Finite Math Examples

$\begin{array}{c} x & y \\ 1 & 3.9 \\ 2 & 8.7 \\ 3 & 16.7 \\ 4 & 27.9 \\ 5 & 42.3 \\ 6 & 59.9 \end{array}$

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.

$\sum_{}^{} x = 1 + 2 + 3 + 4 + 5 + 6$

Step 3

Simplify the expression.

$\sum_{}^{} x = 21$

Step 4

Sum up the $y$ values.

$\sum_{}^{} y = 3.9 + 8.7 + 16.7 + 27.9 + 42.3 + 59.9$

Step 5

Simplify the expression.

$\sum_{}^{} y = 159.4$

Step 6

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

$\sum_{}^{} x y = 1 \cdot 3.9 + 2 \cdot 8.7 + 3 \cdot 16.7 + 4 \cdot 27.9 + 5 \cdot 42.3 + 6 \cdot 59.9$

Step 7

Simplify the expression.

$\sum_{}^{} x y = 753.9$

Step 8

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

$\sum_{}^{} x^{2} = {(1)}^{2} + {(2)}^{2} + {(3)}^{2} + {(4)}^{2} + {(5)}^{2} + {(6)}^{2}$

Step 9

Simplify the expression.

$\sum_{}^{} x^{2} = 91$

Step 10

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

$\sum_{}^{} y^{2} = {(3.9)}^{2} + {(8.7)}^{2} + {(16.7)}^{2} + {(27.9)}^{2} + {(42.3)}^{2} + {(59.9)}^{2}$

Step 11

Simplify the expression.

$\sum_{}^{} y^{2} = 6525.50011987$

Step 12

Fill in the computed values.

$r = \frac{6 (753.9) - 21 \cdot 159.4}{\sqrt{6 (91) - {(21)}^{2}} \cdot \sqrt{6 (6525.5) - {(159.4)}^{2}}}$

Step 13

Simplify the expression.

$r = 0.9789176$

Step 14

Find the critical value for a confidence level of $0$ and $6$ degrees of freedom.

$t = 2.77644509$