Correlation and regression calculator

solution

You entered the following data:

$$\begin{array}{c|cccccccccc}X&6&7&8&9&10&11&12&13&14&15\\Y&109199&108012&115849&103144&100650&93376&94043&94133&98779&92099\end{array}$$

The equation of the regression line is:

$$y~=~123945 ~-~ 2192 \cdot x$$

The graph of the regression line is:

explanation

We will find an equation of the regression line in 4 steps.

Step 1: Find $X \cdot Y$ and $X^2$ as it was done in the table below.

$X$	$Y$	$X\cdot Y$	$X \cdot X$
6	109199	655194	36
7	108012	756084	49
8	115849	926792	64
9	103144	928296	81
10	100650	1006500	100
11	93376	1027136	121
12	94043	1128516	144
13	94133	1223729	169
14	98779	1382906	196
15	92099	1381485	225

Step 2: Find the sum of every column:

$$ \sum{X} = 105 ~,~ \sum{Y} = 1009284 ~,~ \sum{X \cdot Y} = 10416638 ~,~ \sum{X^2} = 1185 $$

Step 3: Use the following equations to find $a$ and $b$:

$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 1009284 \cdot 1185 - 105 \cdot 10416638}{ 10 \cdot 1185 - 105^2} \approx 123945 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 10 \cdot 10416638 - 105 \cdot 1009284 }{ 10 \cdot 1185 - \left( 105 \right)^2} \approx -2192\end{aligned}$$

Step 4: Substitute $a$ and $b$ in regression equation formula

$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~123945 ~-~ 2192 \cdot x\end{aligned}$$