Bibliography - Springer978-1-349-21346-7/1.pdf · Bibliography G. C. Archibald, and Lipsey, R. An...
20
Bibliography G. C. Archibald, and Lipsey, R. G. An Introduction to Mathematical Economics, Weidenfeld, 1973. Alph C. Chiang, Fundamental Methods of Mathemat- ical Economics (2nd edn), McGraw-Hill, 1974. M. Chapman, Plain Figures, HMSO, 1986. D. N. Gujarati, Basic Econometrics (2nd edn), McGraw-Hill, 1988. E. J. Kane, Economic Statistics and Econometrics, Harper & Row, 1969. J. Kmenta, Elements of Econometrics (2nd edn), Mac- millan, 1986. J. M. Pearson, Mathematics for Economists, Longman, 1982. G. W. Summers et al., Basic Statistics for Business and Economics (4th edn), Wadsworth, 1985. 257
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Transcript of Bibliography - Springer978-1-349-21346-7/1.pdf · Bibliography G. C. Archibald, and Lipsey, R. An...
Bibliography
G. C. Archibald, and Lipsey, R. G. An Introduction to Mathematical Economics, Weidenfeld, 1973.
Alph C. Chiang, Fundamental Methods of Mathematical Economics (2nd edn), McGraw-Hill, 1974.
M. Chapman, Plain Figures, HMSO, 1986. D. N. Gujarati, Basic Econometrics (2nd edn),
McGraw-Hill, 1988. E. J. Kane, Economic Statistics and Econometrics,
Harper & Row, 1969.
J. Kmenta, Elements of Econometrics (2nd edn), Macmillan, 1986.
J. M. Pearson, Mathematics for Economists, Longman, 1982.
G. W. Summers et al., Basic Statistics for Business and Economics (4th edn), Wadsworth, 1985.
257
Appendix Statistical Tables
Table I The standardised normal distribution 258
Table II t distribution 259 Table III F distribution 262
Table 1 The standardised normal distribution
Example Pr (0 ~ z ~ 1.96) = 0.4750 Pr (z ~ 1.96) = 0.5 - 0.4750 = 0.025
z .00 .01 .02 .03
0.0 .0000 .0040 .0080 .0120 0.1 .0398 .0438 .0478 .0517 0.2 .0793 .0832 .0871 .0910 0.3 .1179 .1217 .1255 .1293 0.4 .1554 .1591 .1628 .1664 0.5 .1915 .1950 .1985 .2019
0.6 .2257 .2291 .2324 .2357 0.7 .2580 .2611 .2642 .2673 0.8 .2881 .2910 .2939 .2967 0.9 .3159 .3186 .3212 .3238 1.0 .3413 .3438 .3461 .3485
1.1 .3643 .3665 .3686 .3708 1.2 .3849 .3869 .3888 .3907 1.3 .4032 .4049 .4066 .4082 1.4 .4192 .4207 .4222 .4236 1.5 .4332 .4345 .4357 .4370
1.6 .4452 .4463 .4474 .4484 1.7 .4554 .4564 .4573 .4582 1.8 .4641 .4649 .4656 .4664 1.9 .4713 .4719 .4726 .4732 2.0 .4772 .4778 .4783 .4788
2.1 .4821 .4826 .4830 .4834 2.2 .4861 .4864 .4868 .4871 2.3 .4893 .4896 .4898 .4901 2.4 .4918 .4920 .4922 .4925 2.5 .4938 .4940 .4941 .4943
2.6 .4953 .4955 .4956 .4957 2.7 .4965 .4966 .4967 .4968 2.8 .4974 .4975 .4976 .4977 2.9 .4981 .4982 .4982 .4983 3.0 .4987 .4987 .4987 .4988
258
.04
.0160
.0557
.0948
.1331
.1700
.2054
.2389
.2704
.2995
.3264
.3508
.3729
.3925
.4099
.4251
.4382
.4495
.4591
.4671
.4738
.4793
.4838
.4875
.4904
.4927
.4945
.4959
.4969
.4977
.4984
.4988
Table IV Chi-square distribution 266 Table V Binomial distribution 267 Table VI Durbin-Watson distribution 270
l
.05 .06 .07 .08 .09
.0199 .0239 .0279 .0319 .0359
.0596 .0636 .0675 .0714 .0753
.0987 .1026 .1064 .1103 .1141
.1368 .1406 .1443 .1480 .1517
.1736 .1772 .1808 .1844 .1879
.2088 .2123 .2157 .2190 .2224
.2422 .2454 .2486 .2517 .2549
.2734 .2764 .2794 .2823 .2852
.3023 .3051 .3078 .3106 .3133
.3289 .3315 .3340 .3365 .3389
.3531 .3554 .3577 .3599 .3621
.3749 .3770 .3790 .3810 .3830
.3944 .3962 .3980 .3997 .4015
.4115 .4131 .4147 .4162 .4177
.4265 .4279 .4292 .4306 .4319
.4394 .4406 .4418 .4429 .4441
.4505 .4515 .4525 .4535 .4545
.4599 .4608 .4616 .4625 .4633
.4678 .4686 .4693 .4699 .4706
.4744 .4750 .4756 .4761 .4767
.4798 .4803 .4808 .4812 .4817
.4842 .4846 .4850 .4854 .4857
.4878 .4881 .4884 .4887 .4890
.4906 .4909 .4911 .4913 .4916
.4929 .4931 .4932 .4934 .4936
.4946 .4948 .4949 .4951 .4952
.4960 .4961 .4962 .4963 .4964
.4970 .4971 .4972 .4973 .4974
.4978 .4979 .4979 .4980 .4981
.4984 .4985 .4985 .4986 .4986
.4989 .4989 .4989 .4990 .4990
Table ll t distribution
df
1 2 3 4 5
6 7 8 9
10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
31 32 33 34 35
36 37 38 39 40
10.75
1.0000 0.8165 0.7649 0.7407 0.7267
0.7176 0.7111 0.7064 0.7027 0.6998
0.6974 0.6955 0.6938 0.6924 0.6912
0.6901 0.6892 0.6884 0.6876 0.6870
0.6864 0.6858 0.6853 0.6848 0.6844
0.6840 0.6837 0.6834 0.6830 0.6828
0.6825 0 .6822 0.6820 0.6818 0 .6816
0.6814 0.6812 0.6810 0.6808 0.6807
3.0777 1.8856 1.6377 1.5332 1.4759
1.4398 1.4149 1.3968 1.3830 1.3722
1.3634 1.3562 1.3502 1.3450 1.3406
1.3368 1.3334 1.3304 1.3277 1.3253
1.3232 1.3212 1.3195 1.3178 1.3163
1.3150 1.3137 1.3125 1.3114 1.3104
1.3095 1.3086 1.3077 1.3070 1.3062
1.3055 1.3049 1.3042 1.3036 1.3031
10.95
6.3138 2.9200 2.3534 2.1318 2.0150
1.9432 1.8946 1.8595 1.8331 1.8125
1.7959 1.7823 1.7709 1.7613 1.7531
1.7459 1.7396 1. 7341 1.7291 1.7247
1.7207 1.7171 1.7139 1.7109 1.7081
1.7056 1.7033 1.7011 1.6991 1.6973
1.6955 1.6939 1.6924 1.6909 1.6896
1.6883 1.6871 1.6860 1.6849 1.6839
10.975
12.7062 4.3027 3.1824 2.7764 2.5706
2.4469 2.3646 2.3060 2.2622 2.2281
2.2010 2.1788 2.1604 2.1448 2.1315
2.1199 2.1098 2.1009 2.0930 2.0860
2.0796 2.0739 2.0687 2.0639 2.0595
2.0555 2.0518 2.0484 2.0452 2.0423
2.0395 2.0369 2.0345 2.0322 2.0301
2.0281 2.0262 2.0244 2.0227 2.0211
Appendix Statistical Tables 259
t distribution for 10 df
~8125)• 0.95 ,. o!-., •.
::· .· ......
