Statistics Examples

\begin{array}{c} x & y \\ 15 & 12 \\ 15 & 9 \\ 9 & 8 \\ 6 & 10 \\ 11 & 10 \\ 13 & 15 \end{array}

$\begin{array}{c} x & y \\ 15 & 12 \\ 15 & 9 \\ 9 & 8 \\ 6 & 10 \\ 11 & 10 \\ 13 & 15 \end{array}$

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 = 15 + 15 + 9 + 6 + 11 + 13

$\sum_{}^{} x = 15 + 15 + 9 + 6 + 11 + 13$

Step 3

Simplify the expression.

\sum_{}^{} x = 69

$\sum_{}^{} x = 69$

Step 4

Sum up the

y

$y$ values.

\sum_{}^{} y = 12 + 9 + 8 + 10 + 10 + 15

$\sum_{}^{} y = 12 + 9 + 8 + 10 + 10 + 15$

Step 5

Simplify the expression.

\sum_{}^{} y = 64

$\sum_{}^{} y = 64$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum_{}^{} x y = 15 \cdot 12 + 15 \cdot 9 + 9 \cdot 8 + 6 \cdot 10 + 11 \cdot 10 + 13 \cdot 15

$\sum_{}^{} x y = 15 \cdot 12 + 15 \cdot 9 + 9 \cdot 8 + 6 \cdot 10 + 11 \cdot 10 + 13 \cdot 15$

Step 7

Simplify the expression.

\sum_{}^{} x y = 752

$\sum_{}^{} x y = 752$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

\sum_{}^{} x^{2} = {(15)}^{2} + {(15)}^{2} + {(9)}^{2} + {(6)}^{2} + {(11)}^{2} + {(13)}^{2}

$\sum_{}^{} x^{2} = {(15)}^{2} + {(15)}^{2} + {(9)}^{2} + {(6)}^{2} + {(11)}^{2} + {(13)}^{2}$

Step 9

Simplify the expression.

\sum_{}^{} x^{2} = 857

$\sum_{}^{} x^{2} = 857$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum_{}^{} y^{2} = {(12)}^{2} + {(9)}^{2} + {(8)}^{2} + {(10)}^{2} + {(10)}^{2} + {(15)}^{2}

$\sum_{}^{} y^{2} = {(12)}^{2} + {(9)}^{2} + {(8)}^{2} + {(10)}^{2} + {(10)}^{2} + {(15)}^{2}$

Step 11

Simplify the expression.

\sum_{}^{} y^{2} = 714

$\sum_{}^{} y^{2} = 714$

Step 12

Fill in the computed values.

r = \frac{6 (752) - 69 \cdot 64}{\sqrt{6 (857) - {(69)}^{2}} \cdot \sqrt{6 (714) - {(64)}^{2}}}

$r = \frac{6 (752) - 69 \cdot 64}{\sqrt{6 (857) - {(69)}^{2}} \cdot \sqrt{6 (714) - {(64)}^{2}}}$

Step 13

Simplify the expression.

r = 0.35869871

$r = 0.35869871$

Step 14

Find the critical value for a confidence level of

0

$0$ and

6

$6$ degrees of freedom.

t = 2.77644509

$t = 2.77644509$

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