Statistics Examples

\begin{array}{c} x & y \\ 11 & 10 \\ 9 & 8 \\ 9 & 10 \\ 8 & 8 \\ 8 & 9 \\ 9 & 7 \\ 9 & 7 \end{array}

$\begin{array}{c} x & y \\ 11 & 10 \\ 9 & 8 \\ 9 & 10 \\ 8 & 8 \\ 8 & 9 \\ 9 & 7 \\ 9 & 7 \end{array}$

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 = 11 + 9 + 9 + 8 + 8 + 9 + 9

$\sum_{}^{} x = 11 + 9 + 9 + 8 + 8 + 9 + 9$

Step 3

Simplify the expression.

\sum_{}^{} x = 63

$\sum_{}^{} x = 63$

Step 4

Sum up the

y

$y$ values.

\sum_{}^{} y = 10 + 8 + 10 + 8 + 9 + 7 + 7

$\sum_{}^{} y = 10 + 8 + 10 + 8 + 9 + 7 + 7$

Step 5

Simplify the expression.

\sum_{}^{} y = 59

$\sum_{}^{} y = 59$

Step 6

Sum up the values of

x \cdot y

$x \cdot y$ .

\sum_{}^{} x y = 11 \cdot 10 + 9 \cdot 8 + 9 \cdot 10 + 8 \cdot 8 + 8 \cdot 9 + 9 \cdot 7 + 9 \cdot 7

$\sum_{}^{} x y = 11 \cdot 10 + 9 \cdot 8 + 9 \cdot 10 + 8 \cdot 8 + 8 \cdot 9 + 9 \cdot 7 + 9 \cdot 7$

Step 7

Simplify the expression.

\sum_{}^{} x y = 534

$\sum_{}^{} x y = 534$

Step 8

Sum up the values of

x^{2}

$x^{2}$ .

\sum_{}^{} x^{2} = {(11)}^{2} + {(9)}^{2} + {(9)}^{2} + {(8)}^{2} + {(8)}^{2} + {(9)}^{2} + {(9)}^{2}

$\sum_{}^{} x^{2} = {(11)}^{2} + {(9)}^{2} + {(9)}^{2} + {(8)}^{2} + {(8)}^{2} + {(9)}^{2} + {(9)}^{2}$

Step 9

Simplify the expression.

\sum_{}^{} x^{2} = 573

$\sum_{}^{} x^{2} = 573$

Step 10

Sum up the values of

y^{2}

$y^{2}$ .

\sum_{}^{} y^{2} = {(10)}^{2} + {(8)}^{2} + {(10)}^{2} + {(8)}^{2} + {(9)}^{2} + {(7)}^{2} + {(7)}^{2}

$\sum_{}^{} y^{2} = {(10)}^{2} + {(8)}^{2} + {(10)}^{2} + {(8)}^{2} + {(9)}^{2} + {(7)}^{2} + {(7)}^{2}$

Step 11

Simplify the expression.

\sum_{}^{} y^{2} = 507

$\sum_{}^{} y^{2} = 507$

Step 12

Fill in the computed values.

r = \frac{7 (534) - 63 \cdot 59}{\sqrt{7 (573) - {(63)}^{2}} \cdot \sqrt{7 (507) - {(59)}^{2}}}

$r = \frac{7 (534) - 63 \cdot 59}{\sqrt{7 (573) - {(63)}^{2}} \cdot \sqrt{7 (507) - {(59)}^{2}}}$

Step 13

Simplify the expression.

r = 0.39295262

$r = 0.39295262$

Step 14

Find the critical value for a confidence level of

0

$0$ and

7

$7$ degrees of freedom.

t = 2.57058182

$t = 2.57058182$

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