X :::.-: ·:{:~~ :;"~ :>
0
1o.99
31.8207 6.9646 4.5407 3.7469 3.3649
3.1427 2.9980 2.8965 2.8214 2.7638
2.7181 2.6810 2.6503 2.6245 2.6025
2.5835 2.5669 2.5524 2.5395 2.5280
2.5177 2.5083 2.4999 2.4922 2.4851
2.4786 2.4727 2.4671 2.4620 2.4573
2.4528 2.4487 2.4448 2.4411 2.4377
2.4345 2.4314 2.4286 2.4258 2.4233
I =+ 1.8125
t0 .995
63.6574 9.9248 5.8409 4.6041 4.0322
3.7074 3.4995 3.3554 3.2498 3.1693
3.1058 3.0545 3.0123 2.9768 2.9467
2.9208 2.8982 2.8784 2.8609 2.8453
2.8314 2.8188 2.8073 2.7969 2.7874
2.7787 2.7707 2.7633 2.7564 2.7500
2.7440 2.7385 2.7333 2.7284 2.7238
2.7195 2.7154 2.7116 2.7079 2.7045
continued on pages 260 and 261
260 Appendix Statistical Tables
Table II t distribution (continued)
df to.1s to.90 lo.9S lo.97S to.99 to.99S
41 0.6805 1.3025 1.6829 2.0195 2.4208 2.7012 42 0.6804 1.3020 1.6820 2.0181 2.4185 2.6981 43 0.6802 1.3016 1.6811 2.0167 2.4163 2.6951 44 0.6801 1.3011 1.6802 2.0154 2.4141 2.6923 45 0.6800 1.3006 1.6794 2.0141 2.4121 2.6896
46 0.6799 1.3002 1.6787 2.0129 2.4102 2.6870 47 0.6797 1.2998 1.6779 2.0117 2.4083 2.6846 48 0.6796 1.2994 1.6772 2.0106 2.4066 2.6822 49 0.6795 1.2991 1.6766 2.0096 2.4049 2.6800 50 0.6794 1.2987 1.6759 2.0086 2.4033 2.6778
51 0.6793 1.2984 1.6753 2.0076 2.4017 2.6757 52 0.6792 1.2980 1.6747 2.0066 2.4002 2.6737 53 0.6791 1.2977 1.6741 2.0057 2.3988 2.6718 54 0.6791 1.2974 1.6736 2.0049 2.3974 2.6700 55 0.6790 1.2971 1.6730 2.0040 2.3961 2.6682
56 0.6789 1.2969 1.6725 2.0032 2.3948 3.6665 57 0.6788 1.2966 1.6720 2.0025 2.3936 2.6649 58 0.6787 1.2963 1.6716 2.0017 2.3924 2.6633 59 0.6787 1.2961 1.6711 2.0010 2.3912 2.6618 60 0.6786 1.2958 1.6706 2.0003 2.3901 2.6603
61 0.6785 1.2956 1.6702 1.9996 2.3890 2.6589 62 0.6785 1.2954 1.6698 1.9990 2.3880 2.6575 63 0.6784 1.2951 1.6694 1.9983 2.3870 2.6561 64 0.6783 1.2949 1.6690 1.9977 2.3860 2.6549 65 0.6783 1.2947 1.6686 1.9971 2.3851 2.6536
66 0.6782 1.2945 1.6683 1.9966 2.3842 2.6524 67 0.6782 1.2943 1.6679 1.9960 2.3833 2.6512 68 0.6781 1.2941 1.6676 1.9955 2.3824 2.6501 69 0.6781 1.2939 1.6672 1.9949 2.3816 2.6490 70 0.6780 1.2938 1.6669 1.9944 2.3808 2.6479
71 0.6780 1.2936 1.6666 1.9939 2.3800 2.6469 72 0.6779 1.2934 1.6663 1.9935 2.3793 2.6459 73 0.6779 1.2933 1.6660 1.9930 2.3785 2.6449 74 0.6778 1.2931 1.6657 1.9925 2.3778 2.6439 75 0.6778 1.2929 1.6654 1.9921 2.3771 2.6430
76 0.6777 1.2928 1.6652 1.9917 2.3764 2.6421 77 0.6777 1.2926 1.6649 1.9913 2.3758 2.6412 78 0.6776 1.2925 1.6646 1.9908 2.3751 2.6403 79 0.6776 1.2924 1.6644 1.9905 2.3745 2.6395 80 0.6776 1.2922 1.6641 1.9901 2.3739 2.6387
81 0.6775 1.2921 1.6639 1.9897 2.3733 2.6379 82 0.6775 1.2920 1.6636 1.9893 2.3727 2.6371 83 0.6775 1.2918 1.6634 1.9890 2.3721 2.6364 84 0.6774 1.2917 1.6632 1.9886 2.3716 2.6356 85 0.6774 1.2916 1.6630 1.9883 2.3710 2.6349
86 0.6774 1.2915 1.6628 1.9879 2.3705 2.6342 87 0.6773 1.2914 1.6626 1.9876 2.3700 2.6335 88 0.6773 1.2912 1.6624 1.9873 2.3695 2.6329 89 0.6773 1.2911 1.6622 1.9870 2.3690 2.6322
Appendix Statistical Tables 261
Table ll t distribution (continued)
df t0.15 to.90 to.9s to.91s to.99 t0.995
90 0.6772 1.2910 1.6620 1.9867 2.3685 2.6316
91 0.6772 1.2909 1.6618 1.9864 2.3680 2.6309 92 0.6772 1.2908 1.6616 1.9861 2.3676 2.6303 93 0.6771 1.2907 1.6614 1.9858 2.3671 2.6297 94 0.6771 1.2906 1.6612 1.9855 2.3667 2.6291 95 0.6771 1.2905 1.6611 1.9853 2.3662 2.6286
96 0.6771 1.2904 1.6609 1.9850 2.3658 2.6280 97 0.6770 1.2903 1.6607 1.9847 2.3654 2.6275 98 0.6770 1.2902 1.6606 1.9845 2.3650 2.6269 99 0.6770 1.2902 1.6604 1.9842 2.3646 2.6264
100 0.6770 1.2901 1.6602 1.9840 2.3642 2.6259
102 0.6769 1.2899 1.6599 1.9835 2.3635 2.6249 104 0.6769 1.2897 1.6596 1.9830 2.3627 2.6239 106 0.6768 1.2896 1.6594 1.9826 2.3620 2.6230 108 0.6768 1.2894 1.6591 1.9822 2.3614 2.6221 110 0.6767 1.2893 1.6588 1.9818 2.3607 2.6213
112 0.6767 1.2892 1.6586 1.9814 2.3601 2.6204 114 0.6766 1.2890 1.6583 1.9810 2.3595 2.6196 116 0.6766 1.2889 1.6581 1.9806 2.3589 2.6189 118 0.6766 1.2888 1.6579 1.9803 2.3584 2.6181 120 0.6765 1.2886 1.6577 1.9799 2.3578 2.6174
122 0.6765 1.2885 1.6574 1.9796 2.3573 2.6167 124 0.6765 1.2884 1.6572 1.9793 2.3568 2.6161 126 0.6764 1.2883 1.6570 1.9790 2.3563 2.6154 128 0.6764 1.2882 1.6568 1.9787 2.3558 2.6148 130 0.6764 1.2881 1.6567 1.9784 2.3554 2.6142
132 0.6764 1.2880 1.6565 1.9781 2.3549 2.6136 134 0.6763 1.2879 1.6563 1.9778 2.3545 2.6130 136 0.6763 1.2878 1.6561 1.9776 2.3541 2.6125 138 0.6763 1.2877 1.6560 1.9773 2.3537 2.6119 140 0.6762 1.2876 1.6558 1.9771 2.3533 2.6114
142 0.6762 1.2875 1.6557 1.9768 2.3529 2.6109 144 0.6762 1.2875 1.6555 1.9766 2.3525 2.6104 146 0.6762 1.2874 1.6554 1.9763 2.3522 2.6099 148 0.6762 1.2873 1.6552 1.9761 2.3518 2.6095 150 0.6761 1.2872 1.6551 1.9759 2.3515 2.6090
200 0.6757 1.2858 1.6525 1.9719 2.3451 2.6006 300 0.6753 1.2844 1.6499 1.9679 2.3388 2.5923 400 0.6751 1.2837 1.6487 1.9659 2.3357 2.5882 500 0.6750 1.2832 1.6479 1.9647 2.3338 2.5857 600 0.6749 1.2830 1.6474 1.9639 2.3326 2.5840
700 0.6748 1.2828 1.6470 1.9634 2.3317 2.5829 800 0.6748 1.2826 1.6468 1.9629 2.3310 2.5820 900 0.6748 1.2825 1.6465 1.9626 2.3305 2.5813
1000 0.6747 1.2824 1.6464 1.9623 2.3301 2.5808 00 0.6745 1.2816 1.6449 1.9600 2.3263 2.5758
D. B. Owen, Handbook of Statistical Tables (Addison-Wesley Publishing Co., Inc., 1962, pp. 28-30). U.S. Department of Energy. Reprinted with permission of the publisher.
262 Appendix Statistical Tables
Table III F distribution
Right tail of the distribution for P = 0.05 (lightface type), 0.01 (boldface type)
F distribution for 50 and 5 df
P (F ..;: 4.44) - 0.95
m1 = Degrees of Freedom for Numerator
mz 1 2 3 4 5 6 7 8 9 10 11 12
1 161 200 216 225 230 234 237 239 241 242 243 244 4052 4999 5403 5625 5764 5859 5928 5981 6022 6056 6082 6106
2 18.51 19.00 19.16 19.25 19.30 19.33 19.36 19.37 19.38 19.39 19.40 19.41 98.49 99.01 99.17 99.25 99.30 99.33 99.34 99.36 99.38 99.40 99.41 99.42
3 10.13 9.55 9.28 9.12 9.01 8.94 8.88 8.84 8.81 8.78 8.76 8.74 34.12 30.81 29.46 28.71 28.24 27.91 27.67 27.49 27.34 27.23 27.13 27.05
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.93 5.91 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.54 14.45 14.37
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.78 4.74 4.70 4.68 16.26 13.27 12.06 11.39 10.97 10.67 10.45 10.27 10.15 10.05 9.96 9.89
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.03 4.00 13.74 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.79 7.72
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.63 3.60 3.57 12.25 9.55 8.45 7.85 7.46 7.19 7.00 6.84 6.71 6.62 6.54 6.47
... 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.34 3.31 3.28 0
11.26 8.65 7.59 7.01 6.63 6.37 6.19 6.03 5.91 5.82 5.74 5.67 .... "' c 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.13 3.10 3.07 ·a 10.56 8.02 6.99 6.42 6.06 5.80 5.62 5.47 5.35 5.26 5.18 5.11 0 c 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.97 2.94 2.91 11)
0 10.04 7.56 6.55 5.99 5.64 5.39 5.21 5.06 4.95 4.85 4.78 4.71 ... 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.86 2.82 2.79 .2 a 9.65 7.20 6.22 5.67 5.32 5.07 4.88 4.74 4.63 4.54 4.46 4.40 0 12 4.75 3.88 3.49 3.26 3.11 3.00 2.92 2.85 2.80 2.76 2.72 2.69
"0 9.33 6.93 5.95 5.41 5.06 4.82 4.65 4.50 4.39 4.30 4.22 4.16 11)
11) ... 13 4.67 3.80 3.41 3.18 3.02 2.92 2.84 2.77 2.72 2.67 2.63 2.60 ~ ._ 9.07 6.70 5.74 5.20 4.86 4.62 4.44 4.30 4.19 4.10 4.02 3.96 0 ~ 14 4.60 3.74 3.34 3.11 2.96 2.85 2.77 2.70 2.65 2.60 2.56 2.53 11)
8.86 6.51 5.56 5.03 4.69 4.46 4.28 4.14 4.03 3.94 3.86 3.80 11) ... 01) 15 4.54 3.68 3.29 3.06 2.90 2.79 2.70 2.64 2.59 2.55 2.51 2.48 11)
0 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.73 3.67 II 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 2.45 2.42
N
t: 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78 3.69 3.61 3.55 17 4.45 3.59 3.20 2.96 2.81 2.70 2.62 2.55 2.50 2.45 2.41 2.38
8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68 3.59 3.52 3.45 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 2.37 2.34
8.28 6.01 5.09 4.58 4.25 4.01 3.85 3.71 3.60 3.51 3.44 3.37 19 4.38 3.52 3.13 2.90 2.74 2.63 2.55 2.48 2.43 2.38 2.34 2.31
8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52 3.43 3.36 3.30 20 4.35 3.49 3.10 2.87 2.71 2.60 2.52 2.45 2.40 2.35 2.31 2.28
8.10 5.85 4.94 4.43 4.10 3.87 3.71 3.56 3.45 3.37 3.30 3.23 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 2.28 2.25
8.02 5.78 4.87 4.37 4.04 3.81 3.65 3.51 3.40 3.31 3.24 3.17 22 4.30 3.44 3.05 2.82 2.66 2.55 2.47 2.40 2.35 2.30 2.26 2.23
7.94 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35 3.26 3.18 3.12 23 4.28 3.42 3.03 2.80 2.64 2.53 2.45 2.38 2.32 2.28 2.24 2.20
7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30 3.21 3.14 3.07 24 4.26 3.40 3.01 2.78 2.62 2.51 2.43 2.36 2.30 2.26 2.22 2.18
7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.25 3.17 3.09 3.03 25 4.24 3.38 2.99 2.76 2.60 2.49 2.41 2.34 2.28 2.24 2.20 2.16
7.77 5.57 4.68 4.18 3.86 3.63 3.46 3.32 3.21 3.13 3.05 2.99 26 4.22 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 2.18 2.15
7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.17 3.09 3.02 2.96
Appendix Statistical Tables 263
Table III F distribution (continued)
m 1 = Degrees of Freedom for Numerator
14 16 20 24 30 40 50 75 100 200 500 00 mz
245 246 248 249 250 251 252 253 253 254 254 254 1 6142 6169 6208 6234 6258 6286 6302 6323 6334 6352 6361 6366
19.42 19.43 19.44 19.45 19.46 19.47 19.47 19.48 19.49 19.49 19.50 19.50 2 99.43 99.44 99.45 99.46 99.47 99.48 99.48 99.49 99.49 99.49 99.50 99.50
8.71 8.69 8.66 8.64 8.62 8.60 8.58 8.57 8.56 8.54 8.54 8.53 3 26.92 26.83 26.69 26.60 26.50 26.41 26.35 26.27 26.23 26.18 26.14 26.12 5.87 5.84 5.80 5.77 5.74 5.71 5.70 5.68 5.66 5.65 5.64 5.63 4
14.24 14.15 14.02 13.93 13.83 13.74 13.69 13.61 13.57 13.52 13.48 13.46 4.64 4.60 4.56 4.53 4.50 4.46 4.44 4.42 4.40 4.38 4.37 4.36 5 9.77 9.68 9.55 9.47 9.38 9.29 9.24 9.17 9.13 9.07 9.04 9.02 3.96 3.92 3.87 3.84 3.81 3.77 3.75 3.72 3.71 3.69 3.68 3.67 6 7.60 7.52 7.39 7.31 7.23 7.14 7.09 7.02 6.99· 6.94 6.90 6.88 3.52 3.49 3.44 3.41 3.38 3.34 3.32 3.29 3.28 3.25 3.24 3.23 7 6.35 6.27 6.15 6.07 5.98 5.90 5.85 5.78 5.75 5.70 5.67 5.65 3.23 3.20 3.15 3.12 3.08 3.05 3.03 3.00 2.98 2.96 2.94 2.93 8 5.56 5.48 5.36 5.28 5.20 5.11 5.06 5.00 4.96 4.91 4.88 4.86
.... 0 .....
