Use linear regression to find the equation for the line that best fits this data. Round both numbers to two decimal places. Write your final answer in a form of an equation $y=m x+b$
\begin{tabular}{|c|r|r|r|r|r|r|}
\hline $\mathbf{x}$ & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline $\mathbf{y}$ & 71 & 96 & 113 & 137 & 152 & 187 \\
\hline
\end{tabular}
Question Help: $\square$ Video $\square$ Message instructor
Submit Question

Step 1 ：We are given a set of data points and we need to find the equation of the line that best fits these points. The equation of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2 ：We can use the method of linear regression to find the values of \(m\) and \(b\). Linear regression is a statistical method that allows us to study relationships between two continuous (quantitative) variables. One variable, denoted \(x\), is regarded as the predictor, explanatory, or independent variable. The other variable, denoted \(y\), is regarded as the response, outcome, or dependent variable.

Step 3 ：The goal of linear regression is to model the expected value of \(y\) as a linear function of \(x\): \(y = mx + b\).

Step 4 ：Given the data points \(x = [1, 2, 3, 4, 5, 6]\) and \(y = [71, 96, 113, 137, 152, 187]\), we can use the method of least squares to find the slope \(m\) and y-intercept \(b\) of the line that best fits the data.

Step 5 ：After performing the calculations, we find that the slope \(m = 22.06\) and the y-intercept \(b = 48.8\).

Step 6 ：Thus, the equation for the line that best fits the data is \(\boxed{y = 22.06x + 48.8}\).