Enter two data sets and this calculator will find the equation of the regression line and correlation coefficient. The calculator will generate a step by step explanation along with the graphic representation of the data sets and regression line.
solution
You entered the following data:
$$\begin{array}{c|ccccccccccccccccccccccccccc}X&2&1&4&5&23&19&26&29&1&18&3&19&23&27&19&25&17&29&1&1&3&29&29&25&26&1&12\\Y&4050&3433&2910&2701&2560&2501&2400&2400&2400&2400&2400&2301&2300&2250&2212&2200&2200&2200&2119&2110&2107&2100&2100&2100&2089&2019&2000\end{array}$$The equation of the regression line is:
$$y~=~2624 ~-~ 15.05 \cdot x$$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.
$X$ | $Y$ | $X\cdot Y$ | $X \cdot X$ |
2 | 4050 | 8100 | 4 |
1 | 3433 | 3433 | 1 |
4 | 2910 | 11640 | 16 |
5 | 2701 | 13505 | 25 |
23 | 2560 | 58880 | 529 |
19 | 2501 | 47519 | 361 |
26 | 2400 | 62400 | 676 |
29 | 2400 | 69600 | 841 |
1 | 2400 | 2400 | 1 |
18 | 2400 | 43200 | 324 |
3 | 2400 | 7200 | 9 |
19 | 2301 | 43719 | 361 |
23 | 2300 | 52900 | 529 |
27 | 2250 | 60750 | 729 |
19 | 2212 | 42028 | 361 |
25 | 2200 | 55000 | 625 |
17 | 2200 | 37400 | 289 |
29 | 2200 | 63800 | 841 |
1 | 2119 | 2119 | 1 |
1 | 2110 | 2110 | 1 |
3 | 2107 | 6321 | 9 |
29 | 2100 | 60900 | 841 |
29 | 2100 | 60900 | 841 |
25 | 2100 | 52500 | 625 |
26 | 2089 | 54314 | 676 |
1 | 2019 | 2019 | 1 |
12 | 2000 | 24000 | 144 |
Step 2: Find the sum of every column:
$$ \sum{X} = 417 ~,~ \sum{Y} = 64562 ~,~ \sum{X \cdot Y} = 948657 ~,~ \sum{X^2} = 9661 $$Step 3: Use the following equations to find $a$ and $b$:
$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 64562 \cdot 9661 - 417 \cdot 948657}{ 27 \cdot 9661 - 417^2} \approx 2624 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 27 \cdot 948657 - 417 \cdot 64562 }{ 27 \cdot 9661 - \left( 417 \right)^2} \approx -15.05\end{aligned}$$Step 4: Substitute $a$ and $b$ in regression equation formula
$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~2624 ~-~ 15.05 \cdot x\end{aligned}$$Consider the following set of points: ${(-3 , -4), \, (2 , 3), \, (7 , 11)}$
a) Find the regression line for the given data points.
b) Plot the given points and the regression line.
The values of $X$ and their corresponding values of $Y$ are shown in the table below:
$$ \begin{array}{c|ccccc} X & ~1~ & ~2~ & ~3~ & ~4~ & ~5 \\ Y & ~4~ & ~8~ & ~9~ & ~11~& ~16 \end{array} $$Find a Pearson correlation coefficient.
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