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2.55 2.51 2.46 2.42 2.38 2.34 2.32 2.28 2.26 2.24 2.22 2.21 13 Q) .... 3.85 3.78 3.67 3.59 3.51 3.42 3.37 3.30 3.27 3.21 3.18 3.16 .... ._ 2.48 2.44 2.39 2.35 2.31 2.27 2.24 2.21 2.19 2.16 2.14 2.13 14 0
3.43 3.34 3.26 3.21 3.14 3.11 3.06 3.02 3.00 </>
3.70 3.62 3.51 Q) Q)
2.43 2.39 2.33 2.29 2.25 2.21 2.18 2.15 2.12 2.10 2.08 2.07 15 .... 00
3.56 3.48 3.36 3.29 3.20 3.12 3.07 3.00 2.97 2.92 2.89 2.87 Q)
Cl 2.37 2.33 2.28 2.24 2.20 2.16 2.13 2.09 2.07 2.04 2.02 2.01 16 II 3.45 3.37 3.25 3.18 3.10 3.01 2.96 2.89 2.86 2.80 2.77 2.75 N
2.33 2.29 2.23 2.19 2.15 2.11 2.08 2.04 2.02 1.99 1.97 1.96 17 E
3.35 3.27 3.16 3.08 3.00 2.92 2.86 2.79 2.76 2.70 2.67 2.65 2.29 2.25 2.19 2.15 2.11 2.07 2.04 2.00 1.98 1.95 1.93 1.92 18 3.27 3.19 3.07 3.00 2.91 2.83 2.78 2.71 2.68 2.62 2.59 2.57 2.26 2.21 2.15 2.11 2.07 2.02 2.00 1.96 1.94 1.91 1.90 1.88 19 3.19 3.12 3.00 2.92 2.84 2.76 2.70 2.63 2.60 2.54 2.51 2.49 2.23 2.18 2.12 2.08 2.04 1.99 1.96 1.92 1.90 1.87 1.85 1.84 20 3.13 3.05 2.94 2.86 2.77 2.69 2.63 2.56 2.53 2.47 2.44 2.42 2.20 2.15 2.09 2.05 2.00 1.96 1.93 1.89 1.87 1.84 1.82 1.81 21 3.07 2.99 2.88 2.80 2.72 2.63 2.58 2.51 2.47 2.42 2.38 2.36 2.18 2.13 2.07 2.03 1.98 1.93 1.91 1.87 1.84 1.81 1.80 1.78 22 3.02 2.94 2.83 2.75 2.67 2.58 2.53 2.46 2.42 2.37 2.33 2.31 2.14 2.10 2.04 2.00 1.96 1.91 1.88 1.84 1.82 1.79 1.77 1.76 23 2.97 2.89 2.78 2.70 2.62 2.53 2.48 2.41 2.37 2.32 2.28 2.26 2.13 2.09 2.02 1.98 1.94 1.89 1.86 1.82 1.80 1.76 1.74 1.73 24 2.93 2.85 2.74 2.66 2.58 2.49 2.44 2.36 2.33 2.27 2.23 2.21 2.11 2.06 2.00 1.96 1.92 1.87 1.84 1.80 1.77 1.74 1.72 1.71 25 2.89 2.81 2.70 2.62 2.54 2.45 2.40 2.32 2.29 2.23 2.19 2.17 2.10 2.05 1.99 1.95 1.90 1.85 1.82 1.78 1.76 1.72 1.70 1.69 26 2.86 2.77 2.66 2.58 2.50 2.41 2.36 2.28 2.25 2.19 2.15 2.13
continued on pages 264 and 265
264 Appendix Statistical Tables
Table ill F distribution (continued)
m1 = Degrees of Freedom for Numerator
m2 1 2 3 4 5 6 7 8 9 10 11 12
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.30 2.25 2.20 2.16 2.13 7.68 5.49 4.60 4.11 3.79 3.56 3.39 3.26 3.14 3.06 2.98 2.93
28 4.20 3.34 2.95 2.71 2.56 2.44 2.36 2.29 2.24 2.19 2.15 2.12 7.64 5.45 4.57 4.07 3.76 3.53 3.36 3.23 3.11 3.03 2.95 2.90
29 4.18 3.33 2.93 2.70 2.54 2.43 2.35 2.28 2.22 2.18 2.14 2.10 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.08 3.00 2.92 2.87
30 4.17 3.32 2.92 2.69 2.53 2.42 2.34 2.27 2.21 2.16 2.12 2.09 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.06 2.98 2.90 2.84
32 4.15 3.30 2.90 2.67 2.51 2.40 2.32 2.25 2.19 2.14 2.10 2.07 7.50 5.34 4.46 3.97 3.66 3.42 3.25 3.12 3.01 2.94 2.86 2.80
34 4.13 3.28 2.88 2.65 2.49 2.38 2.30 2.23 2.17 2.12 2.08 2.05 7.44 5.29 4.42 3.93 3.61 3.38 3.21 3.08 2.97 2.89 2.82 2.76
36 4.11 3.26 2.86 2.63 2.48 2.36 2.28 2.21 2.15 2.10 2.06 2.03 7.39 5.25 4.38 3.89 3.58 3.35 3.18 3.04 2.94 2.86 2.78 2.72
38 4.10 3.25 2.85 2.62 2.46 2.35 2.26 2.19 2.14 2.09 2.05 2.02 ... 7.35 5.21 4.34 3.86 3.54 3.32 3.15 3.02 2.91 2.82 2.75 2.69 0 -l:'l 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.07 2.04 2.00
·§ 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.88 2.80 2.73 2.66 0 42 4.07 3.22 2.83 2.59 2.44 2.32 2.24 2.17 2.11 2.06 2.02 1.99 = Cl) 7.27 5.15 4.29 3.80 3.49 3.26 3.10 2.96 2.86 2.77 2.70 2.64 0 ... 44 4.06 3.21 2.82 2.58 2.43 2.31 2.23 2.16 2.10 2.05 2.01 1.98 .s 7.24 5.12 4.26 3.78 3.46 3.24 3.07 2.94 2.84 2.75 2.68 2.62
E3 46 4.05 3.20 2.81 2.57 2.42 2.30 2.22 2.14 2.09 2.04 2.00 1.97 0 '0 7.21 5.10 4.24 3.76 3.44 3.22 3.05 2.92 2.82 2.73 2.66 2.60
Cl) Cl) 48 4.04 3.19 2.80 2.56 2.41 2.30 2.21 2.14 2.08 2.03 1.99 1.96 ... ~ 7.19 5.08 4.22 3.74 3.42 3.20 3.04 2.90 2.80 2.71 2.64 2.58 ..... 0 50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.02 1.98 1.95 "' 7.17 5.06 4.20 3.72 3.41 3.18 3.02 2.88 2.78 2.70 2.62 2.56 Cl) Cl) ... 55 4.02 3.17 2.78 2.54 2.38 2.27 2.18 2.11 2.05 2.00 1.97 1.93 till Cl)
7.12 5.01 4.16 3.68 3.37 3.15 2.98 2.85 2.75 2.66 2.59 2.53 0 II 60 4.00 3.15 2.76 2.52 2.37 2.25 2.17 2.10 2.04 1.99 1.95 1.92
€ 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.56 2.50 65 3.99 3.14 2.75 2.51 2.36 2.24 2.15 2.08 2.02 1.98 1.94 1.90
7.04 4.95 4.10 3.62 3.31 3.09 2.93 2.79 2.70 2.61 2.54 2.47 70 3.98 3.13 2.74 2.50 2.35 2.23 2.14 2.07 2.01 1.97 1.93 1.89
7.01 4.92 4.08 3.60 3.29 3.07 2.91 2.77 2.67 2.59 2.51 2.45 80 3.96 3.11 2.72 2.48 2.33 2.21 2.12 2.05 1.99 1.95 1.91 1.88
6.96 4.88 4.04 3.56 3.25 3.04 2.87 2.74 2.64 2.55 2.48 2.41 100 3.94 3.09 2.70 2.46 2.30 2.19 2.10 2.03 1.97 1.92 1.88 1.85
6.90 4.82 3.98 3.51 3.20 2.99 2.82 2.69 2.59 2.51 2.43 2.36 125 3.92 3.07 2.68 2.44 2.29 2.17 2.08 2.01 1.95 1.90 1.86 1.83
6.84 4.78 3.94 3.47 3.17 2.95 2.79 2.65 2.56 2.47 2.40 2.33 150 3.91 3.06 2.67 2.43 2.27 2.16 2.07 2.00 1.94 1.89 1.85 1.82
6.81 4.75 3.91 3.44 3.14 2.92 2.76 2.62 2.53 2.44 2.37 2.30 200 3.89 3.04 2.65 2.41 2.26 2.14 2.05 1.98 1.92 1.87 1.83 1.80
6.76 4.71 3.88 3.41 3.11 2.90 2.73 2.60 2.50 2.41 2.34 2.28 400 3.86 3.02 2.62 2.39 2.23 2.12 2.03 1.96 1.90 1.85 1.81 1.78
6.70 4.66 3.83 3.36 3.06 2.85 2.69 2.55 2.46 2.37 2.29 2.23 1000 3.85 3.00 2.61 2.38 2.22 2.10 2.02 1.95 1.89 1.84 1.80 1.76
6.66 4.62 3.80 3.34 3.04 2.82 2.66 2.53 2.43 2.34 2.26 2.20 00 3.84 2.99 2.60 2.37 2.21 2.09 2.01 1.94 1.88 1.83 1.79 1.75
6.64 4.60 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.24 2.18
Appendix Statistical Tables 265
Table III F distribution (continued)
m1 = Degrees of Freedom for Numerator
14 16 20 24 30 40 50 75 100 200 500 00 m2
2.08 2.03 1.97 1.93 1.88 1.84 1.80 1.76 1.74 1.71 1.68 1.67 27 2.83 2.74 2.63 2.55 2.47 2.38 2.33 2.25 2.21 2.16 2.12 2.10 2.06 2.02 1.96 1.91 1.87 1.81 1.78 1.75 1.72 1.69 1.67 1.65 28 2.80 2.71 2.60 2.52 2.44 2.35 2.30 2.22 2.18 2.13 2.09 2.06 2.05 2.00 1.94 1.90 1.85 1.80 1.77 1.73 1.71 1.68 1.65 1.64 29 2.77 2.68 2.57 2.49 2.41 2.32 2.27 2.19 2.15 2.10 2.06 2.03 2.04 1.99 1.93 1.89 1.84 1.79 1.76 1.72 1.69 1.66 1.64 1.62 30 2.74 2.66 2.55 2.47 2.38 2.29 2.24 2.16 2.13 2.07 2.03 2.01 2.02 1.97 1.91 1.86 1.82 1.76 1.74 1.69 1.67 1.64 1.61 1.59 32 2.70 2.62 2.51 2.42 2.34 2.25 2.20 2.12 2.08 2.02 1.98 1.96 2.00 1.95 1.89 1.84 1.80 1.74 1.71 1.67 1.64 1.61 1.59 1.57 34 2.66 2.58 2.47 2.38 2.30 2.21 2.15 2.08 2.04 1.98 1.94 1.91 1.98 1.93 1.87 1.82 1.78 1.72 1.69 1.65 1.62 1.59 1.56 1.55 36 2.62 2.54 2.43 2.35 2.26 2.17 2.12 2.04 2.00 1.94 1.90 1.87 1.96 1.92 1.85 1.80 1.76 1.71 1.67 1.63 1.60 1.57 1.54 1.53 38 2.59 2.51 2.40 2.32 2.22 2.14 2.08 2.00 1.97 1.90 1.86 1.84 1.95 1.90 1.84 1.79 1.74 1.69 1.66 1.61 1.59 1.55 1.53 1.51 40 2.56 2.49 2.37 2.29 2.20 2.11 2.05 1.97 1.94 1.88 1.84 1.81 1.94 1.89 1.82 1.78 1.73 1.68 1.64 1.60 1.57 1.54 1.51 1.49 42 2.54 2.46 2.35 2.26 2.17 2.08 2.02 1.94 1.91 1.85 1.80 1.78 