Example Compute the total standard deviation the standard

Example : Compute the total standard deviation , the standard error of the estimate , and the correlation coefficient for the date given below , with the least –sequins fit y = 0. 0714 + 0. 839 X

Xi 1 2 3 4 5 6 7 Yi 0. 5 2. 0 4. 0 3. 5 6. 0 5. 5 24. 0 (Yi-Y`)^2 8. 58. 86 2. 04 0. 33. 0051 6. 61 4. 29 22. 71(=Sr) (Yi-0. 0719 -0. 839 Xi)^2 0. 169 0. 563 0. 347 0. 327 0. 59 0. 797 0. 199 2. 991(=St)

Solution : standard deviation (Sy)= (St/(n-1))^0. 5 = ((yi-(y`)^2/(n-1)) ^0. 5 Y` = yi/(n) = 24/7=3. 43 Sy = (22. 71 / (7 -1))^0. 5 = 1. 95 standard error of the estimate Sy/x =( Sr/(n-2))^. 05

Sr = (yi-a. -a 1 xi)^2 Sy/x = 2. 991((7 -2)) ^0. 5 = 0. 774 correlation coefficient (r) =((St –Sr )/St)^0. 5 = ((22. 71 -2. 991)/22. 71)^. 05 = 0. 932 good fit (provided relation is linear)

linearization of nonlinear relationships : linear regression should be used only linear relationships

y = 1 (e)^ 1*X………… take ln for both sides of the equ. : ln(y) = ln ( 1 (e)^ 1*X) = ln ( 1) + ln e^( 1 *X) = ln ( 1) + 1*X*ln (e) = ln ( 1) + 1 *X ln(y) = ln ( 1) + 1 *X ( intercept) (slope)

** y = 2 *(X )^ 2 ……………. . take for both sides of the equ. : log (y) = 2 log (x) + log( 2) slope ** y = 3 = intercept invert the equ to become = +

= * slope + intercept example : the linearization of the power equ. y= 2 * (X)^ 2 yields log (y) = 1. 75 log (x) – 0. 300 find coefficients of the equ : solution : by taking the log of both sides of the equ log (y)= 2 log (X) + log ( 2) = -0. 300 2 = (10)^ -0. 300 = 0. 5 2 = 1. 75 y= 0. 5 (x )^1. 75