(a) Suppose you are given the following (x, y) data pairs: ...

(a) Suppose you are given the following $(x, y)$ data pairs: $$ \begin{array}{l|lll} \hline x & 1 & 3 & 4 \\ \hline y & 2 & 1 & 6 \\ \hline \end{array} $$ Show that the least-squares equation for these data is $y=0.143+1.071 x$ (rounded to three digits after the decimal). (b) Now suppose you are given these $(x, y)$ data pairs: $$ \begin{array}{l|lll} \hline x & 2 & 1 & 6 \\ \hline y & 1 & 3 & 4 \\ \hline \end{array} $$ Show that the least-squares equation for these data is $y=1.595+0.357 x$ (rounded to three digits after the decimal). (c) In the data for parts (a) and (b), did we simply exchange the $x$ and $y$ values of each data pair? (d) Solve $y=0.143+1.071 x$ for $x .$ Do you get the least-squares equation of part (b) with the symbols $x$ and $y$ exchanged? (e) In general, suppose we have the least-squares equation $y=a+b x$ for a set of data pairs $(x, y)$. If we solve this equation for $x$, will we necessarily get the least-squares equation for the set of data pairs $(y, x)$ (with $x$ and $y$ exchanged)? Explain using parts (a) through (d).

(a) Suppose you are given the following $(x, y)$ data pairs:
$$
\begin{array}{l|lll}
\hline x & 1 & 3 & 4 \\
\hline y & 2 & 1 & 6 \\
\hline
\end{array}
$$
Show that the least-squares equation for these data is $y=0.143+1.071 x$ (rounded to three digits after the decimal).
(b) Now suppose you are given these $(x, y)$ data pairs:
$$
\begin{array}{l|lll}
\hline x & 2 & 1 & 6 \\
\hline y & 1 & 3 & 4 \\
\hline
\end{array}
$$
Show that the least-squares equation for these data is $y=1.595+0.357 x$ (rounded to three digits after the decimal).
(c) In the data for parts (a) and (b), did we simply exchange the $x$ and $y$ values of each data pair?
(d) Solve $y=0.143+1.071 x$ for $x .$ Do you get the least-squares equation of part (b) with the symbols $x$ and $y$ exchanged?
(e) In general, suppose we have the least-squares equation $y=a+b x$ for a set of data pairs $(x, y)$. If we solve this equation for $x$, will we necessarily get the least-squares equation for the set of data pairs $(y, x)$ (with $x$ and $y$ exchanged)? Explain using parts (a) through (d).

Key Concepts

Least Squares Regression

Least squares regression is a statistical method used to determine the best-fitting line through a set of data by minimizing the sum of the squares of the vertical (or horizontal) deviations (residuals) from the observed data points to the line. This involves calculating parameters, typically a slope and an intercept, that reduce the overall error in the prediction of dependent variables from independent variables.

Error Minimization in Regression

Central to the least squares method is the minimization of the sum of squared errors between the observed values and the values predicted by the regression line. This minimization ensures that the regression line is the most accurate linear approximation of the relationship inherent in the data, but it does so by treating one variable as dependent and thus subject to error resampling in one direction.

Exchange of Variables in Regression

Exchanging the roles of the independent and dependent variables changes the problem setup because the least squares method minimizes deviations in the direction of the chosen dependent variable only. As a result, the regression line obtained from predicting y using x is generally not the same as the one obtained by predicting x using y, reflecting the asymmetry in the error minimization process.

Inversion of Regression Equations

While inverting a simple linear function algebraically yields another linear equation, doing so does not typically provide the regression line for the reversed data pairs. This is because the original regression minimizes errors in one specific direction (vertical or horizontal), and simply solving for the other variable does not account for a re-minimization of errors when the roles of the variables are exchanged.

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