1.92 1.88 1.81 1.76 1.72 1.66 1.63 1.58 1.56 1.52 1.50 1.48 44 2.52 2.44 2.32 2.24 2.15 2.06 2.00 1.92 1.88 1.82 1.78 1.75 1.91 1.87 1.80 1.75 1.71 1.65 1.62 1.57 1.54 1.51 1.48 1.46 46 2.50 2.42 2.30 2.22 2.13 2.04 1.98 1.90 1.86 1.80 1.76 1.72 1.90 1.86 1.79 1.74 1.70 1.64 1.61 1.56 1.53 1.50 1.47 1.45 48 2.48 2.40 2.28 2.20 2.11 2.02 1.96 1.88 1.84 1.78 1.73 1.70 1.90 1.85 1.78 1.74 1.69 1.63 1.60 1.55 1.52 1.48 1.46 1.44 50 2.46 2.39 2.26 2.18 2.10 2.00 1.94 1.86 1.82 1.76 1.71 1.68 1.88 1.83 1.76 1.72 1.67 1.61 1.58 1.52 1.50 1.46 1.43 1.41 55 2.43 2.35 2.23 2.15 2.06 1.96 1.90 1.82 1.78 1.71 1.66 1.64 1.86 1.81 1.75 1.70 1.65 1.59 1.56 1.50 1.48 1.44 1.41 1.39 60 2.40 2.32 2.20 2.12 2.03 1.93 1.87 1.79 1.74 1.68 1.63 1.60 1.85 1.80 1.73 1.68 1.63 1.57 1.54 1.49 1.46 1.42 1.39 1.37 65 2.37 2.30 2.18 2.09 2.00 1.90 1.84 1.76 1.71 1.64 1.60 1.56 1.84 1.79 1.72 1.67 1.62 1.56 1.53 1.47 1.45 1.40 1.37 1.35 70 2.35 2.28 2.15 2.07 1.98 1.88 1.82 1.74 1.69 1.62 1.56 1.53 1.82 1.77 1.70 1.65 1.60 1.54 1.51 1.45 1.42 1.38 1.35 1.32 80 2.32 2.24 2.11 2.03 1.94 1.84 1.78 1.70 1.65 1.57 1.52 1.49 1.79 1.75 1.68 1.63 1.57 1.51 1.48 1.42 1.39 1.34 1.30 1.28 100 2.26 2.19 2.06 1.98 1.89 1.79 1.73 1.64 1.59 1.51 1.46 1.43 1.77 1.72 1.65 1.60 1.55 1.49 1.45 1.39 1.36 1.31 1.27 1.25 125 2.23 2.15 2.03 1.94 1.85 1.75 1.68 1.59 1.54 1.46 1.40 1.37 1.76 1.71 1.64 1.59 1.54 1.47 1.44 1.37 1.34 1.29 1.25 1.22 150 2.20 2.12 2.00 1.91 1.83 1.72 1.66 1.56 1.51 1.43 1.37 1.33 1.74 1.69 1.62 1.57 1.52 1.45 1.42 1.35 1.32 1.26 1.22 1.19 200 2.17 2.09 1.97 1.88 1.79 1.69 1.62 1.53 1.48 1.39 1.33 1.28 1.72 1.67 1.60 1.54 1.49 1.42 1.38 1.32 1.28 1.22 1.16 1.13 400 2.12 2.04 1.92 1.84 1.74 1.64 1.57 1.47 1.42 1.32 1.24 1.19 1.70 1.65 1.58 1.53 1.47 1.41 1.36 1.30 1.26 1.19 1.13 1.08 1000 2.09 2.01 1.89 1.81 1.71 1.61 1.54 1.44 1.38 1.28 1.19 1.11 1.69 1.64 1.57 1.52 1.46 1.40 1.35 1.28 1.24 1.17 1.11 1.00 00
2.07 1.99 1.87 1.79 1.69 1.59 1.52 1.41 1.36 1.25 1.15 1.00
Reprinted by permission from Statistical Methods, 6th edition, by George W. Snedecor and William G. Cochran. Copyright © 1967 by Iowa State University Press, Ames, Iowa.
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0 .4
000
.450
0 .5
000
.550
0 .6
000
.650
0 .7
000
.750
0 .8
000
.850
0 .9
000
.950
0
2 0
.902
5 .8
100
.722
5 .6
400
.562
5 .4
900
.422
5 .3
600
.302
5 .2
500
.202
5 .1
600
.122
5 .0
900
.062
5 .0
400
.022
5 .0
100
.002
5 1
.095
0 .1
800
.255
0 .3
200
.375
0 .4
200
.455
0 .4
800
.495
0 .5
000
.495
0 .4
800
.455
0 .4
200
.375
0 .3
200
.255
0 .1
800
.095
0 2
.002
5 .0
100
.022
5 .0
400
.062
5 .0
900
.122
5 .1
600
.202
5 .2
500
.302
5 .3
600
.422
5 .4
900
.562
5 .6
400
.722
5 .8
100
.902
5
3 0
.857
4 .7
290
.614
1 .5
120
.421
9 .3
430
.274
6 .2
160
.166
4 .1
250
.091
1 .0
640
.042
9 .0
270
.015
6 .0
080
.003
4 .0
010
.000
1 1
.135
4 .2
430
.325
1 .3
840
.421
9 .4
410
.443
6 .4
320
.408
4 .3
750
.334
1 .2
880
.238
9 .1
890
.140
6 .0
960
.057
4 .0
270
.007
1 2
.007
1 .0
270
.057
4 .0
960
.140
6 .1
890
.238
9 .2
880
.334
1 .3
750
.408
4 .4
320
.443
6 .4
410
.421
9 .3
840
.325
1 .2
430
.135
4 3
.000
1 .0
010
.003
4 .0
080
.015
6 .0
270
.042
9 .0
640
.091
1 .1
250
.166
4 .2
160
.274
6 .3
430
.421
9 .5
120
.614
1 .7
290
.857
4
4 0
.814
5 .6
561
.522
0 .4
096
.316
4 .2
401
.178
5 .1
296
.091
5 .0
625
.041
0 .0
256
.015
0 .0
081
.003
9 .0
016
.000
5 .0
001
.000
0 1
.171
5 .2
916
.368
5 .4
096
.421
9 .4
116
.384
5 .3
456
.299
5 .2
500
.200
5 .1
536
.111
5 .0
756
.046
9 .0
256
.011
5 .0
036
.000
5 2
.013
5 .0
486
.097
5 .1
536
.210
9 .2
646
.310
5 .3
456
.367
5 .3
750
.367
5 .3
456
.310
5 .2
646
.210
9 .1
536
.097
5 .0
486
.013
5 3
.000
5 .0
036
.011
5 .0
256
.046
9 .0
756
.111
5 .1
536
.200
5 .2
500
.299
5 .3
456
.384
5 .4
116
.421
9 .4
096
.386
5 .2
916
.171
5 4
.000
0 .0
001
.000
5 .0
016
.003
9 .0
081
.015
0 .0
256
.041
0 .0
625
.091
5 .1
296
.178
5 .2
401
.316
4 .4
096
.522
0 .6
561
.814
5
5 0
.773
8 .5
905
.443
7 .3
277
.237
3 .1
681
.116
0 .0
778
.050
3 .0
312
.018
5 .0
102
.005
3 .0
024
.001
0 .0
003
.000
1 .0
000
.000
0 1
.203
6 .3
280
.391
5 .4
096
.395
5 .3
602
.312
4 .2
592
.205
9 .1
562
.112
8 .0
768
.048
8 .0
284
.014
6 .0
064
.002
2 .0
004
.000
0 2
.021
4 .0
729
.138
2 .2
048
.263
7 .3
087
.336
4 .3
456
.336
9 .3
125
.275
7 .2
304
.181
1 .1
323
.087
9 .0
512
.024
4 .0
081
.001
1 3
.001
1 .0
081
.024
4 .0
512
.087
9 .1
323
.181
1 .2
304
.275
7 .3
125
.336
9 .3
456
.336
4 .3
087
.263
7 .2
048
.138
2 .0
729
.021
4 4
.000
0 .0
004
.002
2 .0
064
.014
6 .0
284
.048
8 .0
768
.112
8 .1
562
.205
9 .2
592
.312
4 .3
602
.395
5 .4
096
.391
5 .3
280
.203
6 5
.000
0 .0
000
.000
1 .0
003
.001
0 .0
024
.005
3 .0
102
.018
5 .0
312
.050
3 .0
778
.116
0 .1
681
.237
3 .3
277
.443
7 .5
905
.773
8
6 0
.735
1 .5
314
.377
1 .2
621
.178
0 .1
176
.075
4 .0
467
.027
7 .0
156
.008
3 .0
041
.001
8 .0
007
.000
2 .0
001
.000
0 .0
000
.000
0 1
.232
1 .3
543
.399
3 .3
932
.356
0 .3
025
.243
7 .1
866
.135
9 .0
938
.060
9 .0
369
.020
5 .0
102
.004
4 .0
015
.000
4 .0
001
.000
0 2
.030
5 .0
984
.176
2 .2
458
.296
6 .3
241
.328
0 .3
110
.278
0 .2
344
.186
1 .1
382
.095
1 .0
595
.033
0 .0
154
.005
5 .0
012
.000
1 3
.002
1 .0
146
.041
5 .0
819
.131
8 .1
852
.235
5 .2
765
.303
2 .3
125
.303
2 .2
765
.235
5 .1
852
.131
8 .0
819
.041
5 .0
146
.002
1 4
.000
1 .0
012
.005
5 .0
154
.033
0 .0
595
.095
1 .1
382
.186
1 .2
344
.278
0 .3
110
.328
0 .3
241
.296
6 .2
458
.176
2 .0
984
.030
5 5
.000
0 .0
001
.000
4 .0
015
.004
4 .0
102
.020
5 .0
369
.060
9 .0
938
.135
9 .1
866
.243
7 .3
025
.356
0 .3
932
.399
3 .3
543
.232
1 6
.000
0 .0
000
.000
0 .0
001
.000
2 .0
007
.001
8 .0
041
.008
3 .0
156
.027
7 .0
467
.074
5 .1
176
.178
0 .2
621
.377
1 .5
314
.735
1
7 0
.698
3 .4
783
.320
6 .2
097
.133
5 .0
824
.049
0 .0
280
.015
2 .0
078
.003
7 .0
016
.000
6 .0
002
.000
1 .0
000
.000
0 .0
000
.000
0 1
.257
3 .3
720
.396
0 .3
670
.311
5 .2
471
.184
8 .1
306
.087
2 .0
547
.032
0 .0
172
.008
4 .0
036
.001
3 .0
004
.000
1 .0
000
.000
0 2
.040
6 .1
240
.209
7 .2
753
.311
5 .3
177
.298
5 .2
613
.214
0 .1
641
.117
2 .0
774
.046
6 .0
250
.011
5 .0
043
.001
2 .0
002
.000
0 3
.003
6 .0
230
.061
7 .1
147
.173
0 .2
269
.267
9 .2
903
.291
8 .2
734
.238
8 .1
935
.144
2 .0
972
.057
7 .0
287
.010
9 .0
026
.000
2 4
.000
2 .0
026
.010
9 .0
287
.057
7 .0
972
.144
2 .1
935
.238
8 .2
734
.291
8 .2
903
.267
9 .2
269
.173
0 .1
147
.061
7 .0
230
.003
6 5
.000
0 .0
002
.001
2 .0
043
.011
5 .0
250
.046
6 .0
774
.117
2 .1
641
.214
0 .2
613
.298
5 .3
177
.311
5 .2
753
.209
7 .1
240
.040
6 6
.000
0 .0
000
.000
1 .0
004
.001
3 .0
036
.008
4 .0
172
.032
0 .0
547
.087
2 .1
306
.184
8 .2
471
.311
5 .3
670
.396
0 .3
720
.257
3 7
.000
0 .0
000
.000
0 .0
000
.000
1 .0
002
.000
6 .0
016
.003
7 .0
078
.015
2 .0
280
.049
0 .0
824
.133
5 .2
097
.320
6 .4
783
