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:
$$\begin{array}{c|ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc}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end{array}$$The equation of the regression line is:
$$y~=~3545 ~-~ 24.21 \cdot x$$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.
$X$ | $Y$ | $X\cdot Y$ | $X \cdot X$ |
49 | 2010 | 98490 | 2401 |
51 | 2010 | 102510 | 2601 |
39 | 2050 | 79950 | 1521 |
57 | 2085 | 118845 | 3249 |
49 | 2098 | 102802 | 2401 |
37 | 2100 | 77700 | 1369 |
45 | 2100 | 94500 | 2025 |
46 | 2100 | 96600 | 2116 |
62 | 2100 | 130200 | 3844 |
62 | 2119 | 131378 | 3844 |
37 | 2124 | 78588 | 1369 |
49 | 2150 | 105350 | 2401 |
48 | 2200 | 105600 | 2304 |
52 | 2201 | 114452 | 2704 |
57 | 2210 | 125970 | 3249 |
36 | 2238 | 80568 | 1296 |
52 | 2260 | 117520 | 2704 |
59 | 2290 | 135110 | 3481 |
36 | 2301 | 82836 | 1296 |
54 | 2319 | 125226 | 2916 |
38 | 2320 | 88160 | 1444 |
50 | 2422 | 121100 | 2500 |
37 | 2500 | 92500 | 1369 |
54 | 2500 | 135000 | 2916 |
37 | 2600 | 96200 | 1369 |
59 | 2601 | 153459 | 3481 |
43 | 2622 | 112746 | 1849 |
38 | 2671 | 101498 | 1444 |
59 | 2700 | 159300 | 3481 |
36 | 3100 | 111600 | 1296 |
60 | 3200 | 192000 | 3600 |
36 | 3209 | 115524 | 1296 |
44 | 3300 | 145200 | 1936 |
39 | 4195 | 163605 | 1521 |
14 | 2100 | 29400 | 196 |
17 | 2100 | 35700 | 289 |
30 | 2100 | 63000 | 900 |
30 | 2100 | 63000 | 900 |
11 | 2129 | 23419 | 121 |
11 | 2131 | 23441 | 121 |
23 | 2160 | 49680 | 529 |
22 | 2180 | 47960 | 484 |
30 | 2188 | 65640 | 900 |
6 | 2201 | 13206 | 36 |
30 | 2214 | 66420 | 900 |
21 | 2222 | 46662 | 441 |
24 | 2299 | 55176 | 576 |
15 | 2300 | 34500 | 225 |
30 | 2300 | 69000 | 900 |
30 | 2300 | 69000 | 900 |
Step 2: Find the sum of every column:
$$ \sum{X} = 3941 ~,~ \sum{Y} = 404422 ~,~ \sum{X \cdot Y} = 10635996 ~,~ \sum{X^2} = 137735 $$Step 3: Use the following equations to find $a$ and $b$:
$$ \begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 404422 \cdot 137735 - 3941 \cdot 10635996}{ 141 \cdot 137735 - 3941^2} \approx 3545 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 141 \cdot 10635996 - 3941 \cdot 404422 }{ 141 \cdot 137735 - \left( 3941 \right)^2} \approx -24.21\end{aligned}$$Step 4: Substitute $a$ and $b$ in regression equation formula
$$ \begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~3545 ~-~ 24.21 \cdot x\end{aligned}$$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.
Please tell me how can I make this better.