062 .130 .240 .410 .656 .815 .961 4

5 0 .951 .774 .590 .328 .168 .078 .031 .010 .002 0+ 0+ 0+ 0+ 0

1 .048 .204 .328 .410 .360 .259 .156 .077 .028 .006 0+ 0+ 0+ 12 .001 .021 .073 .205 .309 .346 .312 .230 .132 .051 .008 .001 0+ 2

3 0+ .001 .008 .051 .132 .230 .312 .346 .309 .205 .073 .021 .001 3

4 0+ 0+ 0+ .006 .028 .077 .156 .259 .360 .410 .328 .204 .048 4

5 0+ 0+ 0+ 0+ .002 .010 .031 .078 .168 .328 .590 .774 .951 5

6 0 .941 .735 .531 .262 .118 .047 .016 .004 .001 0+ 0+ 0+ 0+ 0

1 .057 .232 .354 .393 .303 .187 .094 .037 .010 .002 0+ 0+ 0+ 1

2 .001 .031 .098 .246 .324 .311 .234 .138 .060 .015 .001 0+ 0+ 2

3 0+ .002 .015 .082 .185 .276 .312 .276 .185 .082 .015 .002 0+ 3

4 0+ 0+ .001 .015 .060 .138 .234 .311 .324 .246 .098 .031 .001 4

5 0+ 0+ 0+ .002 .010 .037 .094 .187 .303 .393 .354 .232 .057 5

6 0+ 0+ 0+ 0+ .001 .004 .016 .047 .118 .262 .531 .735 .941 6

7 0 .932 .698 .478 .210 .082 .028 .008 .002 0+ 0+ 0+ 0+ 0+ 0

1 .066 .257 .372 .367 .247 .131 .055 .017 .004 0+ 0+ 0+ 0+ 1

2 .002 .041 .124 .275 .318 .261 .164 .077 .025 .004 0+ 0+ 0+ 2

3 0+ .004 .023 .115 .227 .290 .273 .194 .097 .029 .003 0+ 0+ 3

4 0+ 0+ .003 .029 .097 .194 .273 .290 .227 .115 .023 .004 0+ 4

5 0+ 0+ 0+ .004 .025 .077 .164 .261 .318 .275 .124 .041 .002 5

6 0+ 0+ 0+ 0+ .004 .017 .055 .131 .247 .367 .372 .257 .066 6

7 0+ 0+ 0+ 0+ 0+ .002 .008 .028 .082 .210 .478 .698 .932 7

8 0 .923 .663 .430 .168 .058 .017 .004 .001 0+ 0+ 0+ 0+ 0+ 0

1 .075 .279 .383 .336 .198 .090 .031 .008 .001 0+ 0+ 0+ 0+ 12 .003 .051 .149 .294 .296 .209 .109 .041 .010 .001 0+ 0+ 0+ 2

3 0+ .005 .033 .147 .254 .279 .219 .124 .047 .009 0+ 0+ 0+ 3

4 0+ 0+ .005 .046 .136 .232 .273 .232 .136 .046 .005 0+ 0+ 4

5 0+ 0+ 0+ .009 .047 .124 .219 .279 .254 .147 .033 .005 0+ 5

6 0+ 0+ 0+ .001 .010 .041 .109 .209 .296 .294 .149 .051 .003 6

7 0+ 0+ 0+ 0+ .001 .008 .031 .090 .198 .336 .383 .279 .075 7

8 0+ 0+ 0+ 0+ 0+ .001 .004 .017 .058 .168 .430 .663 .923 8For specifc details about using this table, see page 110.



Table 2 Binomial Probabilities [(nx) px qn-x](continued)P

n x 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 x

9 0 .914 .630 .387 .134 .040 .010 .002 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .083 .299 .387 .302 .156 .060 .018 .004 0+ 0+ 0+ 0+ 0+ 1

2 .003 .063 .172 .302 .267 .161 .070 .021 .004 0+ 0+ 0+ 0+ 23 0+ .008 .045 .176 .267 .251 .164 .074 .021 .003 0+ 0+ 0+ 3

4 0+ .001 .007 .066 .172 .251 .246 .167 .074 .017 .001 0+ 0+ 4

5 0+ 0+ .001 .017 .074 .167 .246 .251 .172 .066 .007 .001 0+ 5

6 0+ 0+ 0+ .003 .021 .074 .164 .251 .267 .176 .045 .008 0+ 6

7 0+ 0+ 0+ 0+ .004 .021 .070 .161 .267 .302 .172 .063 .003 7

8 0+ 0+ 0+ 0+ 0+ .004 .018 .060 .156 .302 .387 .299 .083 8

9 0+ 0+ 0+ 0+ 0+ 0+ .002 .010 .040 .134 .387 .630 .914 9

10 0 .904 .599 .349 .107 .028 .006 .001 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .091 .315 .387 .268 .121 .040 .010 .002 0+ 0+ 0+ 0+ 0+ 1

2 .004 .075 .194 .302 .233 .121 .044 .011 .001 0+ 0+ 0+ 0+ 2

3 0+ .010 .057 .201 .267 .215 .117 .042 .009 .001 0+ 0+ 0+ 3

4 0+ .001 .011 .088 .200 .251 .205 .111 .037 .006 0+ 0+ 0+ 45 0+ 0+ .001 .026 .103 .201 .246 .201 .103 .026 .001 0+ 0+ 5

6 0+ 0+ 0+ .006 .037 .111 .205 .251 .200 .088 .011 .001 0+ 6

7 0+ 0+ 0+ .001 .009 .042 .117 .215 .267 .201 .057 .010 0+ 7

8 0+ 0+ 0+ 0+ .001 .011 .044 .121 .233 .302 .194 .075 .004 8

9 0+ 0+ 0+ 0+ 0+ .002 .010 .040 .121 .268 .387 .315 .091 9

10 0+ 0+ 0+ 0+ 0+ 0+ .001 .006 .028 .107 .349 .599 .904 10

11 0 .895 .569 .314 .086 .020 .004 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .099 .329 .384 .236 .093 .027 .005 .001 0+ 0+ 0+ 0+ 0+ 1

2 .005 .087 .213 .295 .200 .089 .027 .005 .001 0+ 0+ 0+ 0+ 1

3 0+ .014 .071 .221 .257 .177 .081 .023 .004 0+ 0+ 0+ 0+ 3

4 0+ .001 .016 .111 .220 .236 .161 .070 .017 .002 0+ 0+ 0+ 45 0+ 0+ .002 .039 .132 .221 .226 .147 .057 .010 0+ 0+ 0+ 5

