Answer to Question #262877 in Statistics and Probability for favor

Question #262877

 The following sample observations were randomly selected.

X 4 5 3 6 10

Y 4 6 5 7 7


Determine the regression equation.                                                        (6 marks)




1
Expert's answer
2021-11-09T16:31:00-0500
XYXYX2Y244161616563025363515925674236491077010049Sum=2829173186175\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 4 & 4 & 16 & 16 & 16\\ & 5 & 6 & 30 & 25 & 36\\ & 3 & 5 & 15 & 9 & 25\\ & 6 & 7 & 42 & 36 & 49\\ & 10 & 7 & 70 & 100 & 49\\ Sum= & 28 & 29 & 173 & 186 & 175\\ \end{array}Xˉ=1niXi=285=5.6\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{28}{5}=5.6

Yˉ=1niYi=295=5.8\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{29}{5}=5.8

SSXX=i(XiXˉ)2=29.2SS_{XX}=\sum_i(X_i-\bar{X})^2=29.2

SSYY=i(YiYˉ)2=6.8SS_{YY}=\sum_i(Y_i-\bar{Y})^2=6.8

SSXY=i(XiXˉ)(YiYˉ)=10.6SS_{XY}=\sum_i(X_i-\bar{X})(Y_i-\bar{Y})=10.6

m=slope=SSXYSSXX=10.629.2=0.363014m=slope=\dfrac{SS_{XY}}{SS_{XX}}=\dfrac{10.6}{29.2}=0.363014

n=YˉmXˉn=\bar{Y}-m\bar{X}

=5.80.363014(5.6)=5.8-0.363014(5.6)

=3.767123=3.767123

Therefore, we find that the regression equation is:


Y=3.767123+0.363014XY=3.767123+0.363014X



Correlation coefficient:


r=SSXYSSXXSSYYr=\dfrac{SS_{XY}}{\sqrt{SS_{XX}}\sqrt{SS_{YY}}}

=10.629.26.8=0.752246=\dfrac{10.6}{\sqrt{29.2}\sqrt{6.8}}=0.752246

We have strong positive correlation.

The regression equation is:


Y=3.767123+0.363014XY=3.767123+0.363014X


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