Question

linear regression
use linear regression to find the equation for the linear function that best fits this data. round both numbers to two decimal places. write your final answer in a form of an equation y = mx + b
x 1 2 3 4 5 6
y 82 108 130 144 174 187
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Explanation:

Step1: Calculate sums

Let $n = 6$.
$\sum_{i = 1}^{n}x_{i}=1 + 2+3 + 4+5 + 6=21$
$\sum_{i = 1}^{n}y_{i}=82 + 108+130 + 144+174 + 187 = 825$
$\sum_{i = 1}^{n}x_{i}^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}=1 + 4+9 + 16+25 + 36 = 91$
$\sum_{i = 1}^{n}x_{i}y_{i}=1\times82+2\times108 + 3\times130+4\times144+5\times174+6\times187$
$=82+216+390+576+870+1122 = 3256$

Step2: Calculate slope $m$

The formula for $m$ is $m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}$
Substitute the values:
$m=\frac{6\times3256-21\times825}{6\times91 - 21^{2}}$
$=\frac{19536-17325}{546 - 441}=\frac{2211}{105}\approx21.06$

Step3: Calculate intercept $b$

The formula for $b$ is $b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}$
Substitute $m\approx21.06$, $\sum_{i = 1}^{n}x_{i}=21$ and $\sum_{i = 1}^{n}y_{i}=825$ and $n = 6$
$b=\frac{825-21.06\times21}{6}$
$=\frac{825 - 442.26}{6}=\frac{382.74}{6}=63.79$

Answer:

$y = 21.06x+63.79$