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Cumulative Areas of the Standard Normal Distribution
The entries in this table are the cumulative probabilities for the standard
normal distribution z (that is, the normal distribution with mean 0 and
standard deviation 1). The shaded area under the curve of the standard
normal distribution represents the cumulative probability to the left of a
For specific details about using this table to find probabilities, see pages 272–274, 292–294; p -values, pages 375–377. Table 3 was generated
Cumulative Probability
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Cumulative Areas of the Standard Normal Distribution ( continued )
The entries in this table are the cumulative probabilities for the standard
normal distribution z (that is, the normal distribution with mean 0 and
standard deviation 1). The shaded area under the curve of the standard
normal distribution represents the cumulative probability to the left of a
Table 3 was generated using Minitab.
Cumulative Probability
0 z
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TA B L E 6
Critical Values of Student’s t -Distribution
The entries in this table are the critical values of the Student’s t -distribution, for
which the area under the curve is: a) in the right-hand tail, or b) in two tails. See
the illustrations at the bottom of the page.
Area in One Tail
Area in Two Tails
df 0.50 0.20 0.10 0.05 0.02 0. 3 0.765 1.64 2.35 3.18 4.54 5. 4 0.741 1.53 2.13 2.78 3.75 4. 5 0.727 1.48 2.02 2.57 3.36 4.
6 0.718 1.44 1.94 2.45 3.14 3. 7 0.711 1.41 1.89 2.36 3.00 3. 8 0.706 1.40 1.86 2.31 2.90 3. 9 0.703 1.38 1.83 2.26 2.82 3. 10 0.700 1.37 1.81 2.23 2.76 3.
11 0.697 1.36 1.80 2.20 2.72 3. 12 0.695 1.36 1.78 2.18 2.68 3. 13 0.694 1.35 1.77 2.16 2.65 3. 14 0.692 1.35 1.76 2.14 2.62 2. 15 0.691 1.34 1.75 2.13 2.60 2.
16 0.690 1.34 1.75 2.12 2.58 2. 17 0.689 1.33 1.74 2.11 2.57 2. 18 0.688 1.33 1.73 2.10 2.55 2. 19 0.688 1.33 1.73 2.09 2.54 2. 20 0.687 1.33 1.72 2.09 2.53 2.
21 0.686 1.32 1.72 2.08 2.52 2. 22 0.686 1.32 1.72 2.07 2.51 2. 23 0.685 1.32 1.71 2.07 2.50 2. 24 0.685 1.32 1.71 2.06 2.49 2. 25 0.684 1.32 1.71 2.06 2.49 2.
26 0.684 1.31 1.70 2.05 2.47 2. 27 0.684 1.31 1.70 2.05 2.47 2. 28 0.683 1.31 1.70 2.05 2.47 2. 29 0.683 1.31 1.70 2.05 2.46 2. 30 0.683 1.31 1.70 2.04 2.46 2.
35 0.682 1.31 1.69 2.03 2.44 2. 40 0.681 1.30 1.68 2.02 2.42 2. 50 0.679 1.30 1.68 2.01 2.40 2. 70 0.678 1.29 1.67 1.99 2.38 2. 100 0.677 1.29 1.66 1.98 2.36 2.
df 100 0.675 1.28 1.65 1.96 2.33 2.
For specific details about using this table to find confidence coefficients, see pages 415–416, 418; p -values, pages 421–422; critical values, pages 415, 421. Table 6 was generated using Minitab.
t (df, )
= area of one tail
One-tailed example: df = 9 and = 0. t (df, ) = t (9, 0.10) = 1.
- t (df, /2) 0 + t (df, /2)
= area of two tails
/2 /
Two-tailed example: df = 14, = 0.02, 1 – = 0. t (df, /2) = t (14, 0.01) = 2.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
TA B L E 7
Probability-Values for Student’s t -distribution
The entries in this table are the p -values related to the right-hand tail for the cal-
culated t value for the t -distribution of df degrees of freedom.
