12.3 Linear Regression Equation
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 $\hat{y}=m x+b$
\begin{tabular}{|r|r|r|r|r|r|r|}
\hline $\mathbf{x}$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline $\mathbf{y}$ & 87 & 104 & 118 & 132 & 155 & 171 \\
\hline
\end{tabular}
\[
\hat{y}=
\]
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Step 1 ：First, calculate the necessary sums: \(\Sigma x = 1 + 2 + 3 + 4 + 5 + 6 = 21\), \(\Sigma y = 87 + 104 + 118 + 132 + 155 + 171 = 767\), \(\Sigma xy = 1*87 + 2*104 + 3*118 + 4*132 + 5*155 + 6*171 = 2174\), \(\Sigma x^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 91\), and \(n = 6\).

Step 2 ：Substitute these values into the formulas for m and b: \(m = [6(2174) - 21*767] / [6*91 - 21^2] = 16.60\) and \(b = [767 - 16.60*21] / 6 = 70.60\).

Step 3 ：\(\boxed{\hat{y} = 16.60x + 70.60}\) is the equation of the line that best fits the data. This equation means that for each unit increase in x, we expect y to increase by approximately 16.60, and when x is 0, y is approximately 70.60.