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REFERENCES FOR CHAPTER 21 1109 where m ∗ ij = 1 n ij n ij  k=1 X ijk is the mean over the observations in which the first and second factors acted at levels i and j and m ∗ is the common mean of all observations m ∗ = L 1  i=1 L 2  j=1 n ij  k=1 X ijk  L 1  i=1 L 2  j=1 n ij . The hypothesis H 0 that there is no influence of both factors at all levels must be accepted for a given confidence level γ if φ = s 2∗ 1 /s 2∗ 0 < φ γ ,whereφ γ is the γ-quantile of the F - distribution with parameters L 1  i=1 L 2  j=1 (n ij – 1)andL 1 L 2 – 1. Analysis of variance also permits testing the hypothesis that the theoretical expectation can be represented in the form a ij = a (1) i +a (2) i ,wherea (1) i and a (2) i are unknown expectations introduced by the first and second factors under the action at levels i and j. References for Chapter 21 Arora, P. N. and Anand, S. K., Mathematical and Statistical Tables and Formulae, Anmol Publications Pvt Ltd, Delhi, 2002. Bain,L.J.andEngelhardt,M.,Introduction to Probability and Mathematical Statistics, 2nd Edition (Duxbury Classic), Duxbury Press, Boston, 2000. Bean, M. A., Probability: The Science of Uncertainty with Applications to Investments, Insurance, and Engi- neering, Brooks Cole, Stamford, 2000. Beyer, W. H. (Editor), CRC Standard Probability and Statistics Tables and Formulae, CRC Press, Boca Raton, 1990. Bruning, L. and Kintz, B. L., Computational Handbook of Statistics, 4th Edition, Allyn & Bacon, Boston, 1997. Bulmer,M.G.,Principles of Statistics, Dover Publishers, New York, 1979. Burlington, R. S. and May, D., Handbook of Probability and Statistics With Tables, 2nd Edition, McGraw-Hill, New York, 1970. Craig, R. V. and Hogg, A. T., Introduction to Mathematical Statistics, Macmillan Coll. Div., New York, 1970. DeGroot, M. H. and Schervish, M. J., Probability and Statistics, 3rd Edition, Addison Wesley, Boston, 2001. Devore, J. L., Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac), 6th Edition, Duxbury Press, Boston, 2003. Fisher, R. A., Statistical Tables for Biological, Agricultural and Medical Research, Longman Group United Kingdom, Harlow, 1995. Freedman, D., Pisani, R., and Purves, R., Statistics, 3rd Edition, W. W. Norton & Company, New York, 1997. Graham, A., Teach Yourself Statistics, 2nd Edition, McGraw-Hill, New York, 2003. Hines, W. W., Montgomery, D. C., Goldsman, D. M., and Borror, C. M., Probability and Statistics in Engineering, 4th Edition, Wiley, New York, 2003. Hogg, R. V., Craig, A., and McKean, J. W., Introduction to Mathematical Statistics, 6th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2004. Kapadia, A. S., Chan, W., and Moye, L. A., Mathematical Statistics with Applications (Statistics: Textbooks and Monographs), Chapman & Hall/CRC, Boca Raton, 2005. Kokoska, S. and Zwillinger, D. (Editors), CRC Standard Probability and Statistics Tables and Formulae, Student Edition, Chapman & Hall/CRC, Boca Raton, 2000. Langley, R., Practical Statistics Simply Explained, Rev. Edition (Dover Books Explaining Science), Dover Publishers, New York, 1971. Larsen,R.J.andMarx,M.L.,An Introduction to Mathematical Statistics and Its Applications, 4th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2005. Laurencelle, L. and Dupuis, F., Statistical Tables: Exlained and Applied, World Scientific Publishing Co., Hackensack, New Jersey, 2002. Lindley, D. V. and Scott, W. F., New Cambridge Statistical Tables, 2nd Edition, Cambridge University Press, Cambridge, 1995. Lipschutz, S. and Schiller, J., Schaum’s Outline of Introduction to Probability and Statistics, McGraw-Hill, New York, 1998. 