A textbook of Computer Based Numerical and Statiscal Techniques part 19

A textbook of Computer Based Numerical and Statiscal Techniques part 19. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 166 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES n P n A A2 A3 A4 0 1 1 1 2 2 1 1 2 4 4 2 2 1 3 8 8 4 4 16 From Newton s formula P n P 0 xAP 0 X X 1 A2 P 0 X X -1 x - 2 A3 P 0 x x -1 x - 2 x - 3 A4 P 0 2 3 4 1 x x x - 1 x x -1 x -2 x x -1 x -2 x -3 2 6 24 x2 x x3 3x2 x3 x4 x3 11x2 x 1 x 1 1 2 2 6 6 3 J 24 4 24 4 J x4 x3 11 7 x x 112 7 - x x 1. Ans. 24 12 12 12 Example 6. Find the value of sin 52 from the given table e 45 50 55 60 sin e Sol. Here a 45 h 5 x 52 therefore u x 104 y 104 Ay 104 A2 y 104 A3 y 45 7071 589 50 7660 -57 532 -7 55 8192 -64 468 60 8660 INTERPOLATION WITH EQUAL INTERVAL 167 By forward interpolation formula f a hu f a uAf a u 1 2 f a . 104 f x 7071 589 1-4 0-4 -57 0-4 -6 _7 7880 . . f 52 or sin 52 Example 7. Following are the marks obtained by 492 candidates in a certain examination Marks 0 - 40 40 - 45 45 - 50 50 - 55 55 - 60 60 - 65 No. of Candidates 210 43 54 74 32 79 Find out a No. of candidates if they secure more than 48 but less than 50 marks. b Less than 48 but not less than 45 marks. Sol. First we form the difference table as follows Marks less than No. of Candidates A A2 A3 A 4 A 5 40 210 43 45 253 11 54 9 50 307 20 -71 74 -62 222 55 381 -42 151 32 89 60 413 47 79 65 492 Here h 5 For a x 48 a 40 u 48 _ 40 5 Now on applying Newton s forward difference formula we have f 48 210 x x 11 x 9 L6 X x -71 24 ---------------------------X 222 6 120 168 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 210 - - - - 280 . . No. of students secured more than 48 marks but less than 50 marks 307 - 280 27 for b x 48 a 40 .-. u x x x f 48 210 x ------x 11 ----i x 9 ---------------i -------------- x -71 J 2 6 24 x 222 x 222 120 280 . No. of students secured more than 45 marks but less than 48 marks 280 -253 27 .