linear regression\nuse linear regression to find the equation for the linear function that best fits this…

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:

$y = 21.06x+63.79$