6 0+ 0+ 0+ .010 .057 .147 .226 .221 .132 .039 .002 0+ 0+ 6

7 0+ 0+ 0+ .002 .017 .070 .161 .236 .220 .111 .016 .001 0+ 7

8 0+ 0+ 0+ 0+ .004 .023 .081 .177 .257 .221 .071 .014 0+ 8

9 0+ 0+ 0+ 0+ .001 .005 .027 .089 .200 .295 .213 .087 .005 9

10 0+ 0+ 0+ 0+ 0+ .001 .005 .027 .093 .236 .384 .329 .099 10

11 0+ 0+ 0+ 0+ 0+ 0+ 0+ .004 .020 .086 .314 .569 .895 11

12 0 .886 .540 .282 .069 .014 .002 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .107 .341 .377 .206 .071 .017 .003 0+ 0+ 0+ 0+ 0+ 0+ 1

2 .006 .099 .230 .283 .168 .064 .016 .002 0+ 0+ 0+ 0+ 0+ 2

3 0+ .017 .085 .236 .240 .142 .054 .012 .001 0+ 0+ 0+ 0+ 3

4 0+ .002 .021 .133 .231 .213 .121 .042 .008 .001 0+ 0+ 0+ 45 0+ 0+ .004 .053 .158 .227 .193 .101 .029 .003 0+ 0+ 0+ 5

6 0+ 0+ 0+ .016 .079 .177 .226 .177 .079 .016 0+ 0+ 0+ 6

7 0+ 0+ 0+ .003 .029 .101 .193 .227 .158 .053 .004 0+ 0+ 7

8 0+ 0+ 0+ .001 .008 .042 .121 .213 .231 .133 .021 .002 0+ 8

9 0+ 0+ 0+ 0+ .001 .012 .054 .142 .240 .236 .085 .017 0+ 9

10 0+ 0+ 0+ 0+ 0+ .002 .016 .064 .168 .283 .230 .099 .006 10

11 0+ 0+ 0+ 0+ 0+ 0+ .003 .017 .071 .206 .377 .341 .107 11

12 0+ 0+ 0+ 0+ 0+ 0+ 0+ .002 .014 .069 .282 .540 .886 12



Table 2 Binomial Probabilities [(nx) px qn-x](continued)P

n x 0.01 0.05 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 0.95 0.99 x

13 0 .878 .513 .254 .055 .010 .001 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .115 .351 .367 .179 .054 .011 .002 0+ 0+ 0+ 0+ 0+ 0+ 1

2 .007 .111 .245 .268 .139 .045 .010 .001 0+ 0+ 0+ 0+ 0+ 23 0+ .021 .100 .246 .218 .111 .035 .006 .001 0+ 0+ 0+ 0+ 3

4 0+ .003 .028 .154 .234 .184 .087 .024 .003 0+ 0+ 0+ 0+ 4

5 0+ 0+ .006 .069 .180 .221 .157 .066 .014 .001 0+ 0+ 0+ 5

6 0+ 0+ .001 .023 .103 .197 .209 .131 .044 .006 0+ 0+ 0+ 6

7 0+ 0+ 0+ .006 .044 .131 .209 .197 .103 .023 .001 0+ 0+ 7

8 0+ 0+ 0+ .001 .014 .066 .157 .221 .180 .069 .006 0+ 0+ 8

9 0+ 0+ 0+ 0+ .003 .024 .087 .184 .234 .154 .028 .003 0+ 9

10 0+ 0+ 0+ 0+ .001 .006 .035 .111 .218 .246 .100 .021 0+ 10

11 0+ 0+ 0+ 0+ 0+ .001 .010 .045 .139 .268 .245 .111 .007 11

12 0+ 0+ 0+ 0+ 0+ 0+ .002 .011 .054 .179 .367 .351 .115 12

13 0+ 0+ 0+ 0+ 0+ 0+ 0+ .001 .010 .055 .254 .513 .878 13

14 0 .869 .488 .229 .044 .007 .001 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .123 .359 .356 .154 .041 .007 .001 0+ 0+ 0+ 0+ 0+ 0+ 1

2 .008 .123 .257 .250 .113 .032 .006 .001 0+ 0+ 0+ 0+ 0+ 2

3 0+ .026 .114 .250 .194 .085 .022 .003 0+ 0+ 0+ 0+ 0+ 3

4 0+ .004 .035 .172 .229 .155 .061 .014 .001 0+ 0+ 0+ 0+ 4

5 0+ 0+ .008 .086 .196 .207 .122 .041 .007 0+ 0+ 0+ 0+ 5

6 0+ 0+ .001 .032 .126 .207 .183 .092 .023 .002 0+ 0+ 0+ 6

7 0+ 0+ 0+ .009 .062 .157 .209 .157 .062 .009 0+ 0+ 0+ 7

8 0+ 0+ 0+ .002 .023 .092 .183 .207 .126 .032 .001 0+ 0+ 8

9 0+ 0+ 0+ 0+ .007 .041 .122 .207 .196 .086 .008 0+ 0+ 9

10 0+ 0+ 0+ 0+ .001 .014 .061 .155 .229 .172 .035 .004 0+ 10

11 0+ 0+ 0+ 0+ 0+ .003 .022 .085 .194 .250 .114 .026 .0+ 1112 0+ 0+ 0+ 0+ 0+ .001 .006 .032 .113 .250 .257 .123 .008 12

13 0+ 0+ 0+ 0+ 0+ 0+ .001 .007 .041 .154 .356 .359 .123 13

14 0+ 0+ 0+ 0+ 0+ 0+ 0+ .001 .007 .044 .229 .488 .869 14

15 0 .860 .463 .206 .035 .005 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0

1 .130 .366 .343 .132 .031 .005 0+ 0+ 0+ 0+ 0+ 0+ 0+ 1

2 .009 .135 .267 .231 .092 .022 .003 0+ 0+ 0+ 0+ 0+ 0+ 2

3 0+ .031 .129 .250 .170 .063 .014 .002 0+ 0+ 0+ 0+ 0+ 3

4 0+ .005 .043 .188 .219 .127 .042 .007 .001 0+ 0+ 0+ 0+ 4

5 0+ .001 .010 .103 .206 .186 .092 .024 .003 0+ 0+ 0+ 0+ 5

6 0+ 0+ .002 .043 .147 .207 .153 .061 .012 .001 0+ 0+ 0+ 6

7 0+ 0+ 0+ .014 .081 .177 .196 .118 .035 .003 0+ 0+ 0+ 78 0+ 0+ 0+ .003 .035 .118 .196 .177 .081 .014 0+ 0+ 0+ 8

9 0+ 0+ 0+ .001 .012 .061 .153 .207 .147 .043 .002 0+ 0+ 9

10 0+ 0+ 0+ 0+ .003 .024 .092 .186 .206 .103 .010 .001 0+ 10

11 0+ 0+ 0+ 0+ .001 .007 .042 .127 .219 .188 .043 0.05 0+ 11

12 0+ 0+ 0+ 0+ 0+ .002 .014 .063 .170 .250 .129 .031 0+ 12

13 0+ 0+ 0+ 0+ 0+ 0+ .003 .022 .092 .231 .267 .135 .009 13

14 0+ 0+ 0+ 0+ 0+ 0+ 0+ .005 .031 .132 .343 .366 .130 14

15 0+ 0+ 0+ 0+ 0+ 0+ 0+ 0+ .005 .035 .206 .463 .860 15



Table 3 Areas o the Standard Normal Distribution

0 zz

The entries in this table are the probabilities that a random variable with a

standard normal distribution assumes a value between 0 and z; the probability

is represented by the shaded area under the curve in the accompanying fgure.

