In Inferential Statistics, linear regression is a statistical tool to predict the behavior of a system based on the sample of data given. It finds numerous applications in sales forecasting, quality control, inspection systems etc. Linear Regression makes use of "linear relationship" between sample variables to predict the future outcome.

For a two variable regression, we define the Line of Regression of \(y\) on \(x\) as: \[ (y-\bar{y}) = \hat{b}_{yx}(x-\bar{x})\] where \( \bar{y} , \bar{x} \) are the means of \(y, x\) variables, respectively.Also, \( \hat{b}_{yx} \) is the regression coefficient of y on \(x\).

Similarly, we define the Line of Regression of \(x\) on \(y\) as: \[ (x-\bar{x}) = \hat{b}_{xy}(y-\bar{y})\] Consequently,\( \hat{b}_{yx} \) is the regression coefficient of \(x\) on \(y.\)

In a partially destroyed laboratory record of an analysis of a correlation data, the variance \(\sigma_x^2\) of \(x\) was found to be \(9\).

Also, the Regression Equations were found to be: \[ 8x-10y+66=0 \] \[ 40x-18y-214=0 \] If \(\sigma_{y}^2\) denotes the variance of \(y\) then Calculate \( \bar{x},\bar{y}\) and \( \sigma_{y}\)

Submit your answer as: \( \bar{x}+\bar{y}+\sigma_{y}\).

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