Finite Math Examples

\begin{matrix} x & y 0 & 1 1 & 0 0 & 0 \end{matrix}

$\begin{array}{c} x & y \\ 0 & 1 \\ 1 & 0 \\ 0 & 0 \end{array}$

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 + 0

$\sum_{}^{} x = 0 + 1 + 0$

Step 3

Simplify the expression.

\sum x = 1

$\sum_{}^{} x = 1$

Step 4

Sum up the

y

$y$ values.

\sum y = 1 + 0 + 0

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

Step 5

Simplify the expression.

\sum y = 1

$\sum_{}^{} y = 1$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum x y = 0 \cdot 1 + 1 \cdot 0 + 0 \cdot 0

$\sum_{}^{} x y = 0 \cdot 1 + 1 \cdot 0 + 0 \cdot 0$

Step 7

Simplify the expression.

\sum x y = 0

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

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

\sum x^{2} = {(0)}^{2} + {(1)}^{2} + {(0)}^{2}

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

Step 9

Simplify the expression.

\sum x^{2} = 1

$\sum_{}^{} x^{2} = 1$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum y^{2} = {(1)}^{2} + {(0)}^{2} + {(0)}^{2}

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

Step 11

Simplify the expression.

\sum y^{2} = 1

$\sum_{}^{} y^{2} = 1$

Step 12

Fill in the computed values.

r = \frac{3 (0) - 1 \cdot 1}{\sqrt{3 (1) - {(1)}^{2}} \cdot \sqrt{3 (1) - {(1)}^{2}}}

$r = \frac{3 (0) - 1 \cdot 1}{\sqrt{3 (1) - {(1)}^{2}} \cdot \sqrt{3 (1) - {(1)}^{2}}}$

Step 13

Simplify the expression.

r = - 0.4 ¯ 9

$r = - 0.4\overline{9}$

Step 14

Find the critical value for a confidence level of

0

$0$ and

3

$3$ degrees of freedom.

t = 12.70620454

$t = 12.70620454$