Degrees of Freedom t 3 4 5 6 7 8 10 12 15 18 21 25 29 35 df g 45 0.0 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. 0.1 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. 0.2 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. 0.3 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. 0.4 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.
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. 0.6 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. 0.7 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. 0.8 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. 0.9 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.
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. 1.1 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. 1.2 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. 1.3 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. 1.4 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.
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. 1.6 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. 1.7 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. 1.8 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. 1.9 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.
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. 2.1 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. 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. 2.3 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. 2.4 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.
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. 2.6 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. 2.7 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. 2.8 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. 2.9 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.
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. 3.1 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. 3.2 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. 3.3 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. 3.4 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.
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. 3.6 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 specific details about using this table to find p -values, see pages 421–422.
0 t
p -value
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TA B L E 9 A
Critical Values of the F Distribution
The entries in this table are critical
values of F for which the area under
the curve to the right is equal to
Degrees of Freedom for Numerator
1 2 3 4 5 6 7 8 9 10 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. 3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8. 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5. 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4. 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3. 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3. 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3. 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2. 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2. 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2. 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2. 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2. 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2. 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2. 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2. 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2. 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2. 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2. 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2. 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2. 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2. 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1. 120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1. 10,000 3.84 3.00 2.61 2.37 2.21 2.10 2.01 1.94 1.88 1.
For specific details about using this table to find p -values, see page 527; critical values, pages 523–524. Table 9A was generated using Minitab.
( a 0.05)
Degrees of Freedom for Denominator
F (df n , df d , 0.05)
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TA B L E 9 A
Critical Values of the F Distribution ( continued )
Table 9A was generated using Minitab.
( a 0.05)
Degrees of Freedom for Denominator
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TA B L E 9 B
Critical Values of the F Distribution ( continued )
Table 9B was generated using Minitab.
( a 0.025)
Degrees of Freedom for Denominator
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
TA B L E 9 C
Critical Values of the F Distribution
The entries in the table are critical
values of F for which the area under
the curve to the right is equal to 0.
Degrees of Freedom for Numerator
1 2 3 4 5 6 7 8 9 10 1 4052. 5000. 5403. 5625. 5764. 5859. 5928. 5981. 6022. 6056. 2 98.5 99.0 99.2 99.2 99.3 99.3 99.4 99.4 99.4 99. 3 34.1 30.8 29.5 28.7 28.2 27.9 27.7 27.5 27.3 27. 4 21.2 18.0 16.7 16.0 15.5 15.2 15.0 14.8 14.7 14. 5 16.3 13.3 12.1 11.4 11.0 10.7 10.5 10.3 10.2 10.
6 13.7 10.9 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7. 7 12.2 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6. 8 11.3 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5. 9 10.6 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5. 10 10.0 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.
11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63 4. 12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4. 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19 4. 14 8.86 6.51 5.56 5.04 4.70 4.46 4.28 4.14 4.03 3. 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.
16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3. 17 8.40 6.11 5.19 4.67 4.34 4.10 3.93 3.79 3.68 3. 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60 3. 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3. 20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.
21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40 3. 22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3. 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3. 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3. 25 7.77 5.57 4.68 4.18 3.86 3.63 3.46 3.32 3.22 3.
30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2. 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2. 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2. 120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2. 10,000 6.64 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.
For specific details about using this table to find p -values, see page 527; critical values, pages 523–524. Table 9C was generated using Minitab.
( a 0.01)
Degrees of Freedom for Denominator
0 1^ F (df n , df d ,^ 0.01 )
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TA B L E 1 0
Confidence Belts for the Correlation Coefficient
The numbers on the curves are sample sizes.
(1 a ) 0.
Scale of r (sample correlation)
Scale of
p
(population correlation coefficient)
For specific details about using this table to find confidence intervals, see page 620. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