Areas or negative values oz are obtained by symmetry.

Second Decimal Place inz

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359

0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753

0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141

0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517

0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879

0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224

0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549

0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852

0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133

0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389

1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621

1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.38301.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015

1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177

1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319

1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441

1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545

1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633

1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706

1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767

2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817

2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857

2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.48902.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916

2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936

2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952

2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964

2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974

2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981

2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986

3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

3.1 0.4990 0.4991 0.4991 0.4991 0.4992 0.4992 0.4992 0.4992 0.4993 0.4993

3.2 0.4993 0.4993 0.4994 0.4994 0.4994 0.4994 0.4994 0.4995 0.4995 0.4995

3.3 0.4995 0.4995 0.4995 0.4996 0.4996 0.4996 0.4996 0.4996 0.4996 0.4997

3.4 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.4997 0.49983.5 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998 0.4998

3.6 0.4998 0.4998 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999 0.4999

3.7 0.4999

4.0 0.49997

4.5 0.499997

5.0 0.4999997

For specifc details about using this table to fnd: probabilities, see page 121; confdence coe cients, pages 127128;p-values, page 171; critical values, pages 121123, 127128.



Table 4 Critical Values o Standard Normal Distribution

A One-Tailed Situations

The entries in this table are the critical values or z or which the area

under the curve representing is in the right-hand tail. Critical values

or the let-hand tail are ound by symmetry.

0

= area of one tail

z( )

Amount o in one tail

0.25 0.10 0.05 0.025 0.02 0.01 0.005

z() 0.67 1.28 1.65 1.96 2.05 2.33 2.58

One-tailed example:

= 0.05

z()= z(0.05)= 1.65

B Two-Tailed Situations

The entries in this table are the critical values or z or which the area

under the curve representing is split equally between the two tails.

z( /2) +z( /2)0

= area of two tails

/2 /21

Amount o in two-tails

0.25 0.20 0.10 0.05 0.02 0.01

z(/2) 1.15 1.28 1.65 1.96 2.33 2.58

1- 0.75 0.80 0.90 0.95 0.98 0.99

Area in the center

Two-tailed example:

= 0.05 or 1- = 0.95

/2= 0.025

z(/2)= z(0.025)= 1.96

For specifc details about using:

Table A to fnd: critical values, see page 175.

Table B to fnd: confdence coe cients, see pages 156157; critical values, pages 160, 175.



Table 5 p-Values or Standard Normal Distribution

The entries in this table are thep-values related to the

right-hand tail or the calculated z or the standard

normal distribution.

0 z z

p-value

z p-value z p-value z p-value z p-value

0.00 0.5000 1.00 0.1587 2.00 0.0228 3.00 0.0013

0.05 0.4801 1.05 0.1469 2.05 0.0202 3.05 0.0011

0.10 0.4602 1.10 0.1357 2.10 0.0179 3.10 0.0010

0.15 0.4404 1.15 0.1251 2.15 0.0158 3.15 0.0008

0.20 0.4207 1.20 0.1151 2.20 0.0139 3.20 0.0007

0.25 0.4013 1.25 0.1056 2.25 0.0122 3.25 0.0006

0.30 0.3821 1.30 0.0968 2.30 0.0107 3.30 0.0005

0.35 0.3632 1.35 0.0885 2.35 0.0094 3.35 0.0004

0.40 0.3446 1.40 0.0808 2.40 0.0082 3.40 0.0003

0.45 0.3264 1.45 0.0735 2.45 0.0071 3.45 0.0003

0.50 0.3085 1.50 0.0668 2.50 0.0062 3.50 0.0002

0.55 0.2912 1.55 0.0606 2.55 0.0054 3.55 0.0002

0.60 0.2743 1.60 0.0548 2.60 0.0047 3.60 0.0002

0.65 0.2578 1.65 0.0495 2.65 0.0040 3.65 0.0001

0.70 0.2420 1.70 0.0446 2.70 0.0035 3.70 0.0001

0.75 0.2266 1.75 0.0401 2.75 0.0030 3.75 0.0001

0.80 0.2119 1.80 0.0359 2.80 0.0026 3.80 0.0001

0.85 0.1977 1.85 0.0322 2.85 0.0022 3.85 0.0001

0.90 0.1841 1.90 0.0287 2.90 0.0019 3.90 0+0.95 0.1711 1.95 0.0256 2.95 0.0016 3.95 0+

For specifc details about using this table to fnd p-values, see pages 171, 172.

0 z = 2.30

p-value =P(z> 2.30) = 0.0107



Table 6 Critical Values o Students t-Distribution

The entries in this table, t(d,), are the critical values or Students t-distribution or which the

area under the curve in the right-hand tail is . Critical values or the let-hand tail are ound by

symmetry.

0 t(df, )

= area ofone tail

Amount o in One Tail

0.25 0.10 0.05 0.025 0.01 0.005

Amount o in Two Tails

d 0.50 0.20 0.10 0.05 0.02 0.01

3 0.765 1.64 2.35 3.18 4.54 5.84

4 0.741 1.53 2.13 2.78 3.75 4.60

5 0.729 1.48 2.02 2.57 3.37 4.03

6 0.718 1.44 1.94 2.45 3.14 3.71

7 0.711 1.42 1.89 2.36 3.00 3.50

8 0.706 1.40 1.86 2.31 2.90 3.36

9 0.703 1.38 1.83 2.26 2.82 3.25

10 0.700 1.37 1.81 2.23 2.76 3.17

11 0.697 1.36 1.80 2.20 2.72 3.1112 0.696 1.36 1.78 2.18 2.68 3.05

13 0.694 1.35 1.77 2.16 2.65 3.01

14 0.692 1.35 1.76 2.14 2.62 2.98

15 0.691 1.34 1.75 2.13 2.60 2.95

16 0.690 1.34 1.75 2.12 2.58 2.92

17 0.689 1.33 1.74 2.11 2.57 2.90

18 0.688 1.33 1.73 2.10 2.55 2.88

19 0.688 1.33 1.73 2.09 2.54 2.86

20 0.687 1.33 1.72 2.09 2.53 2.85

21 0.686 1.32 1.72 2.08 2.52 2.83

22 0.686 1.32 1.72 2.07 2.51 2.82

23 0.685 1.32 1.71 2.07 2.50 2.81

24 0.685 1.32 1.71 2.06 2.49 2.80

25 0.684 1.32 1.71 2.06 2.49 2.79

26 0.684 1.32 1.71 2.06 2.48 2.78

27 0.684 1.31 1.70 2.05 2.47 2.77

28 0.683 1.31 1.70 2.05 2.47 2.76

29 0.683 1.31 1.70 2.05 2.46 2.76

30 0.683 1.31 1.70 2.04 2.46 2.75

35 0.682 1.31 1.69 2.03 2.44 2.73

40 0.681 1.30 1.68 2.02 2.42 2.70

50 0.679 1.30 1.68 2.01 2.40 2.68

70 0.678 1.29 1.67 1.99 2.38 2.65100 0.677 1.29 1.66 1.98 2.36 2.63

d>100 0.675 1.28 1.65 1.96 2.33 2.58

0 t(df, )

= area of one tail

One-tailed example:df = 9 and = 0.10

t(df, ) =t(9, 0.10) = 1.38

0t(df, /2) +t(df, /2)

= area of two tails

/2/2

Two-tailed example:df = 14, = 0.02, 1 = 0.98

t(df, /2) =t(14, 0.01) = 2.62

For specifc details about using this table to fnd: confdence coe cients, see pages 186188;p-values, pages 189, 190; critical values, page 186.