.698
3
cont
inue
d on
pag
es 2
68 a
nd 2
69
Tab
le V
. B
inom
ial
dist
ribu
tion
(con
tinue
d)
lt
n X
.5
0 .1
0 .1
5 .2
0 .2
5 .3
0 .3
5 .4
0 .4
5 .5
0 .5
5 .6
0 .6
5 .7
0 .7
5 .8
0 .8
5 .9
0 .9
5
12
0 .5
404
.282
4 .1
422
.068
7 .0
317
.013
8 .0
057
.002
2 .0
008
.000
2 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
1 .3
413
.376
6 .3
012
.206
2 .1
267
.071
2 .0
368
.017
4 .0
075
.002
9 .0
010
.000
3 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
2 .0
988
.230
1 .2
924
.283
5 .2
323
.167
8 .1
088
.063
9 .0
339
.016
1 .0
068
.002
5 .0
008
.000
2 .0
000
.000
0 .0
000
.000
0 .0
000
3 .0
173
.085
2 .1
720
.236
2 .2
581
.239
7 .1
954
.141
9 .0
923
.053
7 .0
277
.012
5 .0
048
.001
5 .0
004
.000
1 .0
000
.000
0 .0
000
4 .0
021
.021
3 .0
683
.132
9 .1
936
.231
1 .2
367
.212
8 .1
700
.120
8 .0
762
.042
0 .0
199
.007
8 .0
024
.000
5 .0
001
.000
0 .0
000
5 .0
002
.003
8 .0
193
.053
2 .1
032
.158
5 .2
039
.227
0 .2
225
.193
4 .1
489
.100
9 .0
591
.029
1 .0
115
.003
3 .0
006
.000
0 .0
000
6 .0
000
.000
5 .0
040
.015
5 .0
401
.079
2 .1
281
.176
6 .2
124
.225
6 .2
124
.176
6 .1
281
.079
2 .0
401
.015
5 .0
040
.000
5 .0
000
7 .0
000
.000
0 .0
006
.003
3 .0
115
.029
1 .0
591
.100
9 .1
489
.193
4 .2
225
.227
0 .2
039
.158
5 .1
032
.053
2 .0
193
.003
8 .0
002
8 .0
000
.000
0 .0
001
.000
5 .0
024
.007
8 .0
199
.042
0 .0
762
.120
8 .1
700
.212
8 .2
367
.231
1 .1
936
.132
9 .0
683
.021
3 .0
021
9 .0
000
.000
0 .0
000
.000
1 .0
004
.001
5 .0
048
.012
5 .0
277
.053
7 .0
923
.141
9 .1
954
.239
7 .2
581
.236
2 .1
720
.085
2 .0
173
10
.000
0 .0
000
.000
0 .0
000
.000
0 .0
002
.000
8 .0
025
.006
8 .0
161
.033
9 .0
639
.108
8 .1
678
.232
3 .2
835
.292
4 .2
301
.098
8 11
.0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
3 .0
010
.002
9 .0
075
.017
4 .0
368
.071
2 .1
267
.206
2 .3
012
.376
6 .3
413
12
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
002
.000
8 .0
022
.005
7 .0
138
.031
7 .0
687
.142
2 .2
824
.540
4
13
0 .5
133
.254
2 .1
209
.055
0 .0
238
.009
7 .0
037
.001
3 .0
004
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
1 .3
512
.367
2 .2
774
.178
7 .1
029
.054
0 .0
259
.011
3 .0
045
.001
6 .0
005
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
2 .1
109
.244
8 .2
937
.268
0 .2
059
.138
8 .0
836
.045
3 .0
220
.009
5 .0
036
.001
2 .0
003
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
3 .0
214
.099
7 .1
900
.245
7 .2
517
.218
1 .1
651
.110
7 .0
660
.034
9 .0
162
.006
5 .0
022
.000
6 .0
001
.000
0 .0
000
.000
0 .0
000
4 .0
028
.027
7 .0
838
.153
5 .2
097
.233
7 .2
222
.184
5 .1
350
.087
3 .0
495
.024
3 .0
101
.003
4 .0
009
.000
1·
.000
0 .0
000
.000
0 5
.000
3 .0
055
.026
6 .0
691
.125
8 .1
803
.215
4 .2
214
.198
9 .1
571
.108
9 .0
656
.003
6 .0
142
.004
7 .0
011
.000
1 .0
000
.000
0 6
.000
0 .0
008
.006
3 .0
230
.055
9 .1
030
.154
6 .1
968
.216
9 .2
095
.177
5 .1
312
.083
3 .0
442
.018
6 .0
058
.001
1 .0
001
.000
0 7
.000
0 .0
001
.001
1 .0
058
.018
6 .0
442
.083
3 .1
312
.177
5 .2
095
.216
9 .1
968
.154
6 .1
030
.055
9 .0
230
.006
3 .0
008
.000
0 8
.000
0 .0
000
.000
1 .0
011
.004
7 .0
142
.033
6 .0
656
.198
9 .1
571
.198
9 .2
214
.215
4 .1
803
.125
8 .0
691
.026
6 .0
055
.000
3 9
.000
0 .0
000
.000
0 .0
001
.000
9 .0
034
.010
1 .0
243
.049
5 .0
873
.135
0 .1
845
.222
2 .2
337
.209
7 .1
535
.083
8 .0
277
.002
8 10
.0
000
.000
0 .0
000
.000
0 .0
001
. 000
6 .0
022
.006
5 .0
162
.034
9 .0
660
.110
7 .1
651
.218
1 .2
517
.245
7 .1
900
.099
7 .0
214
11
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
3 .0
012
.013
6 .0
095
.022
0 .0
453
.083
6 .1
388
.205
9 .2
680
.293
7 .2
448
.110
9 12
.0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
005
.001
6 .0
045
.011
3 .0
259
.054
0 .1
029
.178
7 .2
774
.367
2 .3
512
13
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
4 .0
013
.003
7 .0
097
.023
8 .0
550
.120
9 .2
542
.513
3
14
0 .4
877
.228
8 .1
028
.044
0 .0
178
.006
8 .0
024
.000
8 .0
002
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
1 .3
593
.355
9 .2
539
.153
9 .0
832
.040
7 .0
181
.007
3 .0
027
.000
9 .0
002
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
2 .1
229
.257
0 .2
912
.250
1 .1
802
.113
4 .0
634
.031
7 .0
141
.005
6 .0
019
.000
5 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
3 .0
259
.114
2 .2
056
.250
1 .2
402
.194
3 .1
366
.084
5 .0
462
.022
2 .0
093
.003
3 .0
010
.000
2 .0
000
.000
0 .0
000
.000
0 .0
000
4 .0
037
.034
9 .0
998
.172
0 .2
202
.229
0 .2
022
.154
9 .1
040
.061
1 .0
312
.013
6 .0
049
.001
4 .0
003
.000
0 .0
000
.000
0 .0
000
5 .0
004
.007
8 .0
352
.086
0 .1
468
.196
3 .2
178
.206
6 .1
701
.122
2 .0
762
.040
8 .0
183
.006
6 .0
018
.000
3 .0
000
.000
0 .0
000
6 .0
000
.001
3 .0
093
.032
2 .0
734
.126
2 .1
759
.206
6 .2
088
.183
3 .1
398
.091
8 .0
510
.023
2 .0
082
.002
0 .0
003
.000
0 .0
000
7 .0
000
.000
2 .0
019
.009
2 .0
280
.061
8 .1
082
.157
4 .1
952
.209
5 .1
952
.157
4 .1
082
.061
8 .0
280
.009
2 .0
019
.000
2 .0
000
8 .0
000
.000
0 .0
003
.002
0 .0
082
.023
2 .0
510
.091
8 .1
398
.183
3 .2
088
.206
6 .1
759
.126
2 .0
734
.032
2 .0
093
.001
3 .0
000
9 .0
000
.000
0 .0
000
.000
3 .0
018
.006
6 .0
183
.040
8 .0
762
.122
2 .1
701
.206
6 .2
178
.196
3 .1
468
.086
0 .0
352
.007
8 .0
004
10
.000
0 .0
000
.000
0 .0
000
.000
3 .0
014
.004
9 .0
136
.031
2 .0
611
.104
0 .1
549
.202
2 .2
290
.220
2 .1
720
.099
8 .0
349
.003
7 11
.0
000
.000
0 .0
000
.000
0 .0
000
.000
2 .0
010
.003
3 .0
093
.022
2 .0
462
.084
5 .1
366
.194
3 .2
402
.250
1 .2
056
.114
2 .0
259
12
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
005
.001
9 .0
056
.014
1 .0
317
.063
4 .1
134
.180
2 .2
501
.291
2 .2
570
.122
9 13
.0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
002
.000
9 .0
027
.007
3 .0
181
.040
7 .0
832
.153
9 .2
539
.355
9 .3
593
14
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
2 .0
008
.002
4 .0
068
.017
8 .0
440
.102
8 .2
288
.487
7
Tab
le V
. B
inom
ial
dist
ribu
tion
(con
tinue
d)
:rc n
X
.50
.10
.15
.20
.25
.30
.35
.40
.45
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
15
0 .4
633
.205
9 .0
874
.035
2 .0
134
.004
7 .0
016
.000
5 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
1 .3
658
.343
2 .2
312
.131
9 .0
668
.030
5 .0
126
.004
7 .0
016
.000
5 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
2 .1
348
.266
9 .2
856
.230
9 .1
559
.091
6 .0
476
.021
9 .0
090
.003
2 .0
010
.000
3 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
3 .0
307
.128
5 .2
184
.250
1 .2
252
.170
0 .1
110
.063
4 .0
318
.013
9 .0
052
.001
6 .0
004
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
4 .0
049
.042
8 .1
156
.187
6 .2
252
.218
6 .1
792
.126
8 .0
780
.041
7 .0
191
.007
4 .0
024
.000
6 .0
001
.000
0 .0
000
.000
0 .0
000
5 .0
006
.010
5 .0
449
.103
2 .1
651
.206
1 .2
123
.185
9 .1
404
.091
6 .0
515
.024
5 .0
096
.003
0 .0
007
.000
1.
