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We will have the following:
*Firts: We have that the correlation coefficient is given by:
[tex]r=\frac{\sum ^n_{i=1}X_iY_i-\frac{1}{n}(\sum ^n_{i=1}X_i)(\sum ^n_{i=1}Y_i)}{\sqrt[]{\sum^n_{i=1}X^2_i-\frac{1}{n}(\sum^n_{i=1}X_i)^2}\sqrt[]{\sum ^n_{i=1}Y^2_i-\frac{1}{n}(\sum ^n_{i=1}Y_i)^2}}[/tex]*Second: We calculate the means, that is:
[tex]x_m=4.6[/tex]&
[tex]y_m=\text{0}.82[/tex]*Third: We calculate the sums:
[tex]\sum ^n_{i=1}X^2_i-\frac{1}{n}(\sum ^n_{i=1}X_i)^2=123-\frac{23^2}{5}=17.2[/tex][tex]\sum ^n_{i=1}Y^2_i-\frac{1}{n}(\sum ^n_{i=1}Y_i)^2=4.95-\frac{4.1^2}{5}=1.588[/tex][tex]\sum ^n_{i=1}X_iY_i-\frac{1}{n}(\sum ^n_{i=1}X_i)(\sum ^n_{i=1}Y_i)=13.7-\frac{23\cdot4.1}{5}=-5.16[/tex]Fourth: We replace the data:
[tex]r=\frac{-5.16}{\sqrt[]{17.2\cdot1.588}}\Rightarrow r=-0.987[/tex]Thus making the coefficient r = -0.987.
And this would be the scatterplot:
