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Question Number 17 by user1 last updated on 25/Jan/15

Find the regression coefficient b_(xy) between x and y for the following data: Σx=24, Σy=44, Σxy=306, Σx^2 =164, Σy^2 =574 and n=4.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{regression}\:\mathrm{coefficient}\:{b}_{{xy}} \:\mathrm{between} \\ $$$${x}\:\mathrm{and}\:{y}\:\mathrm{for}\:\mathrm{the}\:\mathrm{following}\:\mathrm{data}: \\ $$$$\Sigma{x}=\mathrm{24},\:\Sigma{y}=\mathrm{44},\:\Sigma{xy}=\mathrm{306},\:\Sigma{x}^{\mathrm{2}} =\mathrm{164}, \\ $$$$\Sigma{y}^{\mathrm{2}} =\mathrm{574}\:\mathrm{and}\:{n}=\mathrm{4}. \\ $$

Answered by user1 last updated on 30/Oct/14

The given data may be written as Σx_i =24, Σy_i =44, Σx_i y_i =306, Σx_i ^2 =164, Σy_i ^2 =574, and n=4 ∴b_(yx) =(({Σx_i y_i −(((Σx_i )(Σy_i ))/n)})/({Σx_i ^2 −(((Σx_i )^2 )/n)})) =(({306−((24×44)/4)})/({164−(((24)^2 )/4)})) =((42)/(20))=2.1

$$\mathrm{The}\:\mathrm{given}\:\mathrm{data}\:\mathrm{may}\:\mathrm{be}\:\mathrm{written}\:\mathrm{as} \\ $$$$\Sigma{x}_{{i}} =\mathrm{24},\:\Sigma{y}_{{i}} =\mathrm{44},\:\Sigma{x}_{{i}} {y}_{{i}} =\mathrm{306},\:\Sigma{x}_{{i}} ^{\mathrm{2}} =\mathrm{164}, \\ $$$$\Sigma{y}_{{i}} ^{\mathrm{2}} =\mathrm{574},\:\mathrm{and}\:{n}=\mathrm{4} \\ $$$$\therefore{b}_{{yx}} =\frac{\left\{\Sigma{x}_{{i}} {y}_{{i}} −\frac{\left(\Sigma{x}_{{i}} \right)\left(\Sigma{y}_{{i}} \right)}{{n}}\right\}}{\left\{\Sigma{x}_{{i}} ^{\mathrm{2}} −\frac{\left(\Sigma{x}_{{i}} \right)^{\mathrm{2}} }{{n}}\right\}} \\ $$$$=\frac{\left\{\mathrm{306}−\frac{\mathrm{24}×\mathrm{44}}{\mathrm{4}}\right\}}{\left\{\mathrm{164}−\frac{\left(\mathrm{24}\right)^{\mathrm{2}} }{\mathrm{4}}\right\}} \\ $$$$=\frac{\mathrm{42}}{\mathrm{20}}=\mathrm{2}.\mathrm{1} \\ $$

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