Finite Math Examples

\begin{matrix} x & g (x) y 0 & 50.2 1 & 49.6 2 & 51.4 3 & 54.5 4 & 57.8 5 & 60.1 6 & 60.3 \end{matrix}

$\begin{array}{c} x & g (x) y \\ 0 & 50.2 \\ 1 & 49.6 \\ 2 & 51.4 \\ 3 & 54.5 \\ 4 & 57.8 \\ 5 & 60.1 \\ 6 & 60.3 \end{array}$

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}}}

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

Step 2

Sum up the

x

$x$ values.

\sum x = 0 + 1 + 2 + 3 + 4 + 5 + 6

$\sum_{}^{} x = 0 + 1 + 2 + 3 + 4 + 5 + 6$

Step 3

Simplify the expression.

\sum x = 21

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

Step 4

Sum up the

y

$y$ values.

\sum y = 50.2 + 49.6 + 51.4 + 54.5 + 57.8 + 60.1 + 60.3

$\sum_{}^{} y = 50.2 + 49.6 + 51.4 + 54.5 + 57.8 + 60.1 + 60.3$

Step 5

Simplify the expression.

\sum y = 383.9

$\sum_{}^{} y = 383.9$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum x y = 0 \cdot 50.2 + 1 \cdot 49.6 + 2 \cdot 51.4 + 3 \cdot 54.5 + 4 \cdot 57.8 + 5 \cdot 60.1 + 6 \cdot 60.3

$\sum_{}^{} x y = 0 \cdot 50.2 + 1 \cdot 49.6 + 2 \cdot 51.4 + 3 \cdot 54.5 + 4 \cdot 57.8 + 5 \cdot 60.1 + 6 \cdot 60.3$

Step 7

Simplify the expression.

\sum x y = 1209.3999

$\sum_{}^{} x y = 1209.3999$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

\sum x^{2} = {(0)}^{2} + {(1)}^{2} + {(2)}^{2} + {(3)}^{2} + {(4)}^{2} + {(5)}^{2} + {(6)}^{2}

$\sum_{}^{} x^{2} = {(0)}^{2} + {(1)}^{2} + {(2)}^{2} + {(3)}^{2} + {(4)}^{2} + {(5)}^{2} + {(6)}^{2}$

Step 9

Simplify the expression.

\sum x^{2} = 91

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

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum y^{2} = {(50.2)}^{2} + {(49.6)}^{2} + {(51.4)}^{2} + {(54.5)}^{2} + {(57.8)}^{2} + {(60.1)}^{2} + {(60.3)}^{2}

$\sum_{}^{} y^{2} = {(50.2)}^{2} + {(49.6)}^{2} + {(51.4)}^{2} + {(54.5)}^{2} + {(57.8)}^{2} + {(60.1)}^{2} + {(60.3)}^{2}$

Step 11

Simplify the expression.

\sum y^{2} = 21181.34971847

$\sum_{}^{} y^{2} = 21181.34971847$

Step 12

Fill in the computed values.

r = \frac{7 (1209.3999) - 21 \cdot 383.9}{\sqrt{7 (91) - {(21)}^{2}} \cdot \sqrt{7 (21181.35) - {(383.9)}^{2}}}

$r = \frac{7 (1209.3999) - 21 \cdot 383.9}{\sqrt{7 (91) - {(21)}^{2}} \cdot \sqrt{7 (21181.35) - {(383.9)}^{2}}}$

Step 13

Simplify the expression.

r = 0.96691702

$r = 0.96691702$

Step 14

Find the critical value for a confidence level of

0

$0$ and

7

$7$ degrees of freedom.

t = 2.57058182

$t = 2.57058182$