.000
0 .0
000
.000
0 6
.000
0 .0
019
.013
2 .0
430
.091
7 .1
472
.190
6 .2
066
.191
4 .1
527
.104
8 .0
612
.029
8 .0
116
.003
4 .0
007
.000
1 .0
000
.000
0 7
.000
0 .0
003
.003
0 .0
138
.039
3 .0
811
.131
9 .1
771
.201
3 .1
964
.164
7 .1
181
.071
0 .0
348
.013
1 .0
035
.000
5 .0
000
.000
0 8
.000
0 .0
000
.000
5 .0
035
.013
1 .0
348
.071
0 .1
181
.164
7 .1
964
.201
3 .1
771
.131
9 .0
811
.039
3 .0
138
.003
0 .0
003
.000
0 9
.000
0 .0
000
.000
1 .0
007
.003
4 .0
116
.029
8 .0
612
.104
8 .1
527
.191
4 .2
066
.190
6 .1
472
.091
7 .0
430
.013
2 .0
019
.000
0 10
.0
000
.000
0 .0
000
.000
1 .0
007
.003
0 .0
096
.024
5 .0
515
.091
6 .1
404
.185
9 .2
123
.206
1 .1
651
.103
2 .0
449
.010
5 .0
006
11
.000
0 .0
000
.000
0 .0
000
.000
1 .0
006
.002
4 .0
074
.019
1 .0
417
.078
0 .1
268
.179
2 .2
186
.225
2 .1
876
.115
6 .0
428
.004
9 12
.0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
004
.001
6 .0
052
.013
9 .0
318
.063
4 .1
110
.170
0 .2
252
.250
1 .2
184
.128
5 .0
307
13
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
003
.001
0 .0
032
.009
0 .0
219
.047
6 .0
916
.155
9 .2
309
.285
6 .2
669
.134
8 14
.0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
5 .0
016
.004
7 .0
126
.030
5 .0
668
.131
9 .2
312
.343
2 .3
658
15
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
005
.001
6 .0
047
.013
4 .0
352
.087
4 .2
059
.463
3
16
0 .4
401
.185
3 .0
743
.028
1 .0
100
.003
3 .0
010
.000
3 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
1 .3
706
.329
4 .2
097
.112
6 .0
535
.022
8 .0
087
.003
0 .0
009
.000
2 .0
001
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
2 .1
463
.274
5 .2
775
.211
1 .1
336
.073
2 .0
353
.015
0 .0
056
.001
8 .0
005
.000
1 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
3 .0
359
.142
3 .2
285
.246
3 .2
079
.146
5 .0
888
.046
8 .0
215
.008
5 .0
029
.000
8 .0
002
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
4 .0
061
.051
4 .1
311
.200
1 .2
252
.204
0 .1
553
.101
4 .0
572
.027
8 .0
115
.004
0 .0
011
.000
2 .0
000
.000
0 .0
000
.000
0 .0
000
5 .0
008
.013
7 .0
555
.120
1 .1
802
.209
9 .2
008
.162
3 .1
123
.066
7 .0
337
.014
2 .0
049
.001
3 .0
002
.000
0 .0
000
.000
0 .0
000
6 .0
001
.002
8 .0
180
.055
0 .1
101
.164
9 .1
982
.198
3 .1
684
.122
2 .0
755
.039
2 .0
167
.005
6 .0
014
.000
2 .0
000
.000
0 .0
000
7 .0
000
.000
4 .0
045
.019
7 .0
524
.101
0 .1
524
.188
9 .1
969
.174
6 .1
318
.084
0 .0
442
.018
5 .0
058
.001
2 .0
001
.000
0 .0
000
8 .0
000
.000
1 .0
009
.005
5 .0
197
.048
7 .0
923
.141
7 .1
812
.196
4 .1
812
.141
7 .0
923
.048
7 .0
197
.005
5 .0
009
.000
1 .0
000
9 .0
000
.000
0 .0
001
.001
2 .0
058
.018
5 .0
442
.084
0 .1
318
.174
6 .1
969
.188
9 .1
524
.101
0 .0
524
.019
7 .0
045
.000
4 .0
000
10
.000
0 .0
000
.000
0 .0
002
.001
4 .0
056
.016
7 .0
392
.075
5 .1
222
.168
4 .1
983
.198
2 .1
649
.110
1 .0
550
.018
0 .0
028
.000
1 11
.0
000
.000
0 .0
000
.000
0 .0
002
.001
3 .0
049
.014
2 .0
337
.066
7 .1
123
.162
3 .2
008
.209
9 .1
802
.120
1 .0
555
.013
7 .0
008
12
.000
0 .0
000
.000
0 .0
000
.000
0 .0
002
.001
1 .0
040
.011
5 .0
278
.057
2 .1
014
.155
3 .2
040
.225
2 .2
001
.131
1 .0
514
.006
1 13
.0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
002
.000
8 .0
029
.008
5 .0
215
.046
8 .0
888
.146
5 .2
079
.246
3 .2
285
.142
3 .0
359
14
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
5 .0
018
.005
6 .0
150
.035
3 .0
732
.133
6 .2
111
.277
5 .2
745
.146
3 15
.0
000
.000
0 . 0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
001
.000
2 .0
009
.003
0 .0
087
.022
8 .0
535
.112
6 .2
097
.329
4 .3
706
16
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
0 .0
000
.000
1 .0
003
.001
0 .0
033
.010
0 .0
281
.074
3 .1
853
.440
1
270 Appendix Statistical Tables
Table VI Durbin-Watson distribution: Significance points of dL and du at 0.05 level of significance
k' = 1 k' = 2 k' = 3 k' = 4 k' = 5
n dL du dL du dL• du dL du dL du
15 1.08 1.36 0.95 1.54 0.82 1.75 0.69 1.97 0.56 2.21 16 1.10 1.37 0.98 1.54 0.86 1.73 0.74 1.93 0.62 2.15 17 1.13 1.38 1.02 1.54 0.90 1.71 0.78 1.90 0.67 2.10 18 1.16 1.39 1.05 1.53 0.93 1.69 0.82 1.87 0.71 2.06 19 1.18 1.40 1.08 1.53 0.97 1.68 0.86 1.85 0.75 2.02 20 1.20 1.41 1.10 1.54 1.00 1.68 0.90 1.83 0.79 1.99 21 1.22 1.42 1.13 1.54 1.03 1.67 0.93 1.81 0.83 1.96 22 1.24 1.43 1.15 1.54 1.05 1.66 0.96 1.80 0.86 1.94 23 1.26 1.44 1.17 1.54 1.08 1.66 0.99 1.79 0.90 1.92 24 1.27 1.45 1.19 1.55 1.10 1.66 1.01 1.78 0.93 1.90 25 1.29 1.45 1.21 1.55 1.12 1.66 1.04 1.77 0.95 1.89 26 1.30 1.46 1.22 1.55 1.14 1.65 1.06 1.76 0.98 1.88 27 1.32 1.47 1.24 1.56 1.16 1.65 1.08 1.76 1.01 1.86 28 1.33 1.48 1.26 1.56 1.18 1.65 1.10 1.75 1.03 1.85 29 1.34 1.48 1.27 1.56 1.20 1.65 1.12 1.74 1.05 1.84 30 1.35 1.49 1.28 1.57 1.21 1.65 1.14 1.74 1.07 1.83 31 1.36 1.50 1.30 1.57 1.23 1.65 1.16 1.74 1.09 1.83 32 1.37 1.50 1.31 1.57 1.24 1.65 1.18 1.73 1.11 1.82 33 1.38 1.51 1.32 1.58 1.26 1.65 1.19 1.73 1.13 1.81 34 1.39 1.51 1.33 1.58 1.27 1.65 1.21 1.73 1.15 1.81 35 1.40 1.52 1.34 1.58 1.28 1.65 1.22 1.73 1.16 1.80 36 1.41 1.52 1.35 1.59 1.29 1.65 1.24 1.73 1.18 1.80 37 1.42 1.53 1.36 1.59 1.31 1.66 1.25 1.72 1.19 1.80 38 1.43 1.54 1.37 1.59 1.32 1.66 1.26 1.72 1.21 1.79 39 1.43 1.54 1.38 1.60 1.33 1.66 1.27 1.72 1.22 1.79 40 1.44 1.54 1.39 1.60 1.34 1.66 1.29 1.72 1.23 1.79 45 1.48 1.57 1.43 1.62 1.38 1.67 1.34 1.72 1.29 1.78 50 1.50 1.59 1.46 1.63 1.42 1.67 1.38 1.72 1.34 1.77 55 1.53 1.60 1.49 1.64 1.45 1.68 1.41 1.72 1.38 1.77 60 1.55 1.62 1.51 1.65 1.48 1.69 1.44 1.73 1.41 1.77 65 1.57 1.63 1.54 1.66 1.50 1.70 1.47 1.73 1.44 1.77 70 1.58 1.64 1.55 1.67 1.52 1.70 1.49 1.74 1.46 1.77 75 1.60 1.65 1.57 1.68 1.54 1.71 1.51 1.74 1.49 1.77 80 1.61 1.66 1.59 1.69 1.56 1.72 1.53 1.74 1.51 1.77 85 1.62 1.67 1.60 1.70 1.57 1.72 1.55 1.75 1.52 1.77 90 1.63 1.68 1.61 1.70 1.59 1.73 1.57 1.75 1.54 1.78 95 1.64 1.69 1.62 1.71 1.60 1.73 1.58 1.75 1.56 1.78
100 1.65 1.69 1.63 1.72 1.61 1.74 1.59 1.76 1.57 1.78
Note: n = number of observations k' = number of explanatory variables excluding constant term
Source: J. Durbin and G. S. Watson, 'Testing for Serial Correlation in Least Squares Regression,' Biometrika, vo!. 38, pp. 159-177, 1951. Reprinted with the permission of the authors and the Biometrika trustees.