1110 MATHEM ATI CA L STATISTIC S Martinez, W. L. and Martinez, A. R., Computational Statistics Handbook with MATLAB, Chapman & Hall/CRC, Boca Raton, 2001. McClave, J.T. and Sincich, T., Statistics, 10th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2005. Mendenhall,W.,Beaver,R.J.,andBeaver,B.M.,Introduction to Probability and Statistics (with CD-ROM), 12th Edition, Duxbury Press, Boston, 2005. Mendenhall, W. and Sincich, T. L., Statistics for Engineering and the Sciences, 4th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 1995. Milton,J.S.andArnold,J.C.,Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences, 2nd Edition, McGraw-Hill, New York, 2002. Miller, I. and Miller, M., John E. Freund’s Mathematical Statistics with Applications, 7th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2003. Montgomery, D. C. and Runger, G. C., Applied Statistics and Probability for Engineers, Student Solutions Manual, 4th Edition, Wiley, New York, 2006. Montgomery,D.C.,Runger,G.C.,andHubele,N.F.,Engineering Statistics, 3th Edition, Wiley, New York, 2003. Murdoch, J. and Barnes, J. A., Statistical Tables for Science, Engineering, Management and Business Studies, Palgrave Macmillan, New York, 1986. Neave, H. R., Statistics Tables: For Mathematicians, Engineers, Economists and the Behavioural Management Sciences (Textbook Binding), Routledge, London, 1998. Owen, D.B., Handbook of Statistical Tables (Addison-Wesley Series in Statistics), Addison Wesley, Boston, 1962. Pestman, W. R. and Alberink, I. B., Mathematical Statistics: Problems and Detailed Solutions (De Gruyter Textbook), Walter de Gruyter, Berlin, New York, 1998. Pham, H. (Editor), Springer Handbook of Engineering Statistics, Springer, New York, 2006. Rice, J. A., Mathematical Statistics and Data Analysis, 2nd Edition, Duxbury Press, Boston, 1994. Rohlf, F. J. and Sokal, R. R., Statistical Tables, 3rd Edition, W. H. Freeman, New York, 1994. Rose, C. and Smith, M. D., Mathematical Statistics with MATHEMATICA, Springer, New York, 2002. Scheaffer, R. L. and McClave, J. T., Probability and Statistics for Engineers, 4th Edition (Statistics), Duxbury Press, Boston, 1994. Seely, J. A., Probability and Statistics for Engineering and Science, 6th Edition, Brooks Cole, Stamford, 2003. Shao, J., Mathematical Statistics, 2nd Edition, Springer, New York, 2003. Shao, J., Mathematical Statistics: Exercises and Solutions, Springer, New York, 2005. Stigler, S. M., Statistics on the Table: The History of Statistical Concepts and Methods, Reprint edition,Harvard University Press, Cambridge, Massachusetts, 2002. Terrell, G. R., Mathematical Statistics: A Unified Introduction (Springer Texts in Statistics), Springer, New York, 1999. Ventsel, H., Th ´ eorie des Probabilit ´ es, Mir, Moscou, 1987. Ventsel, H. and Ovtcharov, L., Probl ´ emes appliqu ´ es de la th ´ eorie des probabilit ´ es, Mir, Moscou, 1988. Wackerly, D., Mendenhall, W., and Scheaffer, R. L., Mathematical Statistics with Applications, 6th Edition, Duxbury Press, Boston, 2001. Walpole, R. E., Myers, R. H., Myers, S. L., and Ye, K., Probability & Statistics for Engineers & Scientists, 8th Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2006. Witte, R. S. and Witte, J. S., Statistics, 