We will need to use the following formulas
[tex]\begin{gathered} X=\frac{1}{N}\sum ^N_{i=1}x_i \\ Y=\frac{1}{N}\sum ^N_{i=1}y_i \\ m=\frac{\sum^N_{i=1}(x_i-X)(y_i-Y)}{\sum^N_{i=1}(x_i-X)^2} \\ \text{and} \\ r=\frac{\sum^N_{i=1}(x_i-X)(y_i-Y)}{\sqrt{\sum^N_{i=1}(x_i-X)^2\sum^N_{i=1}(y_i_{}-Y)}^2^{}} \\ \text{and} \\ Y=mX+b \end{gathered}[/tex]In our case, using a calculator
[tex]X=5.5,Y=555.6\to\text{ means of the values of x and y, respectively}[/tex]Therefore,
[tex]\begin{gathered} SS_x=\sum ^N_{i=1}(x_i-X)^2=82.5 \\ SP=\sum ^N_{i=1}(x_i-X)(y_i-Y)=1153 \\ \Rightarrow m=\frac{1153}{82.5}=13.975\ldots \\ \Rightarrow m\approx14 \end{gathered}[/tex]Finding b,
[tex]\begin{gathered} Y=mX+b \\ \Rightarrow b=Y-mX \\ \Rightarrow b=555.6-\frac{1153}{82.5}\cdot5.5=478.73\ldots \\ \Rightarrow b\approx479 \end{gathered}[/tex]Thus, the equation of the line of best is y=14x+479.
As for the correlation coefficient,
[tex]\begin{gathered} \sum ^N_{i=1}(x_i-X)(y_i-Y)=1153 \\ \sum ^N_{i=1}(x_i-X)^2=SS_x=82.5 \\ and \\ \sum ^N_{i=1}(y_i-Y)^2=SS_y=16118.4 \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} \Rightarrow r=\frac{1153}{\sqrt[]{82.5\cdot16118.4}}=0.9998\ldots\approx1 \\ \Rightarrow r=1 \end{gathered}[/tex]Then, the Correlation Coefficient is r=1.
The r is practically 1, the model is considered a good fit; in fact, it is a perfect fit (strong relationship)
Summation Notation
Consider the expression
[tex]\sum ^N_{i=1}x_i[/tex]Notice that there are 10 values of x in the table, order them in the following way
[tex]x_1=1,x_2=2,\ldots,x_{10}=10[/tex]Then, after setting N=10,
[tex]\sum ^{10}_{i=1}x_i=x_1+x_2+\cdots+x_{10}=1+2+\ldots+10=55[/tex]Another example,
[tex]\begin{gathered} \sum ^N_{i=1}y_i,N=10 \\ \Rightarrow\sum ^{10}_{i=1}y_i=y_1+y_2+\cdots+y_{10}=492+507+\cdots+618 \end{gathered}[/tex]We found that X=5.5 (see above); then,
[tex]\begin{gathered} \sum ^{10}_{i=1}(x_i-X)^2=(x_1-X)^2+(x_2-X)^2+\cdots(x_{10}-X)^2=(1-5.5)^2+(2-5.5)^2+\cdots(10-5.5)^2 \\ \Rightarrow\sum ^{10}_{i=1}(x_i-X)^2=(1-5.5)^2+(2-5.5)^2+\cdots(10-5.5)^2 \end{gathered}[/tex]