In its many guises, the least squares approximation arises...

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example. height and shoe size) in the form of pairs $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)$ The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum. Let the equation of the best-fit line be $y=m x+b,$ where the slope $m$ and the $y$ -intercept $b$ must be determined using the least squares condition. First assume that there are three data points $(1,2),(3,5),$ and $(4,6) .$ Show that the function of $m$ and $b$ that gives the sum of the squares of the vertical distances between the line and the three data points is $$\begin{aligned} E(m, b)=&((m+b)-2)^{2}+((3 m+b)-5)^{2} \\ &+((4 m+b)-6)^{2} \end{aligned}$$ Find the critical points of $E$ and find the values of $m$ and $b$ that minimize $E .$ Graph the three data points and the best-fit line.

In its many guises, the least squares approximation arises in numerous areas of mathematics and statistics. Suppose you collect data for two variables (for example.
height and shoe size) in the form of pairs $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)$ The data may be plotted as a scatterplot in the xy-plane, as shown in the figure. The technique known as linear regression asks the question: What is the equation of the line that "best fits" the data? The least squares criterion for best fit requires that the sum of the squares of the vertical distances between the line and the data points is a minimum.
Let the equation of the best-fit line be $y=m x+b,$ where the slope $m$ and the $y$ -intercept $b$ must be determined using the least squares condition. First assume that there are three data points $(1,2),(3,5),$ and $(4,6) .$ Show that the function of $m$ and $b$ that gives the sum of the squares of the vertical distances between the line and the three data points is $$\begin{aligned} E(m, b)=&((m+b)-2)^{2}+((3 m+b)-5)^{2} \\ &+((4 m+b)-6)^{2} \end{aligned}$$
Find the critical points of $E$ and find the values of $m$ and $b$ that minimize $E .$ Graph the three data points and the best-fit line.

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