Use the Paired Data Set. Find the equation for the least-squares regression line

x      y
5      8
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11  13
17  19

Use the Paired Data Set.  
Find the equation for the least-squares regression line

\(\begin{array}{|c|r|r|r|r|} \hline & x & y & x\cdot y & x\cdot x \\ \hline 1 & 5 & 8 & 40 & 25 \\ \hline 2 & 9 & 10 & 90 & 81 \\ \hline 3 & 11 & 13 & 143 & 121 \\ \hline 4 & 17 & 19 & 323 & 289 \\ \hline \text{sum} & 42 & 50 & 596 & 516 \\ \hline \end{array} \)

We find for the values:

\(\begin{array}{lcr} \hline N &=& 4\\ \sum{X} &=& 42 \\ \sum{Y} &=& 50 \\ \sum{XY} &=& 596 \\ \sum{X^2} &=& 516 \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline \text{Slope(b)} &=& \displaystyle \frac{N\cdot \sum{XY} - (\sum{X})\cdot (\sum{Y}) } { N\cdot \sum{X^2} - (\sum{X})^2 } \\\\ &=& \displaystyle \frac{4\cdot 596 - (42)\cdot (50) } { 4\cdot 516 - (42)^2 } \\\\ &=& \frac{2384 - 2100 } { 2064 - 1764 } \\\\ &=& \frac{ 284 } { 300 } \\\\ &=& 0.94666666667 \\ \hline \end{array} \)

\(\begin{array}{|rcll|} \hline \text{Intercept(a)} &=& \displaystyle \frac{ \sum{Y} - b\cdot (\sum{X}) } { N } \\\\ &=& \displaystyle \frac{ 50 - 0.94666666667\cdot (42) } { 4 } \\\\ &=& \frac{ 50 - 0.94666666667\cdot (42) } { 4 } \\\\ &=& \frac{ 50 - 39.76 } { 4 } \\\\ &=& \frac{ 10.24 } { 4 } \\\\ &=& 2.56 \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline \text{Regression Equation(y)} &=& \displaystyle a + bx \\\\ y &=& 2.56 + 0.94\overline{6}\cdot x \\ \hline \end{array} \)