7th Edition, Wiley, New York, 2003. Wonnacott, T. H. and Wonnacott, R. J., Introductory Statistics, 5th Edition, Wiley, New York, 1990. Zwillinger, D. (Editor), CRC Standard Mathematical Tables and Formulae, 31st Edition, Chapman & Hall/CRC, Boca Raton, 2001. Part II Mathematical Tables Chapter T1 Finite Sums and Infinite Series T1.1. Finite Sums T1.1.1. Numerical Sum T1.1.1-1. Progressions. Arithmetic progression: 1. n–1  k=0 (a + bk)=an + bn(n – 1) 2 . Geometric progression: 2. n  k=1 aq k–1 = a q n – 1 q – 1 . Arithmetic-geometric progression: 3. n–1  k=0 (a + bk)q k = a(1 – q n )–b(n – 1)q n 1 – q + bq(1 – q n–1 ) (1 – q) 2 . T1.1.1-2. Sums of powers of natural numbers having the form  k m . 1. n  k=1 k = n(n + 1) 2 . 2. n  k=1 k 2 = 1 6 n(n + 1)(2n + 1). 3. n  k=1 k 3 = 1 4 n 2 (n + 1) 2 . 4. n  k=1 k 4 = 1 30 n(n + 1)(2n + 1)(3n 2 + 3n – 1). 5. n  k=1 k 5 = 1 12 n 2 (n + 1) 2 (2n 2 + 2n – 1). 6. n  k=1 k m = n m+1 m + 1 + n m 2 + 1 2 C 1 m B 2 n m–1 + 1 4 C 3 m B 4 n m–3 + 1 6 C 5 m B 6 n m–5 + ···. Here the C k m are binomial coefficients and the B 2k are Bernoulli numbers; the last term in the sum contains n or n 2 . 1113 1114 FINITE SUMS AND INFINITE SERIES T1.1.1-3. Alternating sums of powers of natural numbers,  (–1) k k m . 1. n  k=1 (–1) k k =(–1) n  n – 1 2  ;[m] stands for the integer part of m. 2. n  k=1 (–1) k k 2 =(–1) n n(n + 1) 2 . 3. n  k=1 (–1) k k 3 = 1 8  1 +(–1) n (4n 3 + 6n 2 – 1)  . 4. n  k=1 (–1) k k 4 =(–1) n 1 2 (n 4 + 2n 3 – n). 5. n  k=1 (–1) k k 5 = 1 4  –1 +(–1) n (2n 5 + 5n 4 – 5n 2 + 1)  . T1.1.1-4. Other sums containing integers. 1. n  k=0 (2k + 1)=(n + 1) 2 . 2. n  k=0 (2k + 1) 2 = 1 3 (n + 1)(2n + 1)(2n + 3). 3. n  k=1 k(k + 1)= 1 3 n(n + 1)(n + 2). 4. n  k=1 (k + a)(k + b)= 1 6 n(n + 1)(2n + 1 + 3a + 3b)+nab. 5. n  k=1 kk!=(n + 1)! – 1. 6. n  k=0 (–1) k (2k + 1)=(–1) n (n + 1). 7. n  k=0 (–1) k (2k + 1) 2 = 2(–1) n (n + 1) 2 – 1 2  1 +(–1) n  . T1.1.1-5. Sums containing binomial coefficients.  Throughout Paragraph T1.1.1-5, it is assumed that m = 1, 2, 3, 1. n  k=0 C k n = 2 n . T1.1. FINITE SUMS 1115 2. n  k=0 C m m+k = C m+1 n+m+1 . 3. n  k=0 (–1) k C k m =(–1) n C n m–1 . 4. n  k=0 (k + 1)C k n = 2 n–1 (n + 2). 5. n  k=1 (–1) k+1 kC k n = 0. 6. n  k=1 (–1) k+1 k C k n = n  m=1 1 m . 7. n  k=1 (–1) k+1 k + 1 C k n = n n + 1 . 8. n  k=0 1 k + 1 C k n = 2 n+1 – 1 n + 1 . 9. n  k=0 a k+1 k + 1 C k n = (a + 1) n+1 – 1 n + 1 . 10. p  k=0 C k n C p–k m = C p n+m ; m and p are natural numbers. 11. n–p  k=0 C k n C p+k n = (2n)! (n – p)! (n + p)! . 12. n  k=0 (C k n ) 2 = C n 2n . 13. 2n  k=0 (–1) k (C k 2n ) 2 =(–1) n C n 2n . 14. 2n+1  k=0 (–1) k (C k 2n+1 ) 2 = 0. 15. n  k=1 k(C k n ) 2 = (2n – 1)! [(n – 1)!] 2 . T1.1.1-6. Other numerical sums. 1. n–1  k=1 sin πk n =cot π 2n . . expectations introduced by the first and second factors under the action at levels i and j. References for Chapter 21 Arora, P. N. and Anand, S. K., Mathematical and Statistical Tables and Formulae, Anmol Publications. Computational Handbook of Statistics, 4th Edition, Allyn & Bacon, Boston, 1997. Bulmer,M.G.,Principles of Statistics, Dover Publishers, New York, 1979. Burlington, R. S. and May, D., Handbook of Probability. Insurance, and Engi- neering, Brooks Cole, Stamford, 2000. Beyer, W. H. (Editor), CRC Standard Probability and Statistics Tables and Formulae, CRC Press, Boca Raton, 1990. Bruning, L. and Kintz,

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