Finite Math Examples Correlation and Regression

Determine if the Correlation is Significant

$\begin{array}{c} x & y \\ 3 & 12 \\ 8 & 9 \\ 7 & 6 \\ 9 & 10 \\ 5 & 10 \\ 3 & 9 \end{array}$

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.

$\sum_{}^{} x = 3 + 8 + 7 + 9 + 5 + 3$

Step 3

Simplify the expression.

$\sum_{}^{} x = 35$

Step 4

Sum up the $y$ values.

$\sum_{}^{} y = 12 + 9 + 6 + 10 + 10 + 9$

Step 5

Simplify the expression.

$\sum_{}^{} y = 56$

Step 6

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

$\sum_{}^{} x y = 3 \cdot 12 + 8 \cdot 9 + 7 \cdot 6 + 9 \cdot 10 + 5 \cdot 10 + 3 \cdot 9$

Step 7

Simplify the expression.

$\sum_{}^{} x y = 317$

Step 8

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

$\sum_{}^{} x^{2} = {(3)}^{2} + {(8)}^{2} + {(7)}^{2} + {(9)}^{2} + {(5)}^{2} + {(3)}^{2}$

Step 9

Simplify the expression.

$\sum_{}^{} x^{2} = 237$

Step 10

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

$\sum_{}^{} y^{2} = {(12)}^{2} + {(9)}^{2} + {(6)}^{2} + {(10)}^{2} + {(10)}^{2} + {(9)}^{2}$

Step 11

Simplify the expression.

$\sum_{}^{} y^{2} = 542$

Step 12

Fill in the computed values.

$r = \frac{6 (317) - 35 \cdot 56}{\sqrt{6 (237) - {(35)}^{2}} \cdot \sqrt{6 (542) - {(56)}^{2}}}$

Step 13

Simplify the expression.

$r = - 0.3836771$

Step 14

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

$t = 2.77644509$

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Determine if the Correlation is Significant

Correlation and Regression

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