Table 7 Probability-Values or Students t-distribution

The entries in this table are thep-values related to the right-hand tail or

the calculated t value or the t-distribution o d degrees o reedom.

0 t

p-value

Degrees o Freedom

t 3 4 5 6 7 8 10 12 15 18 21 25 29 35 d40.0 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.5000.1 0.463 0.463 0.462 0.462 0.462 0.461 0.461 0.461 0.461 0.461 0.461 0.461 0.461 0.460 0.4600.2 0.427 0.426 0.425 0.424 0.424 0.423 0.423 0.422 0.422 0.422 0.422 0.422 0.421 0.421 0.4210.3 0.392 0.390 0.388 0.387 0.386 0.386 0.385 0.385 0.384 0.384 0.384 0.383 0.383 0.383 0.3830.4 0.358 0.355 0.353 0.352 0.351 0.350 0.349 0.348 0.347 0.347 0.347 0.346 0.346 0.346 0.346

0.5 0.326 0.322 0.319 0.317 0.316 0.315 0.314 0.313 0.312 0.312 0.311 0.311 0.310 0.310 0.3100.6 0.295 0.290 0.287 0.285 0.284 0.283 0.281 0.280 0.279 0.278 0.277 0.277 0.277 0.276 0.2760.7 0.267 0.261 0.258 0.255 0.253 0.252 0.250 0.249 0.247 0.246 0.246 0.245 0.245 0.244 0.2440.8 0.241 0.234 0.230 0.227 0.225 0.223 0.221 0.220 0.218 0.217 0.216 0.216 0.215 0.215 0.2140.9 0.217 0.210 0.205 0.201 0.199 0.197 0.195 0.193 0.191 0.190 0.189 0.188 0.188 0.187 0.186

1.0 0.196 0.187 0.182 0.178 0.175 0.173 0.170 0.169 0.167 0.165 0.164 0.163 0.163 0.162 0.1611.1 0.176 0.167 0.161 0.157 0.154 0.152 0.149 0.146 0.144 0.143 0.142 0.141 0.140 0.139 0.1391.2 0.158 0.148 0.142 0.138 0.135 0.132 0.129 0.127 0.124 0.123 0.122 0.121 0.120 0.119 0.1181.3 0.142 0.132 0.125 0.121 0.117 0.115 0.111 0.109 0.107 0.105 0.104 0.103 0.102 0.101 0.1001.4 0.128 0.117 0.110 0.106 0.102 0.100 0.096 0.093 0.091 0.089 0.088 0.087 0.086 0.085 0.084

1.5 0.115 0.104 0.097 0.092 0.089 0.086 0.082 0.080 0.077 0.075 0.074 0.073 0.072 0.071 0.0701.6 0.104 0.092 0.085 0.080 0.077 0.074 0.070 0.068 0.065 0.064 0.062 0.061 0.060 0.059 0.0581.7 0.094 0.082 0.075 0.070 0.066 0.064 0.060 0.057 0.055 0.053 0.052 0.051 0.050 0.049 0.0481.8 0.085 0.073 0.066 0.061 0.057 0.055 0.051 0.049 0.046 0.044 0.043 0.042 0.041 0.040 0.0391.9 0.077 0.065 0.058 0.053 0.050 0.047 0.043 0.041 0.038 0.037 0.036 0.035 0.034 0.033 0.032

2.0 0.070 0.058 0.051 0.046 0.043 0.040 0.037 0.034 0.032 0.030 0.029 0.028 0.027 0.027 0.0262.1 0.063 0.052 0.045 0.040 0.037 0.034 0.031 0.029 0.027 0.025 0.024 0.023 0.022 0.022 0.021

2.2 0.058 0.046 0.040 0.035 0.032 0.029 0.026 0.024 0.022 0.021 0.020 0.019 0.018 0.017 0.0162.3 0.052 0.041 0.035 0.031 0.027 0.025 0.022 0.020 0.018 0.017 0.016 0.015 0.014 0.014 0.0132.4 0.048 0.037 0.031 0.027 0.024 0.022 0.019 0.017 0.015 0.014 0.013 0.012 0.012 0.011 0.010

2.5 0.044 0.033 0.027 0.023 0.020 0.018 0.016 0.014 0.012 0.011 0.010 0.010 0.009 0.009 0.0082.6 0.040 0.030 0.024 0.020 0.018 0.016 0.013 0.012 0.010 0.009 0.008 0.008 0.007 0.007 0.0062.7 0.037 0.027 0.021 0.018 0.015 0.014 0.011 0.010 0.008 0.007 0.007 0.006 0.006 0.005 0.0052.8 0.034 0.024 0.019 0.016 0.013 0.012 0.009 0.008 0.007 0.006 0.005 0.005 0.005 0.004 0.0042.9 0.031 0.022 0.017 0.014 0.011 0.010 0.008 0.007 0.005 0.005 0.004 0.004 0.004 0.003 0.003

3.0 0.029 0.020 0.015 0.012 0.010 0.009 0.007 0.006 0.004 0.004 0.003 0.003 0.003 0.002 0.0023.1 0.027 0.018 0.013 0.011 0.009 0.007 0.006 0.005 0.004 0.003 0.003 0.002 0.002 0.002 0.0023.2 0.025 0.016 0.012 0.009 0.008 0.006 0.005 0.004 0.003 0.002 0.002 0.002 0.002 0.001 0.0013.3 0.023 0.015 0.011 0.008 0.007 0.005 0.004 0.003 0.002 0.002 0.002 0.001 0.001 0.001 0.0013.4 0.021 0.014 0.010 0.007 0.006 0.005 0.003 0.003 0.002 0.002 0.001 0.001 0.001 0.001 0.001

3.5 0.020 0.012 0.009 0.006 0.005 0.004 0.003 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.0013.6 0.018 0.011 0.008 0.006 0.004 0.004 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0+ 0+3.7 0.017 0.010 0.007 0.005 0.004 0.003 0.002 0.002 0.001 0.001 0.001 0.001 0+ 0+ 0+3.8 0.016 0.010 0.006 0.004 0.003 0.003 0.002 0.001 0.001 0.001 0.001 0+ 0+ 0+ 0+3.9 0.015 0.009 0.006 0.004 0.003 0.002 0.001 0.001 0.001 0.001 0+ 0+ 0+ 0+ 0+4.0 0.014 0.008 0.005 0.004 0.003 0.002 0.001 0.001 0.001 0+ 0+ 0+ 0+ 0+ 0+

For specifc details about using this table to fnd p-values, see pages 190191.



