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Find the equation of the two lines of regression and hence find correlation coefficient from the following data.

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Mumbai University > Mechanical Engineering > Sem 4 > Applied Mathematics IV

Marks: 8M

Year: Dec 2014

1 Answer
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Let a = 68, b = 69, c = 1

Here n = 8

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$$\overline{X}=a+c\overline{x}=a+c.\ \frac{\sum{x}}{n} = 68 + 1. \frac{0}{8} = 68$$ $$\overline{Y}=b+c\overline{y}=a+c.\ \frac{\sum{y}}{n} = 69 + 1. \frac{-2}{8} = 68.75$$ $$bxy=\ \frac{n\sum{xy-\ \sum{x\sum{y}}}}{n\sum{y^2-{(\sum{y)}}^2}} = \frac{8\left(26\right)-\left(0\right)(-2)}{8\left(52\right)-{(-2)}^2} = 0.5049$$ $$byx=\ \frac{n\sum{xy-\ \sum{x\sum{y}}}}{n\sum{x^2-{(\sum{x)}}^2}} = \frac{8\left(26\right)-\left(0\right)(-2)}{8\left(36\right)-{(0)}^2} = 0.7222$$

$\mathrm{\therefore }$ Regression equation of Y on X is $$Y = \overline{Y}=byx\left(X-\ \overline{X}\right)$$

$\mathrm{\therefore }$ Y - 68.75 = 0.7222(X-68)

$\mathrm{\therefore }$ Y = 0.7222 X + 19.6389

$\mathrm{\therefore }$ Regression equation of X on Y is Y = $\overline{X}=bxy\left(Y-\ \overline{Y}\right)$

$\mathrm{\therefore }$ X - 68 = 0.5049(Y - 68.75)

$\mathrm{\therefore }$ X = 0.5409 Y + 33.2913

Now , r = $\pm$ $\sqrt{byx*bxy}$

= $\pm$ $\sqrt{0.5049*0.7222}$

= $\pm$ 0.6039

Since , byx and bxy are both positive , r is positive

$\boldsymbol{\mathrm{\therefore }}$ r = 0.6039

Ans : Lines of regression are Y = 0.7222 X + 19.6389 and X = 0.5049 Y + 33.2913 coefficient of correlation (r) = 0.6039

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