Find the correlation coefficient of the line of best fit for the points (-3,-40), (1,12), (5,72), (7,137). Explain how you guy your answer. Use the coefficient to describe the correlation of this data.
Accepted Solution
A:
Answer:r = 0.9825; good correlation.
Step-by-step explanation:One formula for the correlation coefficient is [tex]r = \dfrac{n\sum{xy} - \sum{x} \sum{y}}{\sqrt{n\left [\sum{x}^{2}-\left (\sum{x}\right )^{2}\right]\left [\sum{y}^{2}-\left (\sum{y}\right )^{2}\right]}}[/tex]The calculation is not difficult, but it is tedious.
1. Calculate the intermediate numbers
We can display them in a table.
x y xy x² y² -3 -40 120 9 1600
1 12 12 1 144
5 72 360 25 5184
7 137 959 49 18769
Σ = 10 181 1451 84 25697
2. Calculate the correlation coefficient
[tex]r = \dfrac{n\sum{xy} - \sum{x} \sum{y}}{\sqrt{\left [n\sum{x}^{2}-\left (\sum{x}\right )^{2}\right]\left [n\sum{y}^{2}-\left (\sum{y}\right )^{2}\right]}}\\\\= \dfrac{4\times 1451 - 10\times 181}{\sqrt{[4\times 84 - 10^{2}][4\times25697 - 181^{2}]}}\\\\= \dfrac{5804 - 1810}{\sqrt{[336 - 100][102788 - 32761]}}\\\\= \dfrac{3994}{\sqrt{236\times70027}}\\\\= \dfrac{3994}{\sqrt{16526372}}\\\\= \dfrac{3994}{4065}\\\\= \mathbf{0.9825}[/tex]The closer the value of r is to +1 or -1, the better the correlation is. The values of x and y are highly correlated.