Table 8 Critical Values o2 (Chi-Square) Distribution

0

area to right

2(df, area to right)

The entries in this table, 2 (d, ), are the critical values or the 2

distribution or which the area under the curve to the right is .

Area to the Right

0.995 0.99 0.975 0.95 0.90 0.75 0.50 0.25 0.10 0.05 0.025 0.01 0.005

Area in Let-hand Tail Median Area in Right-hand Taild 0.005 0.01 0.025 0.05 0.10 0.25 0.50 0.25 0.10 0.05 0.025 0.01 0.005

1 0.0000393 0.000157 0.000982 0.00393 0.0158 0.101 0.455 1.32 2.71 3.84 5.02 6.63 7.88

2 0.0100 0.0201 0.0506 0.103 0.211 0.575 1.39 2.77 4.61 5.99 7.38 9.21 10.6

3 0.0717 0.115 0.216 0.352 0.584 1.21 2.37 4.11 6.25 7.81 9.35 11.3 12.8

4 0.207 0.297 0.484 0.711 1.06 1.92 3.36 5.39 7.78 9.49 11.1 13.3 14.9

5 0.412 0.554 0.831 1.15 1.61 2.67 4.35 6.63 9.24 11.1 12.8 15.1 16.8

6 0.676 0.872 1.24 1.64 2.20 3.45 5.35 7.84 10.6 12.6 14.5 16.8 18.6

7 0.990 1.24 1.69 2.17 2.83 4.25 6.35 9.04 12.0 14.1 16.0 18.5 20.3

8 1.34 1.65 2.18 2.73 3.49 5.07 7.34 10.2 13.4 15.5 17.5 20.1 22.0

9 1.73 2.09 2.70 3.33 4.17 5.90 8.34 11.4 14.7 16.9 19.0 21.7 23.6

10 2.16 2.56 3.25 3.94 4.87 6.74 9.34 12.5 16.0 18.3 20.5 23.2 25.2

11 2.60 3.05 3.82 4.57 5.58 7.58 10.34 13.7 17.3 19.7 21.9 24.7 26.8

12 3.07 3.57 4.40 5.23 6.30 8.44 11.34 14.8 18.5 21.0 23.3 26.2 28.313 3.57 4.11 5.01 5.89 7.04 9.30 12.34 16.0 19.8 22.4 24.7 27.7 29.8

14 4.07 4.66 5.63 6.57 7.79 10.2 13.34 17.1 21.1 23.7 26.1 29.1 31.3

15 4.60 5.23 6.26 7.26 8.55 11.0 14.34 18.2 22.3 25.0 27.5 30.6 32.8

16 5.14 5.81 6.91 7.96 9.31 11.9 15.34 19.4 23.5 26.3 28.8 32.0 34.3

17 5.70 6.41 7.56 8.67 10.1 12.8 16.34 20.5 24.8 27.6 30.2 33.4 35.7

18 6.26 7.01 8.23 9.39 10.9 13.7 17.34 21.6 26.0 28.9 31.5 34.8 37.2

19 6.84 7.63 8.91 10.1 11.7 14.6 18.34 22.7 27.2 30.1 32.9 36.2 38.6

20 7.43 8.26 9.59 10.9 12.4 15.5 19.34 23.8 28.4 31.4 34.2 37.6 40.0

21 8.03 8.90 10.3 11.6 13.2 16.3 20.34 24.9 29.6 32.7 35.5 38.9 41.4

22 8.64 9.54 11.0 12.3 14.0 17.2 21.34 26.0 30.8 33.9 36.8 40.3 42.8

23 9.26 10.2 11.7 13.1 14.8 18.1 22.34 27.1 32.0 35.2 38.1 41.6 44.2

24 9.89 10.9 12.4 13.8 15.7 19.0 23.34 28.2 33.2 36.4 39.4 43.0 45.6

25 10.5 11.5 13.1 14.6 16.5 19.9 24.34 29.3 34.4 37.7 40.6 44.3 46.926 11.2 12.2 13.8 15.4 17.3 20.8 25.34 30.4 35.6 38.9 41.9 45.6 48.3

27 11.8 12.9 14.6 16.2 18.1 21.7 26.34 31.5 36.7 40.1 43.2 47.0 49.6

28 12.5 13.6 15.3 16.9 18.9 22.7 27.34 32.6 37.9 41.3 44.5 48.3 51.0

29 13.1 14.3 16.0 17.7 19.8 23.6 28.34 33.7 39.1 42.6 45.7 49.6 52.3

30 13.8 15.0 16.8 18.5 20.6 24.5 29.34 34.8 40.3 43.8 47.0 50.9 53.7

40 20.7 22.2 24.4 26.5 29.1 33.7 39.34 45.6 51.8 55.8 59.3 63.7 66.8

50 28.0 29.7 32.4 34.8 37.7 42.9 49.33 56.3 63.2 67.5 71.4 76.2 79.5

60 35.5 37.5 40.5 43.2 46.5 52.3 59.33 67.0 74.4 79.1 83.3 88.4 92.0

70 43.3 45.4 48.8 51.7 55.3 61.7 69.33 77.6 85.5 90.5 95.0 100.4 104.2

80 51.2 53.5 57.2 60.4 64.3 71.1 79.33 88.1 96.6 101.9 106.6 112.3 116.3

90 59.2 61.8 65.6 69.1 73.3 80.6 89.33 98.6 107.6 113.1 118.1 124.1 128.3

100 67.3 70.1 74.2 77.9 82.4 90.1 99.33 109.1 118.5 124.3 129.6 135.8 140.2

0

0.900.10

2(28, 0.90)

2(df, area to right) = 2(28, 0.90) = 18.9

2Left-tail example:Find with df = 28; area in left-tail = 0.10.

2(df, area to right) = 2(23, 0.025) = 38.1

0

0.025

2(23, 0.025)

2Right-tail example:Find with df = 23; area in right-tail = 0.025

For specifc details about using this table to fnd:p-values, see pages 202204; critical values, page 200.



Table 9a Critical Values o the F Distribution (= 0.05)

10

= 0.05

F(dfn, df

d, 0.05)

The entries in this table are critical values oFor which the area

under the curve to the right is equal to 0.05.

Degrees o Freedom or Numerator

1 2 3 4 5 6 7 8 9 10

Degreesof

Freedomf

orDenominator

1 161. 200. 216. 225. 230. 234. 237. 239. 241. 242.

2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4

3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79

4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96

5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74

6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06

7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64

8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35

9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98

11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85

12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75

13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67

14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60

15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54

16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49

17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.4518 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41

19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38

20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35

21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32

22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30

23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27

24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25

25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24

30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16

40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08

60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99

120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91

3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83

For specifc details about using this table to fnd: p-values, see page 231; critical values, page 230.