Appendix Statistical Tables 271
Table VI Durbin-Watson distribution: Significance points of dL and dL at 0.01/eve/ of significance (continued)
k' = 1 k' = 2 k' = 3 k' = 4 k' = 5
n dL du dL du dL du dL du dL du
15 0.81 1.07 0.70 1.25 0.59 1.46 0.49 1.70 0.39 1.96 16 0.84 1.09 0.74 1.25 0.63 1.44 0.53 1.66 0.44 1.90 17 0.87 1.10 0.77 1.25 0.67 1.43 0.57 1.63 0.48 1.85 18 0.90 1.12 0.80 1.26 n ,..,,
u.t~ 1.42 0.61 1.60 0.52 1.80 19 0.93 1.13 0.83 1.26 0.74 1.41 0.65 1.58 0.56 1.77 20 0.95 1.15 0.86 1.27 0.77 1.41 0.68 1.57 0.60 1.74 21 0.97 1.16 0.89 1.27 0.80 1.41 0.72 1.55 0.63 1.71 22 1.00 1.17 0.91 1.28 0.83 1.40 0.75 1.54 0.66 1.69 23 1.02 1.19 0.94 1.29 0.86 1.40 0.77 1.53 0.70 1.67 24 1.04 1.20 0.96 1.30 0.88 1.41 0.80 1.53 0.72 1.66 25 1.05 1.21 0.98 1.30 0.90 1.41 0.83 1.52 0.75 1.65 26 1.07 1.22 1.00 1.31 0.93 1.41 0.85 1.52 0.78 1.64 27 1.09 1.23 1.02 1.32 0.95 1.41 0.88 1.51 0.81 1.63 28 1.10 1.24 1.04 1.32 0.97 1.41 0.90 1.51 0.83 1.62 29 1.12 1.25 1.05 1.33 0.99 1.42 0.92 1.51 0.85 1.61 30 1.13 1.26 1.07 1.34 1.01 1.42 0.94 1.51 0.88 1.61 31 1.15 1.27 1.08 1.34 1.02 1.42 0.96 1.51 0.90 1.60 32 1.16 1.28 1.10 1.35 1.04 1.43 0.98 1.51 0.92 1.60 33 1.17 1.29 1.11 1.36 1.05 1.43 1.00 1.51 0.94 1.59 34 1.18 1.30 1.13 1.36 1.07 1.43 1.01 1.51 0.95 1.59 35 1.19 1.31 1.14 1.37 1.08 1.44 1.03 1.51 0.97 1.59 36 1.21 1.32 1.15 1.38 1.10 1.44 1.04 1.51 0.99 1.59 37 1.22 1.32 1.16 1.38 1.11 1.45 1.06 1.51 1.00 1.59 38 1.23 1.33 1.18 1.39 1.12 1.45 1.07 1.52 1.02 1.58 39 1.24 1.34 1.19 1.39 1.14 1.45 1.09 1.52 1.03 1.58 40 1.25 1.34 1.20 1.40 1.15 1.46 1.10 1.52 1.05 1.58 45 1.29 1.38 1.24 1.42 1.20 1.48 1.16 1.53 1.11 1.58 50 1.32 1.40 1.28 1.45 1.24 1.49 1.20 1.54 1.16 1.59 55 1.36 1.43 1.32 1.47 1.28 1.51 1.25 1.55 1.31 1.59 60 1.38 1.45 1.35 1.48 1.32 1.52 1.28 1.56 1.25 1.60 65 1.41 1.47 1.38 1.50 1.35 1.53 1.31 1.57 1.28 1.61 70 1.43 1.49 1.40 1.52 1.37 1.55 1.34 1.58 1.31 1.61 75 1.45 1.50 1.42 1.53 1.39 1.56 1.37 1.59 1.34 1.62 80 1.47 1.52 1.44 1.54 1.42 1.57 1.39 1.60 1.36 1.62 85 1.48 1.53 1.46 1.55 1.43 1.58 1.41 1.60 1.39 1.63 90 1.50 1.54 1.47 1.56 1.45 1.59 1.43 1.61 1.41 1.64 95 1.51 1.55 1.49 1.57 1.47 1.60 1.45 1.62 1.42 1.64
100 1.52 1.56 1.50 1.58 1.48 1.60 1.46 1.63 1.44 1.65
Index
Accept region, in hypothesis tests, 141
Addition rule, 81 Adjusted coefficient of
determination, R2 , 231-3 Alternate hypothesis, 138 Analysis of variance, in regression,
218 Area, of frequency histogram, 24 Arithmetic mean, 43 Asymptotic efficiency, 123--4 Asymptotic properties of
estimators, 123--4 Autocorrelated disturbance term,
191, 242 Autocorrelation
consequences of, 243 detection of, 243-50 meaning and causes of, 242-3 remedial measures, 250--1
Autocorrelation coefficient, 243 Average, 20
Bar charts multiple, 8 simple, 7 stacked, 9
Basic assumptions of simple regression model,
189-92 see also Breakdown of
Best linear unbiased estimation, 194, 199-201
Binomial distribution, 91 mean of, 93 necessary conditions, 91 normal Hpproximation to, 100 probability function, 91 variance of, 93
Breakdown of basic assumptions in regression
constant variance of u, 236-42 independent u, 242-51 linearity, 236 · non-stochastic variables, 251-2 u normally distributed, 236 zero mean of u, 236
Categorical scale, 4 Causal relationship, 187
272
Central limit theorem, 111, 117 Chi-square distribution, 114--15
degrees of freedom of, 114 use of tables of, 115 with distribution of sample
variance, 116 Chow test, 234--5 Class, in frequency table
boundaries, 21 limits, 18 mid-point, 21 width
Classical Normal Linear Regression (CNLR) Model, 189, 228
Cochrane-Orcutt test, 251 Coefficient of determination,
210--12 Coefficient of variation, 66 Compound events, 81 Computer applications
analysis of variance (in regression), 218
autocorrelation, 249 covariance, 183 correlation, 184, 186 cumulative frequency, 14, 30, 68,
69 cumulative probability
distributions, 92 estimation: of difference in
population means, 168, 170; of population mean, 128--9; in regression, 198--9, 201, 207, 230, 232-3
frequency histograms, 31-34 frequency tables, 12, 29-32 hypothesis tests: of difference in
population means, 168--70; of goodness of fit, 157--61; of population mean, 148--149
joint frequency distributions, 33 kurtosis, 67 line graphs, 13 mean, 52-3 measures of location, 51-4 median, 52-3 mode, 53 multicollinearity, 254 normal distribution, 66, 68, 69
ntiles, 54 p-value, 148, 149, 199, 200 percentiles, 54, 67 prediction, in regression, 220--1 probability distributions, 92 quartiles (Q1, Q3), 52 R2 , 199, 201 random sample, 195--6, 202 range, 67 regression, 198--200, 206--7, 230,
232-3 relative frequency, 14, 30 scattergram (using plot), 182-3,
190, 202 skewness, 67 standard deviation, 52, 67 standard error of estimate, 206 standard error of OLS
estimators, 206--7 sums of squares, in regression,
218 trimmed mean, 52 variance, 67
Computer commands - Minitab alternative, 149, 172, 173 binomial, 92 brief, 198, 200 cdf, 68, 69, 92, 100 code, 30 colpercent, 34--5 covariance, 183 describe, 51-4 DW, 249 histogram, 30--1 let, 223--4 mplot, 13 name, 12 normal, 66, 70, 100 nscores, 69-70, 236 outfile, 13 paper, 13 pdf' 66--8' 92 plot, 68--9, 70, 87, 88, 182-3, 190 pooled, 169, 170, 173 predict, 220, 221 read data, 13 read, 12 regression, 198--9, 230, 232-3 residuals, 245, 247 rowpercent, 34
sample, 195--6, 202 table, 12, 30, 33, 34-5 tinterval, 128-129 totpercent, 33-35 ttest, 148 twosample-t, 167, 168, 170, 172 vif, 254 xincrement, 13, 183, 190 xstart, 13, 183, 190 ystart, 183, 190 yincrement, 190
Computer commands- SPSSx, 14 casewise, 245 crosstabs, 35-38 dependent, 198, 201 descriptive, 54, 67 enter, 198 format-notables, 31 frequencies, 31, 54, 67, 79 histogram, 31, 236 normprob, 236 ntiles, 54 Pearson correlation, 184 plot, 198, 202 recode, 31, 3fr8 regression, 198, 201 residuals, 236, 249 statistics, 54, 67, 207 tables, 35, 36, 38
Conditional probability, 82 Confidence bounds, of predictor,
222 Confidence interval, see Estimation Confidence level, 127 Consistent estimators, 123 Constant variance, of regression
disturbance terms, 190-1 Continuous probability
distributions, 96 and probability density functions,
97 Continuous random variables, 85,
96-7 Continuous variables, 5 Correlation, 181 Correlation coefficient simple, 182,
184-7 and coefficient of determination,
212 significance of, 185-7
Covariance, 102-3, 181-3 of disturbance term, 191
Cumulative frequency·, 22 Cumulative frequency diagrams
(ogives), 27 Cumulative probability density
function ( cdf), 99
Cumulative probability distribution, 87
Deciles, 48 Degrees of freedom, 119
of chi-square, 114-15 Dependent events, 82 Dependent variable (in regression),
187 distribution of, 192 total sum of squares of, 208
Descriptive statistics, 3 Deterministic model, 187 Difference in population means
estimation of, 165-9 hypothesis testing and, 169-73
Difference in population proportions
estimation of, 174-5 hypothesis testing and, 175-7
Difference in sample means, distribution of, 164-5
Difference in sample proportions, sampling distribution of, 174
Discrete random variables, 85 mean of, 88-90 probability function of, 86 variance of, 90
Discrete variables, 5 Dispersion, 20 Distribution of dependent variable
in regression, 192 see also Sampling distribution
Disturbance term, 188 assumptions about, 189-91 estimator of variance of, 201
Durbin-Watson test for non-linearity, 236 for autocorrelation, 247-9 h and m tests, 249
Efficient estimators, 123 Elasticity, in non-linear regression,
223 Estimation, 107, 120
and hypothesis testing compared, 156-7
of difference in population means, 165-9
of difference in population proportions, 174-5
of population mean, 124-30 of population proportion, 130--2 of population variance, 133-4 of the predictor, in regression,
220--1 of regression parameters: simple,
Index 273
194-8; multiple, 229-31 of disturbance term variance, 205 of variance of OLS estimators,
205 Estimators, 107
asymptotically efficient, 123 best linear unbiased (blu), 194,
199-200 consistent, 123 efficient, 123 linear, 123 maximum likelihood, 194, 201 minimum variance (best), 122-3 OLS, 194-5 properties of, 120--4 unbiased, 120
Events, in probability, 77 independent, 82 intersection of, 80--1 union of, 79-80
Expected value in goodness of fit test, 159 of a constant, 101 of binomial random variable, 93 of discrete random variable, 89 of sample mean, 111-12 of sample proportion, 116 of sample variance, 113 of sum of two random variables,
102 Expected value operator, 88-89
rules for, 101-2 Experiment (in probability), 75 Explanatory variable, 187
