Question

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.​

Answer 1

r = 0.9825; good correlation.

Explanation:

One formula for the correlation coefficient is

`r = \frac{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]}}`

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

`r = \frac{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]}}\\\\= \frac{4* 1451 - 10* 181}{\sqrt{[4* 84 - 10^(2)][4*25697 - 181^(2)]}}\\\\= (5804 - 1810)/(√([336 - 100][102788 - 32761]))\\\\= (3994)/(√(236*70027))\\\\= (3994)/(√(16526372))\\\\= (3994)/(4065)\\\\= \mathbf{0.9825}`

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.