Question

If the regression equation of x on y is given by mx – y + 10 = 0 and the equation of y on x is given by -2x + 5y = 14, determine the value of ‘m’ if the coefficient of correlation between x and y is 1/√10

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Answer 1

Step-by-step explanation:

${\Large{\mathbf{\orange{Question :}}}}$

If the regression equation of x on y is given by mx – y + 10 = 0 and the equation of y on x is given by -2x + 5y = 14, determine the value of ‘m’ if the coefficient of correlation between x and y is 1/√10

${\Large{\mathbf{\orange{Solution :}}}}$

Two regressions line :

${\bold{\mathrm{x \: \: on \: \: y \: \: is \: \: -}}}$

➮ mx – y + 10 = 0

➮ mx = y - 10

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: x \: = \: \frac{1}{m} \: y \: - \: \frac{10}{m}}}}$

${\bold{: \: ⟹ \: \: \: \: \: }}{\bold{\boxed{\mathrm{\pink{ b_{xy} \: = \: \frac{1}{m}}}}}}$

${\bold{\mathrm{y \: \: on \: \: x \: \: is \: \: -}}}$

➮ -2x + 5y = 14

➮ 5y = 2x + 14

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: y \: = \: \frac{2}{5x} \: + \: \frac{14}{5}}}}$

${\bold{: \: ⟹ \: \: \: \: }}{\bold{\boxed{\mathrm{\pink{b_{yx} \: = \: \frac{2}{5}}}}}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: r \: = \: \frac{1}{ \sqrt{10} }}}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: r \: = \: \sqrt{b_{xy} \: × \: b_{yx}}}}}$

Put the values,

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \frac{1}{\sqrt{10}} \: = \: \sqrt{\frac{1}{m} \: × \: \frac{2}{5}}}}}$

Squaring both the sides,

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: \frac{1}{\sqrt{10}} \: = \: \frac{2}{5m}}}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: 5m \: = \: 20}}}$

${\bold{\mathrm{: \: ⟹ \: \: \: \: \: m \: = \: \frac{20}{5}}}}$

$: \: ⟹ \: \: \: \: \: {\bold{\boxed{\mathrm{\orange{ m \: = \: 4}}}}}$