Table 9a (Continued)


12 15 20 24 30 40 60 120

DegreesofFreedomf

orDenominator

1 244. 246. 248. 249. 250. 251. 252. 253. 254.

2 19.4 19.4 19.4 19.5 19.5 19.5 19.5 19.5 19.5

3 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.534 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63

5 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.37

6 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67

7 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23

8 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93

9 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71

10 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54

11 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40

12 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30

13 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21

14 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13

15 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07

16 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01

17 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96

18 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92

19 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88

20 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84

21 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81

22 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78

23 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76

24 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73

25 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71

30 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62

40 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51

60 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39

120 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25

1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00

From E. S. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, vol. 1 (1958), pp. 159163. Reprinted by permission o the Biometrika Trustees.



Table 9b Critical Values o the F Distribution (= 0.025)

10

= 0.025

F(dfn, df

d, 0.025)

The entries in this table are critical values oFor which the area

under the curve to the right is equal to 0.025.


1 2 3 4 5 6 7 8 9 10

Degreesof

Freedomf

orDenominator

1 648. 800. 864. 900. 922. 937. 948. 957. 963. 969.

2 38.5 39.0 39.2 39.2 39.3 39.3 39.4 39.4 39.4 39.4

3 17.4 16.0 15.4 15.1 14.9 14.7 14.6 14.5 14.5 14.4

4 12.2 10.6 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84

5 10.0 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62

6 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46

7 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76

8 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30

9 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96

10 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72

11 6.72 5.26 4.63 4.28 4.04 3.88 3.76 3.66 3.59 3.53

12 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37

13 6.41 4.97 4.35 4.00 3.77 3.60 3.48 3.39 3.31 3.25

14 6.30 4.86 4.24 3.89 3.66 3.50 3.38 3.28 3.21 3.15

15 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06

16 6.12 4.69 4.08 3.73 3.50 3.34 3.22 3.12 3.05 2.99

17 6.04 4.62 4.01 3.66 3.44 3.28 3.16 3.06 2.98 2.9218 5.98 4.56 3.95 3.61 3.38 3.22 3.10 3.01 2.93 2.87

19 5.92 4.51 3.90 3.56 3.33 3.17 3.05 2.96 2.88 2.82

20 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77

21 5.83 4.42 3.82 3.48 3.25 3.09 2.97 2.87 2.80 2.73

22 5.79 4.38 3.78 3.44 3.22 3.05 2.93 2.84 2.76 2.70

23 5.75 4.35 3.75 3.41 3.18 3.02 2.90 2.81 2.73 2.67

24 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70 2.64

25 5.69 4.29 3.69 3.35 3.13 2.97 2.85 2.75 2.68 2.61

30 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51

40 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39

60 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27

120 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.16

5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05




Table 9b (Continued)


12 15 20 24 30 40 60 120

DegreesofFreedomf

orDenominator

1 977. 985. 993. 997. 1001. 1006. 1010. 1014. 1018.

2 39.4 39.4 39.4 39.5 39.5 39.5 39.5 39.5 39.5

3 14.3 14.3 14.2 14.1 14.1 14.0 14.0 13.9 13.94 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26

5 6.52 6.43 6.33 6.28 6.23 6.18 6.12 6.07 6.02

6 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.90 4.85

7 4.67 4.57 4.47 4.42 4.36 4.31 4.25 4.20 4.14

8 4.20 4.10 4.00 3.95 3.89 3.84 3.78 3.73 3.67

9 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33

10 3.62 3.52 3.42 3.37 3.31 3.26 3.20 3.14 3.08

11 3.43 3.33 3.23 3.17 3.12 3.06 3.00 2.94 2.88

12 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72

13 3.15 3.05 2.95 2.89 2.84 2.78 2.72 2.66 2.60

14 3.05 2.95 2.84 2.79 2.73 2.67 2.61 2.55 2.49

15 2.96 2.86 2.76 2.70 2.64 2.59 2.52 2.46 2.40

16 2.89 2.79 2.68 2.63 2.57 2.51 2.45 2.38 2.32

17 2.82 2.72 2.62 2.56 2.50 2.44 2.38 2.32 2.25

18 2.77 2.67 2.56 2.50 2.44 2.38 2.32 2.26 2.19

19 2.72 2.62 2.51 2.45 2.39 2.33 2.27 2.20 2.13

20 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09

21 2.64 2.53 2.42 2.37 2.31 2.25 2.18 2.11 2.04

22 2.60 2.50 2.39 2.33 2.27 2.21 2.14 2.08 2.00

23 2.57 2.47 2.36 2.30 2.24 2.18 2.11 2.04 1.97

24 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94

25 2.51 2.41 2.30 2.24 2.18 2.12 2.05 1.98 1.91

30 2.41 2.31 2.20 2.14 2.07 2.01 1.94 1.87 1.79

40 2.29 2.18 2.07 2.01 1.94 1.88 1.80 1.72 1.64

60 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48

120 2.05 1.95 1.82 1.76 1.69 1.61 1.53 1.43 1.31

1.94 1.83 1.71 1.64 1.57 1.48 1.39 1.27 1.00

From E. S. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, vol. I (1958), pp. 159-163. R eprinted by persmission o the Biometrika Trustees.



Table 9c Critical Values o the F Distribution (= 0.01)

F(dfn, df

d, 0.01)10

= 0.01

The entries in the table are critical values oFor which the area under the

curve to the right is equal to 0.01


1 2 3 4 5 6 7 8 9 10

Degreesof

Freedomf

orDenominator

1 4052. 5000. 5403. 5625. 5764. 5859. 5928. 5982. 6024. 6056.

2 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99.4

3 34.1 30.8 29.5 28.7 28.2 27.9 27.7 27.5 27.3 27.2

4 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 14.5

5 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.1

6 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87

7 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62

8 11.3 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81

9 10.6 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26

10 10.0 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85

11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4.54

12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30

13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4.10

14 8.86 6.51 5.56 5.04 4.70 4.46 4.28 4.14 4.03 3.94

15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80

16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69

17 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68 3.5918 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3.51

19 8.19 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43

20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37

21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3.31

22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26

23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21

24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17

25 7.77 5.57 4.68 4.18 3.86 3.63 3.46 3.32 3.22 3.13

30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98

40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80

60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63

120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47

6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32




Table 9c (Continued)


12 15 20 24 30 40 60 120

DegreesofFreedomf

orDenominator

1 6106. 6157. 6209. 6235. 6261. 6287. 6313. 6339. 6366.

