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:
$$\begin{array}{c|ccccccccccccccccccccccccccccccccccc}X&12&1&4&26&29&29&25&3&1&1&25&17&29&19&27&23&19&12&17&26&29&1&18&3&26&19&23&5&18&5&17&22&4&1&2\\Y&2000&2019&2047&2089&2100&2100&2100&2107&2110&2119&2200&2200&2200&2212&2250&2300&2301&2319&2350&2400&2400&2400&2400&2400&2500&2501&2560&2600&2700&2701&2800&2900&2910&3433&4050\end{array}$$The equation of the regression line is:
$$y~=~2593 ~-~ 11.14 \cdot x$$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.
$X$ | $Y$ | $X\cdot Y$ | $X \cdot X$ |
12 | 2000 | 24000 | 144 |
1 | 2019 | 2019 | 1 |
4 | 2047 | 8188 | 16 |
26 | 2089 | 54314 | 676 |
29 | 2100 | 60900 | 841 |
29 | 2100 | 60900 | 841 |
25 | 2100 | 52500 | 625 |
3 | 2107 | 6321 | 9 |
1 | 2110 | 2110 | 1 |
1 | 2119 | 2119 | 1 |
25 | 2200 | 55000 | 625 |
17 | 2200 | 37400 | 289 |
29 | 2200 | 63800 | 841 |
19 | 2212 | 42028 | 361 |
27 | 2250 | 60750 | 729 |
23 | 2300 | 52900 | 529 |
19 | 2301 | 43719 | 361 |
12 | 2319 | 27828 | 144 |
17 | 2350 | 39950 | 289 |
26 | 2400 | 62400 | 676 |
29 | 2400 | 69600 | 841 |
1 | 2400 | 2400 | 1 |
18 | 2400 | 43200 | 324 |
3 | 2400 | 7200 | 9 |
26 | 2500 | 65000 | 676 |
19 | 2501 | 47519 | 361 |
23 | 2560 | 58880 | 529 |
5 | 2600 | 13000 | 25 |
18 | 2700 | 48600 | 324 |
5 | 2701 | 13505 | 25 |
17 | 2800 | 47600 | 289 |
22 | 2900 | 63800 | 484 |
4 | 2910 | 11640 | 16 |
1 | 3433 | 3433 | 1 |
2 | 4050 | 8100 | 4 |
Step 2: Find the sum of every column:
$$ \sum{X} = 538 ~,~ \sum{Y} = 84778 ~,~ \sum{X \cdot Y} = 1262623 ~,~ \sum{X^2} = 11908 $$Step 3: Use the following equations to find $a$ and $b$:
$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 84778 \cdot 11908 - 538 \cdot 1262623}{ 35 \cdot 11908 - 538^2} \approx 2593 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 35 \cdot 1262623 - 538 \cdot 84778 }{ 35 \cdot 11908 - \left( 538 \right)^2} \approx -11.14\end{aligned}$$Step 4: Substitute $a$ and $b$ in regression equation formula
$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~2593 ~-~ 11.14 \cdot x\end{aligned}$$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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