Finite Math Examples

$\begin{array}{c} x & y \\ - 3 & 9 \\ - 2 & 4 \\ - 1 & 1 \\ 0 & 0 \\ 1 & 1 \\ 2 & 4 \\ 3 & 9 \end{array}$

Step 1

The linear correlation coefficient measures the relationship between the paired values in a sample.

$r = \frac{n (\sum_{}^{} x y) - \sum_{}^{} x \sum_{}^{} y}{\sqrt{n (\sum_{}^{} x^{2}) - {(\sum_{}^{} x)}^{2}} \cdot \sqrt{n (\sum_{}^{} y^{2}) - {(\sum_{}^{} y)}^{2}}}$

Step 2

Sum up the $x$ values.

$\sum_{}^{} x = - 3 - 2 - 1 + 0 + 1 + 2 + 3$

Step 3

Simplify the expression.

$\sum_{}^{} x = 0$

Step 4

Sum up the $y$ values.

$\sum_{}^{} y = 9 + 4 + 1 + 0 + 1 + 4 + 9$

Step 5

Simplify the expression.

$\sum_{}^{} y = 28$

Step 6

Sum up the values of $x \cdot y$ .

$\sum_{}^{} x y = - 3 \cdot 9 - 2 \cdot 4 - 1 \cdot 1 + 0 \cdot 0 + 1 \cdot 1 + 2 \cdot 4 + 3 \cdot 9$

Step 7

Simplify the expression.

$\sum_{}^{} x y = 0$

Step 8

Sum up the values of $x^{2}$ .

$\sum_{}^{} x^{2} = {(- 3)}^{2} + {(- 2)}^{2} + {(- 1)}^{2} + {(0)}^{2} + {(1)}^{2} + {(2)}^{2} + {(3)}^{2}$

Step 9

Simplify the expression.

$\sum_{}^{} x^{2} = 28$

Step 10

Sum up the values of $y^{2}$ .

$\sum_{}^{} y^{2} = {(9)}^{2} + {(4)}^{2} + {(1)}^{2} + {(0)}^{2} + {(1)}^{2} + {(4)}^{2} + {(9)}^{2}$

Step 11

Simplify the expression.

$\sum_{}^{} y^{2} = 196$

Step 12

Fill in the computed values.

$r = \frac{7 (0) + 0 \cdot 28}{\sqrt{7 (28) - {(0)}^{2}} \cdot \sqrt{7 (196) - {(28)}^{2}}}$

Step 13

Simplify the expression.

$r = 0$

Step 14

Find the critical value for a confidence level of $0$ and $7$ degrees of freedom.

$t = 2.57058182$