Statistics Examples

$\begin{array}{c} x & y \\ 5 & 11 \\ 5 & 10 \\ 5 & 13 \\ 6 & 10 \\ 7 & 11 \end{array}$

Step 1

The slope of the best fit regression line can be found using the formula.

$m = \frac{n (\sum_{}^{} x y) - \sum_{}^{} x \sum_{}^{} y}{n (\sum_{}^{} x^{2}) - {(\sum_{}^{} x)}^{2}}$

Step 2

The y-intercept of the best fit regression line can be found using the formula.

$b = \frac{(\sum_{}^{} y) (\sum_{}^{} x^{2}) - \sum_{}^{} x \sum_{}^{} x y}{n (\sum_{}^{} x^{2}) - {(\sum_{}^{} x)}^{2}}$

Step 3

Sum up the $x$ values.

$\sum_{}^{} x = 5 + 5 + 5 + 6 + 7$

Step 4

Simplify the expression.

$\sum_{}^{} x = 28$

Step 5

Sum up the $y$ values.

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

Step 6

Simplify the expression.

$\sum_{}^{} y = 55$

Step 7

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

$\sum_{}^{} x y = 5 \cdot 11 + 5 \cdot 10 + 5 \cdot 13 + 6 \cdot 10 + 7 \cdot 11$

Step 8

Simplify the expression.

$\sum_{}^{} x y = 307$

Step 9

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

$\sum_{}^{} x^{2} = {(5)}^{2} + {(5)}^{2} + {(5)}^{2} + {(6)}^{2} + {(7)}^{2}$

Step 10

Simplify the expression.

$\sum_{}^{} x^{2} = 160$

Step 11

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

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

Step 12

Simplify the expression.

$\sum_{}^{} y^{2} = 611$

Step 13

Fill in the computed values.

$m = \frac{5 (307) - 28 \cdot 55}{5 (160) - {(28)}^{2}}$

Step 14

Simplify the expression.

$m = - 0.3125$

Step 15

Fill in the computed values.

$b = \frac{(55) (160) - 28 \cdot 307}{5 (160) - {(28)}^{2}}$

Step 16

Simplify the expression.

$b = 12.75$

Step 17

Fill in the values of slope $m$ and y-intercept $b$ into the slope-intercept formula.

$y = - 0.3125 x + 12.75$

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