2 99.4 99.4 99.4 99.5 99.5 99.5 99.5 99.5 99.5

3 27.1 26.9 26.7 26.6 26.5 26.4 26.3 26.2 26.14 14.4 14.2 14.0 13.9 13.8 13.7 13.7 13.6 13.5

5 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02

6 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88

7 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65

8 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86

9 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31

10 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91

11 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60

12 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36

13 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17

14 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00

15 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87

16 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75

17 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65

18 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57

19 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49

20 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42

21 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36

22 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31

23 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26

24 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21

25 2.99 2.85 2.70 2.62 2.53 2.45 2.36 2.27 2.17

30 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01

40 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80

60 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60

120 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38

2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00

From E. S. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, vol. I (1958), pp. 159-163. R eprinted by permission o the Biometrika Trustees.



Table 10 Confdence Belts or the Correlation Coe cient (1-)=0.95

The numbers on the curves are sample sizes.

0.2

3

4

5

6

7

8

10

12

15

20

25

50

100

20040

0

400

200

100

50

25

20

15

12

10

8

7

6

5

4

3

1.0 0.8 0.6 0.4 0.2 +0.2 +0.4 +0.6 +0.80 +1.01.0

0.8

0.6

0.4

0

+0.2

+0.4

+0.6

+0.8

+1.0

Scale ofr (sample correlation)

Scale

of

(populationc

orrela

tion

coefficient)

For specifc details about using this table to fnd confdence intervals, see page 280.



Table 11 Critical Values orWhen= 0

The entries in this table are the critical values or or a two-tailed test at . For simple correlation, d= n - 2,

where n is the number o pairs o data in the sample. For a one-tailed test, the value o shown at the top o the

table is double the value o being used in the hypothesis test.

0 r 1r

/2

1

/2

= area oftwo tails

d 0.10 0.05 0.02 0.01

1 0.988 0.997 1.000 1.000

2 0.900 0.950 0.980 0.990

3 0.805 0.878 0.934 0.959

4 0.729 0.811 0.882 0.917

5 0.669 0.754 0.833 0.874

6 0.621 0.707 0.789 0.834

7 0.582 0.666 0.750 0.798

8 0.549 0.632 0.716 0.765

9 0.521 0.602 0.685 0.735

10 0.497 0.576 0.658 0.70811 0.476 0.553 0.634 0.684

12 0.458 0.532 0.612 0.661

13 0.441 0.514 0.592 0.641

14 0.426 0.497 0.574 0.623

15 0.412 0.482 0.558 0.606

16 0.400 0.468 0.542 0.590

17 0.389 0.456 0.528 0.575

18 0.378 0.444 0.516 0.561

19 0.369 0.433 0.503 0.549

20 0.360 0.423 0.492 0.537

25 0.323 0.381 0.445 0.487

30 0.296 0.349 0.409 0.449

35 0.275 0.325 0.381 0.418

40 0.257 0.304 0.358 0.393

45 0.243 0.288 0.338 0.372

50 0.231 0.273 0.322 0.354

60 0.211 0.250 0.295 0.325

70 0.195 0.232 0.274 0.302

80 0.183 0.217 0.256 0.28390 0.173 0.205 0.242 0.267

100 0.164 0.195 0.230 0.254

From E. S. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, vol. 1 (1962), p. 138. Reprinted by permission o the

Biometrika Trustees.

For specifc details about using this table to fnd:p-values, see page 282; critical values, page 282.



Table 12 Critical Values o the Sign Test

The entries in this table are the critical values or the number o the least requent sign or a two-tailed test at or

the binomialp = 0.5. For a one-tailed test, the value o shown at the top o the table is double the value o be-

ing used in the hypothesis test.

n 0.01 0.05 0.10 0.25 n 0.01 0.05 0.10 0.25

1 51 15 18 19 20

2 52 16 18 19 213 0 53 16 18 20 21

4 0 54 17 19 20 22

5 0 0 55 17 19 20 22

6 0 0 1 56 17 20 21 23

7 0 0 1 57 18 20 21 23

8 0 0 1 1 58 18 21 22 24

9 0 1 1 2 59 19 21 22 24

10 0 1 1 2 60 19 21 23 25

11 0 1 2 3 61 20 22 23 25

12 1 2 2 3 62 20 22 24 25

13 1 2 3 3 63 20 23 24 26

14 1 2 3 4 64 21 23 24 26

15 2 3 3 4 65 21 24 25 27

16 2 3 4 5 66 22 24 25 27

17 2 4 4 5 67 22 25 26 28

18 3 4 5 6 68 22 25 26 28

19 3 4 5 6 69 23 25 27 29

20 3 5 5 6 70 23 26 27 29

21 4 5 6 7 71 24 26 28 30

22 4 5 6 7 72 24 27 28 30

23 4 6 7 8 73 25 27 28 31

24 5 6 7 8 74 25 28 29 31

25 5 7 7 9 75 25 28 29 32

26 6 7 8 9 76 26 28 30 32

27 6 7 8 10 77 26 29 30 32

28 6 8 9 10 78 27 29 31 33

29 7 8 9 10 79 27 30 31 3330 7 9 10 11 80 28 30 32 34

31 7 9 10 11 81 28 31 32 34

32 8 9 10 12 82 28 31 33 35

33 8 10 11 12 83 29 32 33 35

34 9 10 11 13 84 29 32 33 36

35 9 11 12 13 85 30 32 34 36

36 9 11 12 14 86 30 33 34 37

37 10 12 13 14 87 31 33 35 37

38 10 12 13 14 88 31 34 35 38

39 11 12 13 15 89 31 34 36 38

40 11 13 14 15 90 32 35 36 39

41 11 13 14 16 91 32 35 37 39

42 12 14 15 16 92 33 36 37 39

43 12 14 15 17 93 33 36 38 40

44 13 15 16 17 94 34 37 38 40

45 13 15 16 18 95 34 37 38 41

46 13 15 16 18 96 34 37 39 41

47 14 16 17 19 97 35 38 39 42

48 14 16 17 19 98 35 38 40 42

49 15 17 18 19 99 36 39 40 43

50 15 17 18 20 100 36 39 41 43

From Wilred J. Dixon and Frank J. Massey, Jr., Introduction to Statistical Analysis, 3d ed. (New York: McGraw-Hill, 1969), p. 509. Reprinted by permission.

For specifc details about using this table to fnd: p-values, see pages 304306; critical values, pages 303304.



