Finite Math Examples

$\begin{array}{c} x & y \\ 3 & 65 \\ 5 & 80 \\ 2 & 60 \\ 8 & 88 \\ 2 & 66 \\ 4 & 78 \\ 4 & 85 \\ 5 & 90 \\ 6 & 90 \\ 3 & 71 \end{array}$

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.

$\sum_{}^{} x = 3 + 5 + 2 + 8 + 2 + 4 + 4 + 5 + 6 + 3$

Step 3

Simplify the expression.

$\sum_{}^{} x = 42$

Step 4

Sum up the $y$ values.

$\sum_{}^{} y = 65 + 80 + 60 + 88 + 66 + 78 + 85 + 90 + 90 + 71$

Step 5

Simplify the expression.

$\sum_{}^{} y = 773$

Step 6

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

$\sum_{}^{} x y = 3 \cdot 65 + 5 \cdot 80 + 2 \cdot 60 + 8 \cdot 88 + 2 \cdot 66 + 4 \cdot 78 + 4 \cdot 85 + 5 \cdot 90 + 6 \cdot 90 + 3 \cdot 71$

Step 7

Simplify the expression.

$\sum_{}^{} x y = 3406$

Step 8

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

$\sum_{}^{} x^{2} = {(3)}^{2} + {(5)}^{2} + {(2)}^{2} + {(8)}^{2} + {(2)}^{2} + {(4)}^{2} + {(4)}^{2} + {(5)}^{2} + {(6)}^{2} + {(3)}^{2}$

Step 9

Simplify the expression.

$\sum_{}^{} x^{2} = 208$

Step 10

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

$\sum_{}^{} y^{2} = {(65)}^{2} + {(80)}^{2} + {(60)}^{2} + {(88)}^{2} + {(66)}^{2} + {(78)}^{2} + {(85)}^{2} + {(90)}^{2} + {(90)}^{2} + {(71)}^{2}$

Step 11

Simplify the expression.

$\sum_{}^{} y^{2} = 60875$

Step 12

Fill in the computed values.

$r = \frac{10 (3406) - 42 \cdot 773}{\sqrt{10 (208) - {(42)}^{2}} \cdot \sqrt{10 (60875) - {(773)}^{2}}}$

Step 13

Simplify the expression.

$r = 0.84650396$

Step 14

Find the critical value for a confidence level of $0$ and $10$ degrees of freedom.

$t = 2.30600412$