Finite Math Examples

\begin{matrix} x & y 4 & 13 5 & 12 4 & 11 6 & 10 3 & 14 \end{matrix}

$\begin{array}{c} x & y \\ 4 & 13 \\ 5 & 12 \\ 4 & 11 \\ 6 & 10 \\ 3 & 14 \end{array}$

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 = 4 + 5 + 4 + 6 + 3

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

Step 3

Simplify the expression.

\sum x = 22

$\sum_{}^{} x = 22$

Step 4

Sum up the

y

$y$ values.

\sum y = 13 + 12 + 11 + 10 + 14

$\sum_{}^{} y = 13 + 12 + 11 + 10 + 14$

Step 5

Simplify the expression.

\sum y = 60

$\sum_{}^{} y = 60$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum x y = 4 \cdot 13 + 5 \cdot 12 + 4 \cdot 11 + 6 \cdot 10 + 3 \cdot 14

$\sum_{}^{} x y = 4 \cdot 13 + 5 \cdot 12 + 4 \cdot 11 + 6 \cdot 10 + 3 \cdot 14$

Step 7

Simplify the expression.

\sum x y = 258

$\sum_{}^{} x y = 258$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

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

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

Step 9

Simplify the expression.

\sum x^{2} = 102

$\sum_{}^{} x^{2} = 102$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum y^{2} = {(13)}^{2} + {(12)}^{2} + {(11)}^{2} + {(10)}^{2} + {(14)}^{2}

$\sum_{}^{} y^{2} = {(13)}^{2} + {(12)}^{2} + {(11)}^{2} + {(10)}^{2} + {(14)}^{2}$

Step 11

Simplify the expression.

\sum y^{2} = 730

$\sum_{}^{} y^{2} = 730$

Step 12

Fill in the computed values.

r = \frac{5 (258) - 22 \cdot 60}{\sqrt{5 (102) - {(22)}^{2}} \cdot \sqrt{5 (730) - {(60)}^{2}}}

$r = \frac{5 (258) - 22 \cdot 60}{\sqrt{5 (102) - {(22)}^{2}} \cdot \sqrt{5 (730) - {(60)}^{2}}}$

Step 13

Simplify the expression.

$r = - 0.83205029$

Step 14

Find the critical value for a confidence level of

$0$ and

$5$ degrees of freedom.

$t = 3.18244628$

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