Table 13 Critical Values o U in the Mann-Whitney Test

A. The entries are the critical values oUor a one-tailed test at 0.025 or or a two-tailed test at 0.05.

n1

n2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1

2 0 0 0 0 1 1 1 1 1 2 2 2 2

3 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 84 0 1 2 3 4 4 5 6 7 8 9 10 11 11 12 13 13

5 0 1 2 3 5 6 7 8 9 11 12 13 14 15 17 18 19 20

6 1 2 3 5 6 8 10 11 13 14 16 17 19 21 22 24 25 27

7 1 3 5 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34

8 0 2 4 6 8 10 13 15 17 19 22 24 26 29 31 34 36 38 41

9 0 2 4 7 10 12 15 17 20 23 26 28 31 34 37 39 42 45 48

10 0 3 5 8 11 14 17 20 23 26 29 33 36 39 42 45 48 52 55

11 0 3 6 9 13 16 19 23 26 30 33 37 40 44 47 51 55 58 62

12 1 4 7 11 14 18 22 26 29 33 37 41 45 49 53 57 61 65 69

13 1 4 8 12 16 20 24 28 33 37 41 45 50 54 59 63 67 72 76

14 1 5 9 13 17 22 26 31 36 40 45 50 55 59 64 67 74 78 8315 1 5 10 14 19 24 29 34 39 44 49 54 59 64 70 75 80 85 90

16 1 6 11 15 21 26 31 37 42 47 53 59 64 70 75 81 86 92 98

17 2 6 11 17 22 28 34 39 45 51 57 63 67 75 81 87 93 99 105

18 2 7 12 18 24 30 36 42 48 55 61 67 74 80 86 93 99 106 112

19 2 7 13 19 25 32 38 45 52 58 65 72 78 85 92 99 106 113 119

20 2 8 13 20 27 34 41 48 55 62 69 76 83 90 98 105 112 119 127

B. The entries are the critical values oUor a one-tailed test at 0.05 or or a two-tailed test at 0.10.

n1

n2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1 0 0

2 0 0 0 1 1 1 1 2 2 2 3 3 3 4 4 4

3 0 0 1 2 2 3 3 4 5 5 6 7 7 8 9 9 10 11

4 0 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18

5 0 1 2 4 5 6 8 9 11 12 13 15 16 18 19 20 22 23 25

6 0 2 3 5 7 8 10 12 14 16 17 19 21 23 25 26 28 30 32

7 0 2 4 6 8 11 13 15 17 19 21 24 26 28 30 33 35 37 39

8 1 3 5 8 10 13 15 18 20 23 26 28 31 33 36 39 41 44 47

9 1 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54

10 1 4 7 11 14 17 20 24 27 31 34 37 41 44 48 51 55 58 62

11 1 5 8 12 16 19 23 27 31 34 38 42 46 50 54 57 61 65 69

12 2 5 9 13 17 21 26 30 34 38 42 47 51 55 60 64 68 72 77

13 2 6 10 15 19 24 28 33 37 42 47 51 56 61 65 70 75 80 84

14 2 7 11 16 21 26 31 36 41 46 51 56 61 66 71 77 82 87 92

15 3 7 12 18 23 28 33 39 44 50 55 61 66 72 77 83 88 94 100

16 3 8 14 19 25 30 36 42 48 54 60 65 71 77 83 89 95 101 107

17 3 9 15 20 26 33 39 45 51 57 64 70 77 83 89 96 102 109 115

18 4 9 16 22 28 35 41 48 55 61 68 75 82 88 95 102 109 116 123

19 0 4 10 17 23 30 37 44 51 58 65 72 80 87 94 101 109 116 123 130

20 0 4 11 18 25 32 39 47 54 62 69 77 84 92 100 107 115 123 130 138

Reproduced rom the Bulletin of the Institute of Educational Research at Indiana University, vol. 1, no. 2; with the permission o the author and the publisher.

For specifc details about using this table to fnd:p-values, see pages 311312; critical values, page 311.



Table 14 Critical Values or Total Number o Runs (V)

The entries in this table are the critical values or a two-tailed test using =0.05. For a one-tailed test, =0.025 an

use only one o the critical values: the smaller critical value or a let-hand critical region, the larger or a right-hand

critical region.

The larger on1

and n2

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

T

hesmallerofn1

andn2

2 2 2 2 2 2 2 2 2 2

6 6 6 6 6 6 6 6 6

3 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3

8 8 8 8 8 8 8 8 8 8 8 8 8 8 8

4 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4

9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10

5 2 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5

10 10 11 11 12 12 12 12 12 12 12 12 12 12 12 12

6 3 3 3 4 4 4 4 5 5 5 5 5 5 6 6

11 12 12 13 13 13 13 14 14 14 14 14 14 14 14

7 3 4 4 5 5 5 5 5 6 6 6 6 6 613 13 14 14 14 14 15 15 15 16 16 16 16 16

8 4 5 5 5 6 6 6 6 6 7 7 7 7

14 14 15 15 16 16 16 16 17 17 17 17 17

9 5 5 6 6 6 7 7 7 7 8 8 8

15 16 16 16 17 17 18 18 18 18 18 18

10 6 6 7 7 7 7 8 8 8 8 9

16 17 17 18 18 18 19 19 19 20 20

11 7 7 7 8 8 8 9 9 9 9

17 18 19 19 19 20 20 20 21 21

12 7 8 8 8 9 9 9 10 10

19 19 20 20 21 21 21 22 2213 8 9 9 9 10 10 10 10

20 20 21 21 22 22 23 23

14 9 9 10 10 10 11 11

21 22 22 23 23 23 24

15 10 10 11 11 11 12

22 23 23 24 24 25

16 11 11 11 12 12

23 24 25 25 25

17 11 12 12 13

25 25 26 26

18 12 13 1326 26 27

19 13 13

27 27

20 14

28

From C. Eisenhart and F. Swed, Tables or testing randomness o grouping in a sequence o alternatives,Annals of Statistics, vol. 14 (1943): 6687. Reprinted by permission.

For specifc details about using this table to fnd: p-values, see pages 315316; critical values, page 315.


23/23

Table 15 Critical Values o Spearmans Rank Correlation Coe cient

The entries in this table are the critical values orsor a two-tailed test at .

For a one-tailed test, the value o shown at the top o the table is double

the value o being used in the hypothesis test.

0 rs

1rs

1

/2 /2

= area of

two tails

n = 0.10 = 0.05 = 0.02 = 0.01

5 0.900

6 0.829 0.886 0.943

7 0.714 0.786 0.893

8 0.643 0.738 0.833 0.881

9 0.600 0.700 0.783 0.833

10 0.564 0.648 0.745 0.794

11 0.536 0.618 0.736 0.818

12 0.497 0.591 0.703 0.780

13 0.475 0.566 0.673 0.745

14 0.457 0.545 0.646 0.716

15 0.441 0.525 0.623 0.689

16 0.425 0.507 0.601 0.666

17 0.412 0.490 0.582 0.645

18 0.399 0.476 0.564 0.625

19 0.388 0.462 0.549 0.608

20 0.377 0.450 0.534 0.591

21 0.368 0.438 0.521 0.576

22 0.359 0.428 0.508 0.56223 0.351 0.418 0.496 0.549

24 0.343 0.409 0.485 0.537

25 0.336 0.400 0.475 0.526

26 0.329 0.392 0.465 0.515

27 0.323 0.385 0.456 0.505

28 0.317 0.377 0.448 0.496

29 0.311 0.370 0.440 0.487

30 0.305 0.364 0.432 0.478

From E. G. Olds, Distribution o sums o squares o rank dierences or small numbers o individuals,Annals of Statistics, vol. 9 (1938), pp. 138148, and amended, vol. 20 (1949), pp. 117118.

Reprinted by permission.For specifc details about using this table to fnd:p-values, see page 320; critical values, page 318.

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