Subject: Applied Mathematics 4

Topic: Correlation & Regression

Difficulty: Medium

Change of scale: u = x-8; v = y-14

$ \bar{u} = \frac{1}{n}(\sum u) = \frac{1}{7}(0) = 0 \\ \bar{v} = \frac{1}{n}(\sum v) = \frac{1}{7}(1) = 0.142857 \\ \sigma_u^2 = \frac{1}{n}(\sum u^2) - (\bar{u})^2 = \frac{1}{7}(28) - 0^2 = 4 \\ \sigma_u = 2 \\ \sigma_v^2 = \frac{1}{n}(\sum v^2) - (\bar{v})^2 = \frac{1}{7}(27) - 1^2 = 2.857142857 \\ \sigma_v = 1.690308 $

$ cov(u,v) = \frac{1}{n}(\sum uv) - \bar{u}\bar{v} = \frac{1}{7}(19) - (0)\frac{1}{7} = 2.714285 \\ r_{uv} = \frac{cov(u,v)}{\sigma_u \sigma_v} = \frac{2.714285}{(2)(1.690308)} = 0.802896 \\ r_{uv} = r_{xy} = r = 0.802896 $

$ b_{uv} = r \frac{\sigma_u}{\sigma_v} = (0.802896) \frac{2}{1.690308} = 0.9499996 \\ b_{uv} = b_{xy} = 0.9499996 \\ b_{vu} = r \frac{\sigma_v}{\sigma_u} = (0.802896) \frac{1.690308}{2} = 0.678570766 \\ b_{yx} = b_{vu} = 0.678570766 $

$ u = x - 8 \\ \bar{u} = \bar{x} - 8 \\ \bar{x} = \bar{u} + 8 = 0 + 8 = 8 \\ v = y - 14 \\ \bar{v} = \bar{y} - 14 \\ \bar{y} = \bar{v} + 14 = 0.142857 + 14 = 14.142857 $

There are two equations of lines of regression:

(i) Equation of line of regression of 'y' on 'x' is given by:

$ (y - \bar{y}) = byx(x - \bar{x}) \\ (y - 14.142857) = 0.678570766(x - 8) \\ y - 14.142857 = 0.678570766x - 5.428566128 \\ y = 0.678570766x + 8.714290872 $

(ii) Equation of line of regression of 'x' on 'y' is given by:

$ (x - \bar{x}) = bxy(y - \bar{y}) \\ (x-8) = (0.9499996)(y - 14.142857) \\ x - 8 = 0.9499996y - 13.43570849 \\ x = 0.9499996y - 13.43570849 + 8 \\ x = 0.9499996y + 5.435708493 $