HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 622

Additional equations for linear regression The summary statistics such as Σx, Σx2, etc., can be used to define the following quantities: ∑ ∑ ∑ Sxx = n ( xi i =1 − x)2 = (n − 1) ⋅ s 2 x = n i =1 xi 2 − 1  n n i =1 xi   ∑ ∑ ∑ Sy = n ( yi i =1 − y)2 = (n − 1) ⋅ s 2 y = n i =1 yi 2 − 1  n n i =1 yi  2  Sxy = n ( xi i =1 − x)( yi − y)2 = (n − 1) ⋅ sxy = n i =1 xi yi − 1  n n  i=1 xi n i =1 yi   From which it follows that the standard deviations of x and y, and the covariance of x,y are given, respectively, by sx = S xx n −1 , sy = S yy , and n −1 sxy = S yx n −1 Also, the sample correlation coefficient is rxy = S xy . S xx ⋅ S yy In terms of x, y, Sxx, Syy, and Sxy, the solution to the normal equations is: a = y − bx , b= S xy S xx = s xy s 2 x Prediction error The regression curve of Y on x is defined as Y x + ε. If we have a set of n data points (xi, yi), then we can write Yi xi + εI, (i = 1,2,...,n), where Yi = independent, normally distributed random variables with mean (Α + Β⋅xi) and the common variance σ2; εi = independent, normally distributed random variables with mean zero and the common variance σ2. Page 18-51