F distribution, 215-17 Frequency curve, 26 Frequency density, 24 Frequency diagram, 17 Frequency distributions, 11
with grouped data, 18 with single valued classes, 16
Frequency histogram, 23 Frequency polygon, 24 Frequency tables, 11
and class, 11 and open-ended classes, 19
Frequency, defined, 11 cumulative frequency, 22 joint frequency, 28 relative frequency, 21
Gamma distribution, 114 Generalised least squares, 241 Geometric means, 50--1 Goldfeld and Quandt test, 238-40 Goodness of fit
274 Index
hypothesis test of, 157-61 in regression, 207, 210-12
h test, 249 Heteroskedastic disturbance term,
191 Heteroskedasticity
consequences of, 237-8 detection of, 238-41 meaning and causes of, 237 remedial measures and, 241-2
Histogram, frequency, 23 Homoskedastic disturbance term,
190 breakdown of assumption of, 236
Hypotheses, 108 null, 138 types of, 138-9
Hypothesis testing, 120, 137-8 and estimation compared, 156--7 Chow test, 234-5 of difference in population
means, 169-73 of difference in population
proportions, 175-7 of goodness of fit, 157-61 of population mean, 141-52; and
standardised test statistic, 141; and p-value method, 143--4
of population proportion, 152-4 of population variance, 155-6 power of test, 151 of R2, 217 of relationship between variables
in regression, 215-16, 234 of regression parameters: simple,
212-16; multiple, 233--5 significance level and, 139 types of error with, 139-40
Independence, of regression disturbance term, 191
Independent events, 82 Independent random variables, 102 Independent samples, 173 Independent variable, 187 Inferential statistics, see Statistical
inference Interquartile range, 57-8 Intersection of events, 80-1 Interval estimation, see Estimation Interval/ratio scale, 4
Kurtosis, 26
Large (asymptotic) sample properties of estimators, 123--4
Less-than cumulative frequency, 23 Linearity, assumption of in
regression model, 189 breakdown of, 236
Line graphs, 10 Linear estimators, 123
m test, 249 Matched samples, 173 Maximum likelihood estimation, in
regression, 194, 201 Mean
arithmetic, 43 of binomial random variable, 93 of discrete random variable, 89 of distribution of sample means,
111-12 of distribution of sample
proportion, 116 of distribution of sample
variance, 113 (population) estimation of,
124-30 (population) hypothesis tests
and, 141-52 Mean deviation, 45 Mean square error, 122 Mean value prediction, in
regression, 219-20 Measures of dispersion, 41-2 Measures of location, 41
comparison of, 49-50 Measuring scales, 4 Median, 46
from ogives, 29 of grouped data, 46--7 of ungrouped data, 46
Minimum variance ('best') estimators, 122-3
Minitab, 12 see also Computer commands -
Mini tab Mode, 49 More-than cumulative frequency,
23 Multicollinearity, and basic
assumption, 228 consequences of, 253--4 detection of, 254-5 meaning and causes of, 253 remedial measures and, 255
Multiple regression, see Regression Multiplication rule, 82-3
and independent events, 83 Mutually exclusive events (addition
rule), 81-2
n-tiles, 48 Norninalscale, 4 Non-linear regression, see
Regression model Non-stochastic explanatory
variable, 191 breakdown of assumption of,
251-2 Normal approximation to binomial,
100, 118 Normal distribution, 26
goodness of fit test and, 158-61 Normal equations, in OLS
regression, 195 Normal frequency curve, 26 Null hypothesis, 138
Ogives, 27 one-tailed hypothesis tests, 139 Open-ended classes, 19 ordinal scale, 4 ordinary least squares estimation,
194-8 ordinary Least Squares estimators
properties of, 203 sampling distribution of, 202-4
p-value, 143 calculation of, 143--4, 147-8, 153,
154, 155-6, 168, 170, 171-2, 173
Parameter, 4 Parameters, of population, 4, 107 Partial regression parameters,
227-8 Percentage (relative) frequency, 21 Percentiles, 48
from ogives, 28 Permutations, 110 Pie diagrams, 9 Point estimators, see Estimation Pooled variance, 168, 169 Population mean
estimation of, 124-30 hypothesis tests and, 141-52 ungrouped data, 44
Population multiple regression function, 227
Population parameters, 4, 107 Population proportion
estimation of, 130-3 hypothesis tests and, 152-4
Population regression function (PRF), 189, 227
Population regression line, 192, 227 Population regression parameters,
188-9, 227
Population variance estimation of, 133-4 hypothesis tests and, 155--6
Populations, 3 Power of hypothesis test, 151 Prais-Winsten transformation, 251 Prediction, in regression, 219-222 Probability
calculation of, 76--7 conditional, 82 of type I and type II errors,
139-40 of type II errors, 149
Probability density, 97 Probability density function (pdf),
97 cumulative pdf, 99
Probability diagram, 87 Probability distributions, 85
binomial, 91 continuous, 96 cumulative, 87, 92 discrete, 86--87 normal, 97-8
Probability function, 86 binomial, 91
Properties of estimators in general, 120-4 of OLS estimators, 203; with
autocorrelation, 243; with heteroskedasticity, 237-8; with multicollinearity, 253-4; with stochastic explanatory variables, 252
of sample proportion, 131 of sample mean, 124
Proportion hypothesis tests and, 152--4
. (population) estimation of, 130-2
Qualitative data, 5 Quantitative data, 5. Quartiles, 47-8
R2 (coefficient of determination), 210-12
in multiple regression, 231 significance test of, 217
R2 (adjusted coefficient of determination), 231-3
Random sample, 3 Random variables, 85 Range, 17, 56 Realisation, of random variable,
129 Regression model, simple linear,
181-222
basic assumptions of, 189-92 estimation of, 194-8 goodness of fit in, 207-12 hypothesis testing with, 212-17
Regression model, multiple, 227-55 basic assumptions of, 229 estimation of, 229-31 hypothesis tests and, 233-5
Regression model, simple non-linear, 223-4
Regression parameters, see Population regression parameters
Regression sum of squares (SSR), 209
Reject region, in hypothesis tests, 141
Relationship, between variables, 181
Relative frequency, 21 with SPSSx, 30 method, for calculating
probability, 76
Sample covariance, see Covariance Sample mean
distribution of, 109-113 grouped data, 44. ungrouped data, 44
Sample point, 75 Sample regression function (SRF),
193 Sample regression line, 193, 229 Sample size
required in estimation: of population mean, 130; of population proportion, 132
required in hypothesis tests, of population mean, 151
Sample space, 75 Sample statistics, 4, 107 Sample variance, 58-9
distribution of, 113-16 Samples, 3 Sampling distribution
of difference in sample means, 164-5
of OLS estimators: variance of, 200-1' 204-5
of regression disturbance term, 191
of sample mean, 109-13; mean (expected value) of, 110, 111-12; standard error of, 110, 111; variance of, 111-12
of sample proportion, 116--18 of sample statistics, 109
Index 275
of sample variance, 113-16 Sampling error, 3, 75, 120, 121, 127
and required sample size (mean), 130
and required sample size (proportion), 132-3
Scattergram, 182 Significance level, 139 Simple regression, see Regression Skewness, 26 Small sample properties of
estimators, 122-3 SPSSx, 13
see also Computer commands -SPSSx
Standard deviation of binomial random variable, 93 of discrete random variable, 90 grouped data, 59 interpretation, 60, 64 population, 60 sample, 58 ungrouped data, 59
Standard error of distribution of difference in
sample means, 167 of distribution of sample mean,
110, 111 of distribution of sample
proportion, 116 of OLS estimators, 205 of the estimate, in regression,
205 of the predictor, in regression,
220-1 Standard normal distribution, 61-6 Standardised test statistic, 141 Statistical inference, 3, 75, 107
with two populations, 165-77 Statistical Package for the Social
Sciences, see SPSSx Stochastic explanatory variable (in
regression), 191 Stochastic model, 188 Summation operator, 43-4 Sums of squares, in regression,
208-9
t distribution, 125--6 Testing, see Hypothesis testing Total sum of squares (SST), 208 Trimmed mean, 45 Two-tailed hypothesis tests, 139 Type II error and power of test,
151 Types of error in hypothesis tests,
139
276 Index
Unbiased estimators, 120, 122 Unexplained sum of squares (SSE),
209 Union of events, 79-80
Variables, types of, 5 Variance
of a random variable, 121 of binomial random variable, 93
of discrete random variable, 90, 101
of distribution of sample mean, 110-12
of distribution of sample proportion, 116
of OLS estimators, 203-5 of regression disturbance term,
191
population, 59 (population) estimation of, 133-4 sample, 58--59
Variance-covariance matrix, 183 Variation, in regression, 207, 211
z distribution, 61--6 z table, use of, 63--6
