A textbook of Computer Based Numerical and Statiscal Techniques part 57. 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. | 546 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Under the null hypothesis H0 i ox2 oy2 o2 . the population variances are equal or ii two independent estimates of the population variances are homogeneous then the statistic F is given by F Sx2 Sy2 where i n1 -1 ni 2 Z X - X 2 i i SX2 Sy2 i n2 -1 2 2 Z y -y j i It follows Snedecor s F-distribution with . v1 n1 -1 and v2 n2 -1. Also greater of two variances Sf and Sy2 is to be taken in the numerator and n1 corresponds to the greater variance. The critical values of F for left tail test H0 o2 of against H1 o2 2 are given by F Fn1-1 n2 -1 1-a and for the two tailed test H0 o o against H1 o A oj are given by F Fn1 -1 n2-2 ffl and F Fn1 -1 n2 -2 Fisher s Z-test To test the significance of an observed sample correlation coefficient from an uncorrelated bivariate normal population f-test is used. But in random sample of size n from a normal bivariate population in which P A 0 it is proved that the distribution of r is by no means normal and in the neighbourhood of p 1 its probability curve is extremely skewed even for large n. If p A 0 Fisher s suggested the transformation. Z loge 1T7 tanh r and proved that for small samples the distribution of Z is approximately normal with mean 1 1 p Z n 2 loge 1-p tanh 4 p and variance 1 n - 3 and for large values of n n 50 the approximation is very good. Example 21. Two independent sample of sizes 7 and 6 had the following values Sample A 28 30 32 33 31 29 34 Sample B 29 30 30 24 27 28 Examine whether the samples have been drawn from normal populations having the same variance. Sol. H0 The variance are equal. . G 02 . the samples have been drawn from normal populations with same variance. Hp 02 A o2 TESTING OF HYPOTHESIS 547 Under null hypothesis the test statistic F -2 s s2 s2 Computations for s2 and s2 X1 X - X1 X1- X1 2 X2 X 2- X 2 X2- X 2 2 28 - 3 9 29 1 1 30 - 1 1 30 2 4 32 1 1 30 2 4 33 2 4 24 - 4 16 31 0 0 27 - 1 1 29 - 2 4 28 0 0 34 3 9 28 26 X 1 31 n1
